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at P is TP.V / D fP 0 2 V.kŒ"/ j P 0 7! P under V.kŒ"/! V.k/g: Thus an element of TP.V / is a homomorphism of k-algebras ˛W kŒV! kŒ" whose "7!0! k is the point P. To say that kŒV! k is the point P means that composite with kŒ" its kernel is mP, and so mP D ˛1.."//. PROPOSITION 4.26. Let V be an algebraic subset of A equipped with its canonical structure of an affine algebraic variety. Let P 2 V. Then O n, and let V 0 D.V; V / be V TP.V / (as defined in 4.22)'TP.V 0/ (as defined in 4.25). PROOF. Let I.V / D a and let P D.a1; : : : ; an/. On rewriting a polynomial F.X1; : : : ; Xn/ in terms of the variables Xi ai, we obtain the (trivial Taylor) formula, F.X1; : : : ; Xn/ D F.a1; : : : ; an/ CX @F @Xi ˇ ˇ ˇ ˇa.Xi ai / C R with R a finite sum of products of at least two terms.Xi ai /. According to 4.25, TP.V 0/ consists of the elements a C "b of kŒ"n D kn ˚ kn" lying in V.kŒ"/. Let F 2 a. On setting Xi equal to ai C "bi in the above formula, we obtain: F.a1 C "b1; : : : ; an C "bn/ D " X @F @Xi ˇ ˇ ˇ ˇa bi : Thus,.a1 C "b1; : : : ; an C "bn/ lies in V.kŒ"/ if and only if.b1; : : : ; bn/ 2 Ta.V /. We can restate this as follows. Let V be an affine algebraic variety, and let P 2 V. Choose
an embedding V,! A n, and let P map to.a1; : : : ; an/. Then the point.a1; : : : ; an/ C.b1; : : : ; bn/" n.kŒ"/ is an element of TP.V / (definition 4.25) if and only if.b1; : : : ; bn/ is an element of A of TP.V / (definition 4.22). PROPOSITION 4.27. Let V be an affine variety, and let P 2 V. There is a canonical isomorphism TP.V /'Hom. O P ; kŒ"/ (local homomorphisms of local k-algebras). PROOF. By definition, an element of TP.V / is a homomorphism ˛W kŒV! kŒ" such that ˛1.."// D mP. Therefore ˛ maps elements of kŒV X mP into.kŒ" X."// D kŒ", and so ˛ extends (uniquely) to a homomorphism ˛0W P! kŒ". By construction, ˛0 is a local homomorphism of local k-algebras, and clearly every such homomorphism arises in this way from an element of TP.V /. O f. Tangent spaces to affine algebraic varieties 91 Derivations DEFINITION 4.28. Let A be a k-algebra and M an A-module. A k-derivation is a map DW A! M such that (a) D.c/ D 0 for all c 2 k; (b) D.f C g/ D D.f / C D.g/; (c) D.fg/ D f Dg C g Df (Leibniz’s rule). Note that the conditions imply that D is k-linear (but not A-linear). We write Derk.A; M / for the k-vector space of all k-derivations A! M. For example, let A be a local k-algebra with maximal ideal m, and assume that A=m D k. For f 2 A, let f.m/ denote the image of f in A=m. Then f f.m/ 2 m,
and the map f 7! df defD f f.m/ mod m2 is a k-derivation A! m=m2 because, mod m2, 0 D.f f.m//.g g.m// D fg C f.m/g.m/ C f.g g.m// C g.f f.m// D d.fg/ C f dg C g df: PROPOSITION 4.29. Let.A; m/ be as above. There are canonical isomorphisms Homlocal k-algebra.A; kŒ"/! Derk.A; k/! Homk-linear.m=m2; k/: c7!c! A PROOF. The composite k k-vector space, A decomposes into f 7!f.m/! k is the identity map, and so, when regarded as A D k ˚ m; f $.f.m/; f f.m//: Let ˛W A! kŒ" be a local homomorphism of k-algebras, and write ˛.a/ D a0 C D˛.a/". Because ˛ is a homomorphism of k-algebras, a0 D a.m/. We have ˛.ab/ D.ab/0 C D˛.ab/"; and ˛.a/˛.b/ D.a0 C D˛.a/"/.b0 C D˛.b/"/ D a0b0 C.a0D˛.b/ C b0D˛.a//": On comparing these expressions, we see that D˛ satisfies Leibniz’s rule, and therefore is a k-derivation A! k. Conversely, if DW A! k is a k-derivation, then ˛W a 7! a.m/ C D.a/" is a local homomorphism of k-algebras A! kŒ", and all such homomorphisms arise in this way. A derivation DW A! k is zero on k and on m2 (by Leibniz’s rule). It therefore defines a k-linear map m=m2! k. Conversely, a k-linear map m=m2! k defines
a derivation by composition f 7!df! m=m2! k: A 92 4. LOCAL STUDY Tangent spaces and differentials We now summarize the above discussion in the context of affine algebraic varieties. 4.30. Let V be an affine algebraic variety, and let P be a point on V. Write mP for the corresponding maximal ideal in kŒV and nP for the maximal ideal mP V;P in the local ring at P. There are canonical isomorphisms O TP.V / Derk.kŒV ; k/ Homk-linear.mP =m2 P ; k/ (24) Homlocal k-algebra. P ; kŒ"/ Derk. P ; k/ O O In the middle term on the top row, kŒV acts on k through kŒV! kŒV =mP'k,1 and on P =nP'k. The maps have the following the bottom row descriptions. P acts on k through P! O O O Homk-linear.nP =n2 P ; k/: (a) By definition, TP.V / is the fibre of V.kŒ"/! V.k/ over P. To give an element of TP.V / amounts to giving a homomorphism ˛W kŒV! kŒ" such that ˛1.."// D mP. (b) The homomorphism ˛ in (a) can be decomposed, ˛.f / D f.mP / ˚ D˛.f /"; f 2 kŒV, f.mP / 2 k, D˛.f / 2 k:! k. The map D˛ is a k-derivation kŒV! k, and D˛ induces a k-linear map mP =m2 P (c) The homomorphism ˛W kŒV! kŒ" in (a) extends uniquely to a local homomorphism P! kŒ". Similarly, a k-derivation kŒV! k extends uniquely to a k-derivation P! k. (d) The two right hand groups are related through the isomorphism mP =m2 P! nP =n2 P O O of (1.15
). 4.31. A regular map 'W V! W defines a map '.kŒ"/W V.kŒ"/! W.kŒ"/. If Q D '.P /, then'maps the fibre over P to the fibre over Q, i.e., it defines a map d'W TP.V /! TQ.W /: This map of tangent spaces is called the differential of'at P. (a) When V and W are embedded as closed subvarieties of A p. 89. TP.V / V.kŒ"/ d''TQ.W / W.kŒ"/ "7!0 "7!0'V.k/ W.k/ n, d' has the description in (b) As a map Hom. (c) As a map Hom.mP =m2 O P ; kŒ"/! Hom. Q; kŒ"/, d' is induced by '/! Hom.mQ=m2 Q; k/, d' is induced by the map mQ=m2 Q! O mP =m2 P defined by 'W kŒW! kŒV. EXAMPLE 4.32. Let E be a finite dimensional vector space over k. Then To.A.E//'E: ASIDE 4.33. A map Spm.kŒ"/! V should be thought of as a curve in V but with only the first infinitesimal structure retained. Thus, the descriptions of the tangent space provided by the terms in the top row of (24) correspond to the three standard descriptions of the tangent space in differential geometry (Wikipedia: TANGENT SPACE). 1Thus, Derk.kŒV ; k/ depends on P. g. Tangent cones 93 g. Tangent cones Let V be an algebraic subset of km, and let a D I.V /. Assume that P D.0; : : : ; 0/ 2 V. Define a to be the ideal generated by the polynomials F for F 2 a, where F is the leading form of F (see p. 83). The geometric tangent cone at P, CP.V / is V.a/, and the tangent cone is the pair.V.a/; kŒX1; : : : ; Xn=a
/. Obviously, CP.V / TP.V /. CAUTION. If a is principal, say a D.F /, then a D.F/, but if a D.F1; : : : ; Fr /, then it need not be true that a D.F1; : : : ; Fr/. Consider for example a D.XY; XZ C Z.Y 2 Z2//. One can show that this is an intersection of prime ideals, and hence is radical. As the polynomial Y Z.Y 2 Z2/ D Y.XZ C Z.Y 2 Z2// Z.XY / lies in a and is homogeneous, it lies in a, but it is not in the ideal generated by XY, XZ. In fact, a is the ideal generated by XY; XZ; Y Z.Y 2 Z2/: Let A be a local ring with maximal ideal n. The associated graded ring is gr.A/ D M ni =ni C1: i 0 Note that if A D Bm and n D mA, then gr.A/ D L mi =mi C1 (because of 1.15). PROPOSITION 4.34. The map kŒX1; : : : ; Xn=a! gr. kŒX1; : : : ; Xn=a to the class of Xi in gr. P / is an isomorphism. O PROOF. Let m be the maximal ideal in kŒX1; : : : ; Xn=a corresponding to P. Then O P / sending the class of Xi in gr mi =mi C1.X1; : : : ; Xn/i =.X1; : : : ; Xn/i C1 C a \.X1; : : : ; Xn/i.X1; : : : ; Xn/i =.X1; : : : ; Xn/i C1 C ai ; where ai is the homogeneous piece of a of degree i (that is, the subspace of a consisting of homogeneous polynomials of degree i ). But.X1; : : : ; Xn/i =.X1; : : : ; Xn/i C1 C ai D i th homogeneous piece of kŒX1; : : : ; Xn=a: For an aff
ine algebraic variety V and P 2 V, we define the geometric tangent cone P / by its P /red is the quotient of gr. CP.V / of V at P to be Spm.gr. nilradical, and we define the tangent cone to be.CP.V /; gr. P /red/, where gr. P //. O O O As in the case of a curve, the dimension of the geometric tangent cone at P is the same as the dimension of V (because the Krull dimension of a noetherian local ring is equal to P / is a polynomial ring in dim V variables if and that of its graded ring). Moreover, gr. P / is a only if polynomial ring in d variables, in which case CP.V / D TP.V /. P is regular. Therefore, P is nonsingular (see below) if and only if gr. O O O O A regular map 'W V! W sending P to Q induces a homomorphism gr. and hence a map CP.V /! CQ.V / of the geometric tangent cones. CAUTION. The map on the rings kŒX1; : : : ; Xn=a defined by a map of algebraic varieties is not the obvious one, i.e., it is not necessarily induced by the same map on polynomial rings as the original map. To see what it is, it is necessary to use Proposition 4.34, i.e., it is necessary to work with the rings gr. Q/! gr. P /, O O P /. O 94 4. LOCAL STUDY h. Nonsingular points; the singular locus DEFINITION 4.35. A point P on an affine algebraic variety V is said to be nonsingular or smooth if it lies on a single irreducible component W of V and dim TP.V / D dim W ; otherwise the point is said to be singular. A variety is nonsingular if all of its points are nonsingular. The set of singular points of a variety is called its singular locus. Thus, on an irreducible variety V of dimension d, P is nonsingular ” dimk TP.V / D d ” dimk.nP =n2 P / D d. PROPOSITION 4.36. Let V be an irreduc
ible variety of dimension d, and let P be a nonsingular point on V. Then there exist d regular functions f1; : : : ; fd defined in an open neighbourhood U of P such that P is the only common zero of the fi on U. PROOF. Suppose that P is nonsingular. Let f1; : : : ; fd generate the maximal ideal nP in P. Then f1; : : : ; fd are all defined on some open affine neighbourhood U of P, and I O claim that P is an irreducible component of the zero set V.f1; : : : ; fd / of f1; : : : ; fd in U. If not, there will be some irreducible component Z ¤ P of V.f1; : : : ; fd / passing through P. Write Z D V.p/ with p a prime ideal in kŒU. Because V.p/ V.f1; : : : ; fd / and because Z contains P and is not equal to it, we have.f1; : : : ; fd / p ¤ mP (ideals in kŒU /: On passing to the local ring P D kŒU mP, we find (using 1.14) that O.f1; : : : ; fd / p P ¤ nP O (ideals in P /: O This contradicts the assumption that the fi generate nP. Hence P is an irreducible component of V.f1; : : : ; fd /. On removing the remaining irreducible components of V.f1; : : : ; fd / from U, we obtain an open neighbourhood of P with the required property. Let P be a point on an irreducible variety V, and let f1; : : : ; fr generate the maximal ideal nP in P. The proof of the proposition shows that P is an irreducible component of V.f1; : : : ; fr /, and so r d (see 3.45). Nakayama’s lemma (1.3) shows that f1; : : : ; fr P span it. Thus dim TP.V / dim V, with generate nP if and only if their images in nP =n2 equality if and only if P is
nonsingular. O A point P on V is nonsingular if and only if there exists an open affine neighbourhood U of P and functions f1; : : : fd on U such that.f1; : : : ; fd / is the ideal of all regular functions on U zero at P. THEOREM 4.37. The set of nonsingular points of an affine algebraic variety is dense and open. PROOF. Let V be an irreducible component of the variety. It suffices to show that the singular locus of V is a proper closed subset.2 2Let V1; : : : ; Vr be the irreducible components of V. Then Vi \.T j ¤i Vj / is a proper closed subset of Vi. We show that.Vi /sing is a proper closed subset of Vi. It follows that Vi \ Vsing is the union of two proper closed subsets of Vi, and so it is proper and closed in Vi. Hence the points of Vi that are nonsingular on V form a nonempty open (hence dense) subset of Vi. h. Nonsingular points; the singular locus 95 We first show that it is closed. We may suppose that V D V.a/ A n. Let P1; : : : ; Pr generate a. Then the singular locus is the zero set of the ideal generated by the.n d /.n d / minors of the matrix Jac.P1; : : : ; Pr /.a/ D 0 B B @ @P1 @X1 ::: @Pr @X1.a/ : : :.aa/ @P1 @Xn ::: @Pr @Xn.a/ which is closed. We now show that the singular locus is not equal to V. According to 3.36 and 3.37 some nonempty open affine subset of V is isomorphic to a nonempty open affine subset of an d C1, and so we may suppose that V itself is an irreducible irreducible hypersurface in A d C1, say, equal to the zero set of the nonconstant irreducible polynomial hypersurface in A F.X1; : : : ; Xd C1/. By 2.64, dim V D d. The singular locus is the set of common zeros of the po
lynomials F ; @F @X1 ; : : : ; @F @Xd C1 ; and so it will be proper unless the polynomials @F=@Xi are identically zero on V. As in the proof of 4.7, if @F=@Xi is identically zero on V.F /, then it is the zero polynomial, and so F is a polynomial in X1; : : : ; Xi 1; Xi C1; : : : Xd C1 (characteristic zero) or in X1; : : : ; X p i ; : : : ; Xd C1 (characteristic p). Therefore, if the singular locus equals V, then F is constant (characteristic 0) or a pth power (characteristic p), which contradicts the hypothesis. COROLLARY 4.38. If V is irreducible, then dim V D min P 2V dim TP.V /. PROOF. By definition dim TP.V / dim V, with equality if and only if P is nonsingular. As there exists a nonsingular P, dim V is the minimum value of dim TP.V /. This formula can be useful in computing the dimension of a variety. COROLLARY 4.39. An irreducible algebraic variety is nonsingular if and only if the tangent spaces TP.V /, P 2 V, have constant dimension. PROOF. The constant dimension is the dimension of V, and so all points are nonsingular. COROLLARY 4.40. Every variety on which a group acts transitively by regular maps is nonsingular. PROOF. The group must act by isomorphisms, and so the tangent spaces have constant dimension. In particular, every group variety (see p. 109) is nonsingular. 96 Examples 4. LOCAL STUDY 4.41. For the surface Z3 D XY, the only singular point is.0; 0; 0/. The tangent cone at.0; 0; 0/ has equation XY D 0, and so it is the union of two planes intersecting in the z-axis. 4.42. For the surface V W Z3 D X 2Y, the singular locus is the line X D 0 D Z (and the singularity at.0; 0/ is very bad: for example, it lies
in the singular set of the singular set.3 The intersection of the surface with the surface Y D c is the cuspidal curve X 2 D Z3=c: 3, and let W be the zero set of 4.43. Let V be the union of the coordinate axes in A XY.X Y / in A 2. Each of V and W is a union of three lines meeting at the origin. Are they isomorphic as algebraic varieties? Obviously, the origin o is the only singular point on V or W. An isomorphism V! W would have to send the singular point o to the singular point o and map To.V / isomorphically onto To.W /. But V D V.XY; Y Z; XZ/, and so To.V / has dimension 3, whereas ToW has dimension 2. Therefore, V and W are not isomorphic. i. Nonsingularity and regularity THEOREM 4.44. Let P be a point on an irreducible variety V. Every generating set for the maximal ideal nP of P has at least d elements, and there exists a generating set with d elements if and only if P is nonsingular. O PROOF. If f1; : : : ; fr generate nP, then the proof of 4.36 shows that P is an irreducible component of V.f1; : : : ; fr / in some open neighbourhood U of P. Therefore 3.45 shows that 0 d r, and so r d. The rest of the statement has already been noted. COROLLARY 4.45. A point P on an irreducible variety is nonsingular if and only if regular. P is O PROOF. This is a restatement of the second part of the theorem. According to CA 22.3, a regular local ring is an integral domain. If P lies on two P is not P is not an integral domain (3.14), and so irreducible components of a V, then regular. Therefore, the corollary holds also for reducible varieties. O O 3In fact, it belongs to the worst class of singularities (sx2848895, KReiser). 0.10.512y j. Examples of tangent spaces 97 j. Examples of tangent spaces The description of the tangent space in terms of dual numbers is particularly convenient when our variety is given to us in
terms of its points functor. For example, let Mn be the set of n n matrices, and let I be the identity matrix. Write e for I when it is to be regarded as the identity element of GLn. 4.46. A matrix I C "A has inverse I "A in Mn.kŒ"/, and so lies in GLn.kŒ"/. In fact, Te.GLn/ D fI C "A j A 2 Mng'Mn.k/: 4.47. On expanding det.I C "A/ as a sum of signed products and using that "2 D 0, we find that det.I C "A/ D I C "trace.A/: Hence Te.SLn/ D fI C "A j trace.A/ D 0g'fA 2 Mn.k/ j trace.A/ D 0g: 4.48. Assume that the characteristic ¤ 2, and let On be the orthogonal group: On D fA 2 GLn j Atr A D I g: (Atr denotes the transpose of A). This is the group of matrices preserving the quadratic form n. The determinant defines a surjective regular homomorphism detW On! f˙1g, X 2 1 whose kernel is defined to be the special orthogonal group SOn. For I C "A 2 Mn.kŒ"/,.I C "A/tr.I C "A/ D I C "Atr C "A; C C X 2 and so Te.On/ D Te.SOn/ D fI C "A 2 Mn.kŒ"/ j A is skew-symmetricg'fA 2 Mn.k/ j A is skew-symmetricg: ASIDE 4.49. On the tangent space Te.GLn/'Mn of GLn, there is a bracket operation ŒM; N defD MN NM which makes Te.GLn/ into a Lie algebra. For any closed algebraic subgroup G of GLn, Te.G/ is stable under the bracket operation on Te.GLn/ and is a sub-Lie-algebra of Mn, which we denote Lie.G/. The Lie algebra structure on Lie.G/ is independent of the embedding of G into GLn (in fact, it has an intrinsic definition in
terms of left invariant derivations), and G 7! Lie.G/ is a functor from the category of linear group varieties to that of Lie algebras. This functor is not fully faithful, for example, every ´etale homomorphism G! G0 defines an isomorphism Lie.G/! Lie.G0/, but it is nevertheless very useful. Assume that k has characteristic zero. A connected algebraic group G is said to be semisimple if it has no closed connected solvable normal subgroup (except feg). Such a group G may have a finite nontrivial centre Z.G/, and we call two semisimple groups G and G0 locally isomorphic if G=Z.G/ G0=Z.G0/. For example, SLn is semisimple, with centre n, the set of diagonal matrices diag.; : : : ; /, n D 1, and SLn =n D PSLn. A Lie algebra is semisimple if it has no commutative ideal (except f0g). One can prove that G is semisimple ” Lie.G/ is semisimple; and the map G 7! Lie.G/ defines a one-to-one correspondence between the set of local isomorphism classes of semisimple algebraic groups and the set of isomorphism classes of Lie algebras. The classification of semisimple algebraic groups can be deduced from that of semisimple Lie algebras and a study of the finite coverings of semisimple algebraic groups — this is quite similar to the relation between Lie groups and Lie algebras. 98 4. LOCAL STUDY Exercises 4-1. Find the singular points, and the tangent cones at the singular points, for each of (a 3Y 2X C 3X 2Y C 2XY I (b) X 4 C Y 4 X 2Y 2 (assume that the characteristic is not 2). 4-2. Let V A n be an irreducible affine variety, and let P be a nonsingular point on V. Let n (i.e., the subvariety defined by a linear equation P ai Xi D d with H be a hyperplane in A not all ai zero) passing through P but not containing TP.V /.
Show that P is a nonsingular point on each irreducible component of V \ H on which it lies. (Each irreducible component has codimension 1 in V — you may assume this.) Give an example with H TP.V / and P singular on V \ H. Must P be singular on V \ H if H TP.V /? 4-3. Given a smooth point on a variety and a tangent vector at the point, show that there is a smooth curve passing through the point with the given vector as its tangent vector (see mo111467). 4-4. Let P and Q be points on varieties V and W. Show that T.P;Q/.V W / D TP.V / ˚ TQ.W /: 4-5. For each n, show that there is a curve C and a point P on C such that the tangent space to C at P has dimension n (hence C can’t be embedded in A 0 I 4-6. Let I be the n n identity matrix, and let J be the matrix. The symplectic group Spn is the group of 2n 2n matrices A with determinant 1 such that Atr J A D J. (It is the group of matrices fixing a nondegenerate skew-symmetric form.) Find the tangent space to Spn at its identity element, and also the dimension of Spn. n1 ). I 0 4-7. Find a regular map ˛W V! W which induces an isomorphism on the geometric tangent cones CP.V /! C˛.P /.W / but is not ´etale at P. 4-8. Show that the cone X 2 C Y 2 D Z2 is a normal variety, even though the origin is singular (characteristic ¤ 2). See p. 174. n. Suppose that a ¤ I.V /, and for a 2 V, let T 0 n/ defined by the equations.df /a D 0, f 2 a. Clearly, T 0 a a be the subspace Ta.V /, but need they 4-9. Let V D V.a/ A of Ta.A always be different? 4-10. Let W be a finite-dimensional k-vector space, and let RW D k ˚W endowed with the k-algebra structure for which
W 2 D 0. Let V be an affine algebraic variety over k. Show that the elements of V.RW / defD Homk-algebra.kŒV ; RW / are in natural one-to-one correspondence with the pairs.P; t / with P 2 V and t 2 W ˝ TP.V / (cf. Mumford, Lectures on curves..., 1966, p25). CHAPTER 5 Algebraic Varieties An algebraic variety is a ringed space that is locally isomorphic to an affine algebraic variety, just as a topological manifold is a ringed space that is locally isomorphic to an open subset of R n. We require both to satisfy a separation axiom. a. Algebraic prevarieties As motivation, recall the following definitions. DEFINITION 5.1. (a) A topological manifold of dimension n is a ringed space.V; V / such that V is Hausdorff and every point of V has an open neighbourhood U for which V jU / is isomorphic to the ringed space of continuous functions on an open subset of.U; R n (cf. 3.2). O O (b) A differentiable manifold of dimension n is a ringed space such that V is Hausdorff V jU / is isomorphic to and every point of V has an open neighbourhood U for which.U; the ringed space of smooth functions on an open subset of R n (cf. 3.3). O (c) A complex manifold of dimension n is a ringed space such that V is Hausdorff and V jU / is isomorphic to the every point of V has an open neighbourhood U for which.U; ringed space of holomorphic functions on an open subset of C O n (cf. 3.4). These definitions are easily seen to be equivalent to the more classical definitions in terms of charts and atlases.1 Often one imposes additional conditions on V, for example, that it be connected or that it have a countable base of open subsets. DEFINITION 5.2. An algebraic prevariety over k is a k-ringed space.V; is quasicompact and every point of V has an open neighbourhood U for which.U; is isomorphic to the ringed space of regular functions on an algebraic set over k. V / such that V V
jU / O O V / is an algebraic prevariety over k if there exists a finite open Thus, a ringed space.V; covering V D S Vi such that.Vi ; V jVi / is an affine algebraic variety over k for all i. An algebraic variety will be defined to be an algebraic prevariety satisfying a certain separation condition. O O An open subset U of an algebraic prevariety V such that.U, V jU / is an affine algebraic variety is called an open affine (subvariety) in V. Because V is a finite union of open affines, and in each open affine the open affines (in fact the basic open subsets) form a base for the topology, it follows that the open affines form a base for the topology on V. O 1Provided the latter are stated correctly, which is frequently not the case. 99 100 5. ALGEBRAIC VARIETIES Let.V; V / be an algebraic prevariety, and let U be an open subset of V. The functions O f W U! k lying in.U; V / are called regular. Note that if.Ui / is an open covering of V by affine varieties, then f W U! k is regular if and only if f jUi \ U is regular for all i (by 3.1(c)). Thus understanding the regular functions on open subsets of V amounts to understanding the regular functions on the open affine subvarieties and how these subvarieties fit together to form V. O EXAMPLE 5.3. (Projective space). Let P relation n denote knC1 X foriging modulo the equivalence.a0; : : : ; an/.b0; : : : ; bn/ ”.a0; : : : ; an/ D.cb0; : : : ; cbn/ some c 2 k: Thus the equivalence classes are the lines through the origin in knC1 (with the origin omitted). Write.a0W : : : W an/ for the equivalence class containing.a0; : : : ; an/. For each i, let Ui D f.a0 W : : : W ai W : : : W an/ 2 P n j ai ¤ 0g: Then P n D S Ui, and
the map.a0W : : : W an/ 7! a0 ai ; : : : ; bai ai ; : : : ; an ai W Ui ui! A n (the term ai =ai is omitted) is a bijection. In Chapter 6 we shall show that there is a unique n for which each Ui is an open affine structure of a (separated) algebraic variety on P subvariety of P n and each map ui is an isomorphism of algebraic varieties. b. Regular maps In each of the examples (5.1a,b,c), a morphism of manifolds (continuous map, smooth map, holomorphic map respectively) is just a morphism of ringed spaces. This motivates the following definition. O Let.V; V / and.W; W / be algebraic prevarieties. A map 'W V! W is said to be regular if it is a morphism of k-ringed spaces. In other words, a continuous map 'W V! W is regular if f 7! f ı'sends a regular function on an open subset U of W to a regular function on '1.U /. A composite of regular maps is again regular (this is a general fact about morphisms of ringed spaces). O Note that we have three categories: (affine varieties) (algebraic prevarieties) (ringed spaces). Each subcategory is full, i.e., the morphisms Mor.V; W / are the same in the three categories. W / be prevarieties, and let 'W V! W be a PROPOSITION 5.4. Let.V; O continuous map (of topological spaces). Let W D S Wj be a covering of W by open affines, and let '1.Wj / D S Vj i be a covering of '1.Wj / by open affines. Then'is regular if and only if its restrictions V / and.W; O are regular for all i; j. 'jVj i W Vj i! Wj PROOF. We assume that'satisfies this condition, and prove that it is regular. Let f be a regular function on an open subset U of W. Then f jU \ Wj is regular for each Wj (sheaf condition 3.1(b)), and so f ı
'j'1.U / \ Vj i is regular for each j; i (this is our assumption). It follows that f ı'is regular on '1.U / (sheaf condition 3.1(c)). Thus'is regular. The converse is even easier. c. Algebraic varieties 101 ASIDE 5.5. A differentiable manifold of dimension n is locally isomorphic to an open subset of n. In particular, all manifolds of the same dimension are locally isomorphic. This is not true for R algebraic varieties, for two reasons: (a) We are not assuming our varieties are nonsingular (see Chapter 4). (b) The inverse function theorem fails in our context: a regular map that induces an isomorphism on the tangent space at a point P need not induce an isomorphism in a neighbourhood of P. However, see 5.55 below. c. Algebraic varieties In the study of topological manifolds, the Hausdorff condition eliminates such bizarre possibilities as the line with the origin doubled in which a sequence tending to the origin has two limits (see 5.10 below). It is not immediately obvious how to impose a separation axiom on our algebraic varieties, because even affine algebraic varieties are not Hausdorff. The key is to restate the Hausdorff condition. Intuitively, the significance of this condition is that it prevents a sequence in the space having more than one limit. Thus a continuous map into the space should be determined by its values on a dense subset, i.e., if '1 and '2 are continuous maps Z! V that agree on a dense subset U of Z, then they should agree on the whole of Z.2 Equivalently, the set where two continuous maps '1; '2W Z U agree should be closed. Surprisingly, affine varieties have this property, provided '1 and '2 are required to be regular maps. LEMMA 5.6. Let '1; '2W Z V regular maps of affine algebraic varieties. The subset of Z on which '1 and '2 agree is closed. PROOF. There are regular functions xi on V such that P 7!.x1.P /; : : : ; xn.P // identifies n (take the xi to be any set of generators for kŒV as a k-algebra). V with a
closed subset of A Now xi ı '1 and xi ı '2 are regular functions on Z, and the set where '1 and '2 agree is Tn iD1 V.xi ı '1 xi ı '2/, which is closed. DEFINITION 5.7. An algebraic prevariety V is said to be separated if it satisfies the following additional condition: Separation axiom: for every pair of regular maps '1; '2W Z V with Z an affine algebraic variety, the set fz 2 Z j '1.z/ D '2.z/g is closed in Z. An algebraic variety over k is a separated algebraic prevariety over k.3 PROPOSITION 5.8. Let '1 and '2 be regular maps Z V from an algebraic prevariety Z to a separated prevariety V. The subset of Z on which '1 and '2 agree is closed. PROOF. Let W be the set on which '1 and '2 agree. For any open affine U of Z, W \ U is the subset of U on which '1jU and '2jU agree, and so W \ U is closed. This implies that W is closed because Z is a finite union of open affines. 2Let z 2 Z, and let z D lim un with un 2 U. Then '1.z/ D lim '1.un/ because '1 is continuous, and lim '1.un/ D lim '2.un/ D '2.z/. 3These are sometimes called “algebraic varieties in the sense of FAC” (Serre, Jean-Pierre. Faisceaux alg´ebriques coh´erents. Ann. of Math. (2) 61, (1955). 197–278; 34). In Grothendieck’s language, they are separated and reduced schemes of finite type over k (assumed to be algebraically closed), except that we omit the nonclosed points; cf. EGA IV, 10.10. Some authors use a more restrictive definition — they may require a variety to be connected, irreducible, or quasi-projective — usually because their foundations do not allow for a more flexible definition. 102 5. ALGEBRAIC VARIETIES EXAMPLE 5.9. The open subspace U D
an affine k-algebra A to the k-algebra of regular functions on V. For any P 2 V, f 7! ˛.f /.P / is a k-algebra homomorphism A! k, and so its kernel '.P / is a maximal ideal in A. In this way, we get a map O 'W V! spm.A/ which is easily seen to be regular. Conversely, from a regular map 'W V! Spm.A/, we get a k-algebra homomorphism f 7! f ı 'W A!.V; V /. Since these maps are inverse, we have proved the following result. O 4This is the algebraic analogue of the standard example of a non Hausdorff topological space. Let R denote the real line with the origin removed but with two points a ¤ b added. The topology is generated by the open intervals in R together with the sets of the form.u; 0/ [ fag [.0; v/ and.u; 0/ [ fbg [.0; v/, where u < 0 < v. Then X is not Hausdorff because a and b cannot be separated by disjoint open sets. Every sequence that converges to a also converges to b; for example, 1=n converges to both a and b. e. Subvarieties 103 PROPOSITION 5.12. For an algebraic variety V and an affine k-algebra A, there is a canonical one-to-one correspondence Mor.V; Spm.A//'Homk-algebra.A;.V; V //: O Let V be an algebraic variety such that.V; V / is an affine k-algebra. The proposition O shows that the regular map 'W V! Spm..V; V // defined by id.V;OV / has the following universal property: every regular map from V to an affine algebraic variety U factors uniquely through ': O'V Spm..V; V // O 9Š U: CAUTION 5.13. For a nonaffine algebraic variety V,.V; ated as a k-algebra. O V / need not be finitely gener- e. Subvarieties Let.V; O V / be an algebraic variety over k. Open subvarieties Let
U be an open subset of V. Then U is a union of open affines, and it follows that V jU / is a variety, called an open subvariety of V. A regular map 'W W! V is an open.U; immersion if '.W / is open in V and'defines an isomorphism W! '.W / of varieties. O Closed subvarieties Let Z be a closed subset of V. A function f on an open subset U of Z is regular if, for every P 2 U, there exists a germ.U 0; f 0/ of a regular function at P on V such that f 0jU 0 \ U D f jU 0 \ U. This defines a ringed structure Z/ is a variety it ZjU \ Z/ suffices to check that, for every open affine U V, the ringed space.U \ Z; is an affine algebraic variety, but this is only an exercise (Exercise 3-2 to be precise). Such Z/ is called a closed subvariety of V. A regular map 'W W! V is a closed a pair.Z; immersion if '.W / is closed in V and'defines an isomorphism W! '.W / of varieties. Z on Z. To show that.Z; O O O O Subvarieties A subset W of a topological space V is said to be locally closed if every point P in W has an open neighbourhood U in V such that W \ U is closed in U. Equivalent conditions: W is the intersection of an open and a closed subset of V ; W is open in its closure. A locally closed subset W of a variety V acquires a natural structure as a variety: write it as the intersection W D U \ Z of an open and a closed subset; Z is a variety, and W (being open in Z/ therefore acquires the structure of a variety. This structure on W has the following characterization: the inclusion map W,! V is regular, and a map 'W V 0! W with V 0 a variety is regular if and only if it is regular when regarded as a map into V. With this structure, W is called a subvariety of V. A regular map 'W W! V is an immersion if it induces an isomorphism of W onto a subvariety of V. Every immersion is the composite of an open immersion with a closed immersion (in both
orders). 104 5. ALGEBRAIC VARIETIES Application PROPOSITION 5.14. A prevariety V is separated if and only if two regular maps from a prevariety to V agree on the whole prevariety whenever they agree on a dense subset of it. PROOF. If V is separated, then the set on which a pair of regular maps '1; '2W Z V agree is closed, and so must be the whole of the Z. Conversely, consider a pair of maps '1; '2W Z V, and let S be the subset of Z on which they agree. We assume that V has the property in the statement of the proposition, and show that S is closed. Let NS be the closure of S in Z. According to the above discussion, NS has the structure of a closed prevariety of Z and the maps '1j NS and '2j NS are regular. Because they agree on a dense subset of NS they agree on the whole of NS, and so S D NS is closed. f. Prevarieties obtained by patching PROPOSITION 5.15. Suppose that the set V is a finite union V D S i 2I Vi of subsets Vi and that each Vi is equipped with ringed space structure. Assume that the following “patching” condition holds: for all i; j, Vi \ Vj is open in both Vi and Vj and jVi \ Vj D Vi O jVi \ Vj. Vj O Then there is a unique structure of a ringed space on V for which (a) each inclusion Vi,! V is a homeomorphism of Vi onto an open set, and (b) for each i 2 I, V jVi D Vi. O O If every Vi is an algebraic prevariety, then so also is V, and to give a regular map from V to a prevariety W amounts to giving a family of regular maps 'i W Vi! W such that 'i jVi \ Vj D 'j jVi \ Vj : PROOF. One checks easily that the subsets U V such that U \ Vi is open for all i are the open subsets for a topology on V satisfying (a), and that this is the only topology to satisfy Vi.U \ Vi / V.U / to be the set of functions f W U! k such that f jU \
Vi 2 (a). Define for all i. Again, one checks easily that V is a sheaf of k-algebras satisfying (b), and that it is the only such sheaf. O O O For the final statement, if each.Vi ; Vi / is a finite union of open affines, so also V /. Moreover, to give a map 'W V! W amounts to giving a family of maps is.V; 'i W Vi! W such that 'i jVi \ Vj D 'j jVi \ Vj (obviously), and'is regular if and only 'jVi is regular for each i. O O Clearly, the Vi may be separated without V being separated (see, for example, 5.10). In 5.29 below, we give a condition on an open affine covering of a prevariety sufficient to ensure that the prevariety is separated. g. Products of varieties Let V and W be objects in a category C. A triple.V W; pW V W! V; qW V W! W / g. Products of varieties 105 is said to be the product of V and W if it has the following universal property: for every pair of morphisms Z! V, Z! W in C, there exists a unique morphism Z! V W making the diagram Z 9Š commute. In other words, the triple is a product if the map p V V W q W'7!.p ı '; q ı '/W Hom.Z; V W /! Hom.Z; V / Hom.Z; W / is a bijection. The product, if it exists, is uniquely determined up to a unique isomorphism by its universal property. For example, the product of two sets (in the category of sets) is the usual cartesian product of the sets, and the product of two topological spaces (in the category of topological spaces) is the product of the underlying sets endowed with the product topology. We shall show that products exist in the category of algebraic varieties. Suppose, for the 0; Z/ is the underlying set of Z; 0! Z with image z is regular, and these are all moment, that V W exists. For any prevariety Z, Mor.A more precisely, for any z 2 Z, the map A the regular maps (cf. 3.28). Thus, from the definition of products
we have (underlying set of V W /'Mor.A'Mor.A'(underlying set of V / (underlying set of W /: 0; V W / 0; V / Mor.A 0; W / Hence, our problem can be restated as follows: given two prevarieties V and W, define on the set V W the structure of a prevariety such that (a) the projection maps p; qW V W V; W are regular, and (b) a map 'W T! V W of sets (with T an algebraic prevariety) is regular if its compo- nents p ı '; q ı'are regular. There can be at most one such structure on the set V W. Products of affine varieties EXAMPLE 5.16. Let a and b be ideals in kŒX1; : : : ; Xm and kŒXmC1; : : : ; XmCn respectively, and let.a; b/ be the ideal in kŒX1; : : : ; XmCn generated by the elements of a and b. Then there is an isomorphism f ˝ g 7! fgW kŒX1; : : : ; Xm a ˝k kŒXmC1; : : : ; XmCn b! kŒX1; : : : ; XmCn.a; b/ : Again this comes down to checking that the natural map Homk-alg.kŒX1; : : : ; XmCn=.a; b/; R/ Homk-alg.kŒX1; : : : ; Xm=a; R/ Homk-alg.kŒXmC1; : : : ; XmCn=b; R/ is a bijection. But the three sets are respectively 106 5. ALGEBRAIC VARIETIES V.a; b/ D zero set of.a; b/ in RmCn; V.a/ D zero set of a in Rm; V.b/ D zero set of b in Rn; and so this is obvious. The tensor product of two k-algebras A and B has the universal property to be a product in the category of k-alge
bras, but with the arrows reversed. Because of the category antiequivalence (3.25), this shows that Spm.A ˝k B/ will be the product of Spm A and Spm B in the category of affine algebraic varieties once we have shown that A ˝k B is an affine k-algebra. PROPOSITION 5.17. Let A and B be k-algebras with A finitely generated. (a) If A and B are reduced, then so also is A ˝k B. (b) If A and B are integral domains, then so also is A ˝k B. PROOF. Let ˛ 2 A ˝k B. Then ˛ D Pn is a linear combination of the remaining bi, say, bn D Pn1 bilinearity of ˝, we find that i D1 ai ˝ bi, some ai 2 A, bi 2 B. If one of the bj i D1 ci bi, ci 2 k, then, using the ˛ D n1 X i D1 ai ˝ bi C n1 X i D1 ci an ˝ bi D n1 X i D1.ai C ci an/ ˝ bi : Thus we can suppose that in the original expression of ˛, the bi are linearly independent over k. Now assume A and B to be reduced, and suppose that ˛ is nilpotent. Let m be a maximal ideal of A. From a 7! NaW A! A=m D k we obtain homomorphisms a ˝ b 7! Na ˝ b 7! NabW A ˝k B! k ˝k B '! B: The image P Nai bi of ˛ under this homomorphism is a nilpotent element of B, and hence is zero (because B is reduced). As the bi are linearly independent over k, this means that the Nai are all zero. Thus, the ai lie in all maximal ideals m of A, and so are zero (see 2.18). Hence ˛ D 0, and we have shown that A ˝k B is reduced. Now assume that A and B are integral domains, and let ˛, ˛0 2
A ˝k B be such that i with the sets fb1; b2; : : :g 2; : : :g each linearly independent over k. For each maximal ideal m of A, we know i / D 0. Thus either all ˛˛0 D 0. As before, we can write ˛ D P ai ˝bi and ˛0 D P a0 i 1; b0 and fb0.P Nai bi /.P Na0 the ai 2 m or all the a0 i i / D 0 in B, and so either.P Nai bi / D 0 or.P Na0 2 m. This shows that ˝b0 i b0 i b0 spm.A/ D V.a1; : : : ; am/ [ V.a0 1; : : : ; a0 n/: As spm.A/ is irreducible (see 2.27), it follows that spm.A/ equals either V.a1; : : : ; am/ or V.a0 n/. In the first case ˛ D 0, and in the second ˛0 D 0. 1; : : : ; a0 REMARK 5.18. The proof of 5.17 fails when k is not algebraically closed, because then A=m may be a finite extension of k over which the bi become linearly dependent (see sx599391). The following examples show that the statement of 5.17 also fails in this case. (a) Suppose that k is nonperfect of characteristic p, so that there exists an element ˛ in an algebraic closure of k such that ˛ … k but ˛p 2 k. Let k0 D kŒ˛, and let ˛p D a. Then g. Products of varieties 107.˛ ˝ 1 1 ˝ ˛/ ¤ 0 in k0 ˝k k0 (in fact, the elements ˛i ˝ ˛j, 0 i; j p 1, form a basis for k0 ˝k k0 as a k-vector space), but.˛ ˝ 1 1 ˝ ˛/p D.a ˝ 1 1 ˝ a/ D.1 ˝ a 1
˝ a/ D 0:.because a 2 k/ Thus k0 ˝k k0 is not reduced, even though k0 is a field. (b) Let K be a finite separable extension of k and let ˝ be a second field containing k. By the primitive element theorem (FT 5.1), K D kŒ˛ D kŒX=.f.X//; for some ˛ 2 K and its minimal polynomial f.X/. Assume that ˝ is large enough to split f, say, f.X/ D Q i.X ˛i / with ˛i 2 ˝. Because K=k is separable, the ˛i are distinct, and so ˝ ˝k K'˝ŒX=.f.X//'Y ˝ŒX=.X ˛i /; (1.58(b)) (1.1) which is not an integral domain. For example, C ˝ R C'CŒX=.X i/ CŒX=.X C i/'C C: The proposition allows us to make the following definition. DEFINITION 5.19. The product of the affine varieties V and W is.V W; O V W / D Spm.kŒV ˝k kŒW / with the projection maps p; qW V W! V; W defined by the homomorphisms f 7! f ˝ 1W kŒV! kŒV ˝k kŒW g 7! 1 ˝ gW kŒW! kŒV ˝k kŒW : PROPOSITION 5.20. Let V and W be affine varieties. (a) The variety.V W; V W / is the product of.V; W / in the category of affine algebraic varieties; in particular, the set V W is the product of the sets V and W and p and q are the projection maps. (b) If V and W are irreducible, then so also is V W. V / and.W; O O O PROOF. (a) As noted at the start of the subsection, the first statement follows from 5.17(a), and the
second statement then follows by the argument on p. 105. (b) This follows from 5.17(b) and 2.27. COROLLARY 5.21. Let V and W be affine varieties. For every prevariety T, a map 'W T! V W is regular if p ı'and q ı'are regular. PROOF. If p ı'and q ı'are regular, then 5.20 implies that'is regular when restricted to any open affine of T, which implies that it is regular on T. 108 5. ALGEBRAIC VARIETIES The corollary shows that V W is the product of V and W in the category of prevarieties (hence also in the categories of varieties). EXAMPLE 5.22. (a) It follows from 1.57 that A mCn endowed with the projection maps m p A mCn q! A n; A p.a1; : : : ; amCn/ D.a1; : : : ; am/ q.a1; : : : ; amCn/ D.amC1; : : : ; amCn/; is the product of A m and A n. (b) It follows from 5.16 that is the product of V.a/ and V.b/. V.a/ p V.a; b/ q! V.b/ CAUTION. When V and W have dimension > 0, the topology on V W is strictly finer 1, every than product topology. For example, for the product topology on A proper closed subset is contained in a finite union of vertical and horizontal lines, whereas A 2 has many more closed subsets (see 2.68). If V is affine, then the diagonal in V V is closed for the Zariski topology. Therefore, if the Zariski topology on V V is equal to the product topology, then V is Hausdorff. We deduce that the Zariski topology on V V is the product topology if and only if V is finite. 2 D A 1 A Products in general We now define the product of two algebraic prevarieties V and W. Write V as a union of open affines V D S Vi, and note that V can be regarded as the Vi /; in particular, this covering satisfies the patching variety obtained
by patching the.Vi ; condition (5.15). Similarly, write W as a union of open affines W D S Wj. Then O V W D [ Vi Wj and the.Vi Wj ; W; O O V W / to be the variety obtained by patching the.Vi Wj ; Vi Wj /. O Vi Wj / satisfy the patching condition. Therefore, we can define.V V W just defined, V W becomes the PROPOSITION 5.23. With the sheaf of k-algebras product of V and W in the category of prevarieties. In particular, the structure of prevariety on V W defined by the coverings V D S Vi and W D S Wj are independent of the coverings. O PROOF. Let T be a prevariety, and let 'W T! V W be a map of sets such that p ı'and q ı'are regular. Then 5.21 implies that the restriction of'to '1.Vi Wj / is regular. As these open sets cover T, this shows that'is regular. PROPOSITION 5.24. If V and W are separated, then so also is V W. PROOF. Let '1; '2 be two regular maps U! V W. The set where '1; '2 agree is the intersection of the sets where p ı '1; p ı '2 and q ı '1; q ı '2 agree, which is closed. PROPOSITION 5.25. If V and W are connected, then so also is V W. h. The separation axiom revisited 109 PROOF. For v0 2 V, we have continuous maps W'v0 W closed V W: Similarly, for w0 2 W, we have continuous maps V'V w0 closed V W: The images of V and W in V W intersect in.v0; w0/ and are connected, which shows that.v0; w/ and and.v; w0/ lie in the same connected component of V W for all v 2 V and w 2 W. Since v0 and w0 were arbitrary, this shows that any two points lie in the same connected component. Group varieties A group variety is an algebraic variety G together with a group structure defined by regular maps mW G G! G; invW G
'2/W Z! V V; z 7!.'1.z/; '2.z// is regular because its components '1 and '2 are regular (see p. 105). In particular, it is continuous, and so.'1; '2/1.V / is closed, but this is exactly the subset on which '1 and '2 agree. Conversely, V is the set on which the two projection maps V V! V agree, and so it is closed if V is separated. COROLLARY 5.27. For any prevariety V, the diagonal is a locally closed subset of V V. PROOF. Let P 2 V, and let U be an open affine neighbourhood of P. Then U U is an open neighbourhood of.P; P / in V V, and V \.U U / D U, which is closed in U U because U is separated (5.6). Thus V is always a subvariety of V V, and it is closed if and only if V is separated. The graph'of a regular map 'W V! W is defined to be f.v; '.v// 2 V W j v 2 V g: COROLLARY 5.28. For any morphism 'W V! W of prevarieties, the graph'of'is locally closed in V W, and it is closed if W is separated. The map v 7!.v; '.v// is an isomorphism of V onto'(as algebraic prevarieties). PROOF. The map.v; w/ 7!.'.v/; w/W V W! W W is regular because its composites with the projections are'and idW which are regular. In particular, it is continuous, and as'is the inverse image of W under this map, this proves the first statement. The second statement follows from the fact that the regular map ',! V W p! V is an inverse to v 7!.v; '.v//W V! '. THEOREM 5.29. The following three conditions on a prevariety V are equivalent: (a) V is separated; (b) for every pair of open affines U and U 0 in V, U \ U 0 is an open affine, and the map f ˝ g 7! f jU \U 0 gjU \U 0W kŒU
˝k kŒU 0! kŒU \ U 0 is surjective; 5Recall that the topology on V V is not the product topology. Thus the statement does not contradict the fact that V is not Hausdorff. VΓϕvWϕ(v)(v,ϕ(v)) i. Fibred products 111 (c) the condition in (b) holds for the sets in some open affine covering of V. PROOF. Let U and U 0 be open affines in V. We shall prove that (i) if is closed then U \ U 0 affine, (ii) when U \ U 0 is affine,.U U 0/ \ is closed ” kŒU ˝k kŒU 0! kŒU \ U 0 is surjective: Assume (a); then these statements imply (b). Assume that (b) holds for the sets in an open affine covering.Ui /i 2I of V. Then.Ui Uj /.i;j /2I I is an open affine covering of V V, and V \.Ui Uj / is closed in Ui Uj for each pair.i; j /, which implies (a). Thus, the statements (i) and (ii) imply the theorem. Proof of (i): The graph of the inclusion U \ U 0,! V is the subset.U U 0/ \ of.U \ U 0/ V: If V is closed, then.U U 0/ \ V is a closed subvariety of an affine variety, and hence is affine. Now 5.28 implies that U \ U 0 is affine. Proof of (ii): Assume that U \ U 0 is affine. Then.U U 0/ \ V is closed in U U 0 ” v 7!.v; v/W U \ U 0! U U 0 is a closed immersion ” kŒU U 0! kŒU \ U 0 is surjective (3.34). Since kŒU U 0 D kŒU ˝k kŒU 0, this completes the proof of (ii). In more down-to-earth terms, condition (b) says that U \ U 0 is affine and every regular function on U \ U
0 is a sum of functions of the form P 7! f.P /g.P / with f and g regular functions on U and U 0. EXAMPLE 5.30. (a) Let V D P 5.3). Then U0 \ U1 D A U0 \ U1,! Ui are 1, and let U0 and U1 be the standard open subsets (see 1 X f0g, and the maps on rings corresponding to the inclusions f.X/ 7! f.X/W kŒX! kŒX; X 1 f.X/ 7! f.X 1/W kŒX! kŒX; X 1: Thus the sets U0 and U1 satisfy the condition in (b). (b) Let V be A lower copies of A inclusions U0 \ U1,! Ui are 1 with the origin doubled (see 5.10), and let U and U 0 be the upper and 1 in V. Then U \ U 0 is affine, but the maps on rings corresponding to the X 7! XW kŒX! kŒX; X 1 X 7! XW kŒX! kŒX; X 1: Thus the sets U0 and U1 fail the condition in (b). (c) Let V be A 2 in V. Then U \ U 0 is not affine (see 3.33). of A 2 with the origin doubled, and let U and U 0 be the upper and lower copies i. Fibred products Let 'W V! S and W W! S be regular maps of algebraic varieties. The set V S W defD f.v; w/ 2 V W j '.v/ D.w/g 112 5. ALGEBRAIC VARIETIES is closed in V W; because it is the set where'ı p and ı q agree, and so it has a canonical structure of an algebraic variety (see p. 103). The algebraic variety V S W is called the fibred product of V and W over S. Note that if S consists of a single point, then V S W D V W. Write '0 for the map.v; w/ 7! wW V S W! W and 0 for the map.v; w/ 7! vW V S W! V. We then have a commutative
diagram: '0 V S W 0'V W S: The system.V S W; '0; 0/ has the following universal property: for any regular maps ˛W T! V, ˇW T! W such that '˛ D ˇ, there is a unique regular map.˛; ˇ/W T! V S W such that the following diagram T ˇ.˛; ˇ / V S W ˛ 0 V '0'W S commutes. In other words, Hom.T; V S W /'Hom.T; V / Hom.T;S/ Hom.T; W /: Indeed, there is a unique such map of sets, namely, t 7!.˛.t/; ˇ.t//, which is regular because it is as a map into V W. The map '0 in the above diagrams is called the base change of'with respect to. For any point P 2 S, the base change of 'W V! S with respect to P,! S is the map '1.P /! P induced by ', which is called the fibre of V over P. EXAMPLE 5.31. If f W V! S is a regular map and U is a subvariety of S, then V S U is the inverse image of U in V. Notes 5.32. Since a tensor product of rings A ˝R B has the opposite universal property to that of a fibred product, one might hope that Spm.A/ Spm.R/ Spm.B/ ‹‹D Spm.A ˝R B/: This is true if A ˝R B is an affine k-algebra, but in general it may have nonzero nilpotent elements. For example, let k have characteristic p, let R D kŒX, and consider the kŒXalgebras kŒX! k; X 7! a kŒX! kŒX; X 7! X p: j. Dimension 113 Then A ˝R B'k ˝kŒX p kŒX'kŒX=.X p a/; which contains the nilpotent element x a 1 p. The correct statement is Spm.A/ Spm.R/ Spm
.B/'Spm.A ˝R B=N/; (25) where N is the ideal of nilpotent elements in A ˝R B. To prove this, note that for any algebraic variety T, Mor.T; Spm.A ˝R B=N//'Hom.A ˝R B=N; T.T // O T.T //'Hom.A ˝R B;'Hom.A; O T.T // O'Mor.T; Spm.A// Hom.R;OT.T // Mor.T;Spm.R// Hom.B; T.T // O Mor.T; Spm.B// (5.12).5.12). For the second isomorphism we used that the ring isomorphism, we used the universal property of A ˝R B. O T.T / is reduced, and for the third 5.33. Fibred products may differ depending on whether we are working in the category of algebraic varieties or algebraic schemes. For example, Spec.A/ Spec.R/ Spec.B/'Spec.A ˝R B/ in the category of schemes. Consider the map x 7! x2W A 1 (see 5.49). The fibre '1.a/ consists of two points if a ¤ 0, and one point if a D 0. Thus '1.0/ D Spm.kŒX=.X//. However, the scheme-theoretic fibre is Spec.kŒX=.X 2//, which reflects the fact that 0 is “doubled” in the fibre over 0. 1 '! A 5.34. Fibred products exist also for prevarieties. In this case, V S W is only locally closed in V W. j. Dimension Recall p. 46 that, in an irreducible topological space, every nonempty open subset is dense and irreducible. Let V be an irreducible algebraic variety V, and let U and U 0 be nonempty open affines in V. Then U \ U 0 is also a nonempty open affine (5.29), which is dense in U, and so the V.U \ U 0/ is injective. Therefore restriction map V.U /! O O kŒU k
ŒU \ U 0 k.U /; where k.U / is the field of fractions of kŒU, and so k.U / is also the field of fractions of kŒU \ U 0 and of kŒU 0. Thus, attached to V there is a field k.V /, called the function field of V or the field of rational functions on V, which is the field of fractions of kŒU for any open affine U in V. The dimension of V is defined to be the transcendence degree of k.V / over k. Note the dim.V / D dim.U / for any open subset U of V. In particular, dim.V / D dim.U / for U an open affine in V. It follows that some of the results in 2 carry over — for example, if Z is a proper closed subvariety of V, then dim.Z/ < dim.V /. 114 5. ALGEBRAIC VARIETIES PROPOSITION 5.35. Let V and W be irreducible varieties. Then dim.V W / D dim.V / C dim.W /: PROOF. We may suppose V and W to be affine. Write kŒV D kŒx1; : : : ; xm kŒW D kŒy1; : : : ; yn; where the x and y have been chosen so that fx1; : : : ; xd g and fy1; : : : ; yeg are maximal algebraically independent sets of elements of kŒV and kŒW. Then fx1; : : : ; xd g and fy1; : : : ; yeg are transcendence bases of k.V / and k.W / (see 1.63), and so dim.V / D d and dim.W / D e. Now6 kŒV W defD kŒV ˝k kŒW kŒx1; : : : ; xd ˝k kŒy1; : : : ; ye; which is a polynomial ring in the symbols x1 ˝1; : : : ; xd ˝1; 1˝y1; : : : ; 1˝ye (see 1.57). In particular,
the elements x1 ˝1; : : : ; xd ˝1; 1˝y1; : : : ; 1˝ye are algebraically independent in kŒV ˝k kŒW. Obviously kŒV W is generated as a k-algebra by the elements xi ˝ 1, 1 ˝ yj, 1 i m, 1 j n, and all of them are algebraic over kŒx1; : : : ; xd ˝k kŒy1; : : : ; ye. Thus the transcendence degree of k.V W / is d C e. We extend the definition of dimension to an arbitrary variety V as follows. An algebraic variety is a finite union of noetherian topological spaces, and so is noetherian. Consequently (see 2.31), V is a finite union V D S Vi of its irreducible components, and we define dim.V / D max dim.Vi /. When all the irreducible components of V have dimension n; V is said to be pure of dimension n (or to be of pure dimension n). PROPOSITION 5.36. Let V and W be closed subvarieties of A ducible component Z of V \ W, n; for any (nonempty) irre- that is, dim.Z/ dim.V / C dim.W / nI codim.Z/ codim.V / C codim.W /: PROOF. In the course of the proof of Theorem 5.29, we saw that V \ W is isomorphic to \.V W /, and this is defined by the n equations Xi D Yi in V W. Thus the statement follows from 3.45. REMARK 5.37. (a) The subvariety X 2 C Y 2 D Z2 Z D 0 3 is the curve X 2 C Y 2 D 0, which is the pair of lines Y D ˙iX if k D C; in particular, of A the codimension is 2. Note however, that real locus is f.0; 0/g, which has codimension 3. Thus, Proposition 5.36 becomes false if one looks only at real points (and the pictures we draw can mislead). 6In general, it is not true that if M 0 and N 0 are
R-submodules of M and N, then M 0 ˝R N 0 is an Rsubmodule of M ˝R N. However, this is true if R is a field, because then M 0 and N 0 will be direct summands of M and N, and tensor products preserve direct summands. k. Dominant maps 115 (b) Proposition 5.36 becomes false if A n is replaced by an arbitrary affine variety. Consider for example the affine cone V It contains the planes, X1X4 X2X3 D 0: Z W X2 D 0 D X4I Z0 W X1 D 0 D X3I Z D f.; 0; ; 0/g Z0 D f.0; ; 0; /g and Z \ Z0 D f.0; 0; 0; 0/g. Because V is a hypersurface in A of Z and Z0 has dimension 2. Thus 4, it has dimension 3, and each codim Z \ Z0 D 3 — 1 C 1 D codim Z C codim Z0: The proof of 5.36 fails because the diagonal in V V cannot be defined by 3 equations 4) — the diagonal is not a set-theoretic (it takes the same 4 that define the diagonal in A complete intersection. k. Dominant maps As in the affine case, a regular map 'W V! W is said to be dominant if the image of'is dense in W. Suppose V and W are irreducible. If V 0 and W 0 are open affine subsets of V and W such that '.V 0/ W 0, then 3.34 implies that the map f 7! f ı 'W kŒW 0! kŒV 0 is injective. Therefore it extends to a map on the fields of fractions, k.W /! k.V /, and this map is independent of the choice of V 0 and W 0. l. Rational maps; birational equivalence Loosely speaking, a rational map from a variety V to a variety W is a regular map from a dense open subset of V to W, and a birational map is a rational map admitting a rational inverse. Let V and W be varieties over k, and consider pairs.U; 'U /, where U is a dense open subset of V and 'U is a regular map U! W. Two such pairs.
U; 'U / and.U 0; 'U 0/ are said to be equivalent if 'U and 'U 0 agree on U \ U 0. An equivalence class of pairs is called a rational map 'W V Ü W. A rational map'is said to be defined at a point v of V if v 2 U for some.U; 'U / 2 '. The set U1 of v at which'is defined is open, and there is a regular map '1W U1! W such that.U1; '1/ 2'— clearly, U1 D S.U;'U /2' U and we can define '1 to be the regular map such that '1jU D 'U for all.U; 'U / 2 '. Hence, in the equivalence class, there is always a pair.U; 'U / with U largest (and U is called “the open subvariety on which'is defined”). PROPOSITION 5.38. Let V and V 0 be irreducible varieties over k. A regular map 'W U 0! U from an open subset U 0 of V 0 onto an open subset U of V defines a k-algebra homomorphism k.V /! k.V 0/, and every such homomorphism arises in this way. PROOF. The first part of the statement is obvious, so let k.V /,! k.V 0/ be a k-algebra homomorphism. We identify k.V / with a subfield of k.V 0/. Let U (resp. U 0) be an open affine subset of V (resp. U 0). Let kŒU D kŒx1; : : : ; xm. Each xi 2 k.V 0/, which is the field of fractions of kŒU 0, and so there exists a nonzero d 2 kŒU 0 such that dxi 2 kŒU 0 116 5. ALGEBRAIC VARIETIES for all i. After inverting d, i.e., replacing U 0 with basic open subset, we may suppose that kŒU kŒU 0. Thus, the inclusion k.V /,! k.V 0/ is induced by a dominant regular map 'W U 0! U. According to Theorem 9.1 below, the image of'contains
an open subset U0 of U. Now '1.U0/ '! U0 is the required map. A rational (or regular) map 'W V Ü W is birational if there exists a rational map '0W W Ü V such that '0 ı'D idV and'ı '0 D idW as rational maps. Two varieties V and V 0 are birationally equivalent if there exists a birational map from one to the other. In this case, there exist dense open subsets U and U 0 of V and V 0 respectively such that U U 0. PROPOSITION 5.39. Two irreducible varieties V and V 0 are birationally equivalent if and only if their function fields are isomorphic over k. PROOF. Assume that k.V / k.V 0/. We may suppose that V and W are affine, in which case the existence of U U 0 is proved in 3.36. This proves the “if” part, and the “only if” part is obvious. PROPOSITION 5.40. Every irreducible algebraic variety of dimension d is birationally equivalent to a hypersurface in A d C1. PROOF. Let V be an irreducible variety of dimension d. According to Proposition 3.38, there exist x1; : : : ; xd ; xd C1 2 k.V / such that k.V / D k.x1; : : : ; xd ; xd C1/. Let f 2 kŒX1; : : : ; Xd C1 be an irreducible polynomial satisfied by the xi, and let H be the hypersurface f D 0. Then k.V / k.H /. m. Local study Everything in Chapter 4, being local, extends mutatis mutandis, to general algebraic varieties. 5.41. The tangent space TP.V / at a point P on an algebraic variety V is the fibre of V.kŒ"/! V.k/ over P. There are canonical isomorphisms TP.V /'Derk. P ; k/'Homk-linear.nP =n2 P ; k/; where nP is the maximal ideal of O P. O 5.42. A point P on an algebraic variety V is nonsingular (or
smooth) if it lies on a single irreducible component W and dim TP.V / D dim W. A point P is nonsingular if and only if the local ring P is regular. The singular points form a proper closed subvariety, called the O singular locus. 5.43. A variety is nonsingular (or smooth) if every point is nonsingular. n. ´Etale maps DEFINITION 5.44. A regular map 'W V! W of smooth varieties is ´etale at a point P of V if the map.d'/P W TP.V /! T'.P /.W / is an isomorphism;'is ´etale if it is ´etale at all points of V. n. ´Etale maps 117 Examples 5.45. A regular map 'W A n! A n, a 7!.P1.a1; : : : ; an/; : : : ; Pn.a1; : : : ; an// is ´etale at a if and only if rank Jac.P1; : : : ; Pn/.a/ D n, because the map on the tangent spaces has matrix Jac.P1; : : : ; Pn/.a/. Equivalent condition: det ¤ 0..a/ @Pi @Xj 5.46. Let V D Spm.A/ be an affine variety, and let f D P ci X i 2 AŒX be such that AŒX=.f.X// is reduced. Let W D Spm.AŒX=.f.X///, and consider the map W! V corresponding to the inclusion A,! AŒX=.f /. Thus AŒX=.f / AŒX W A V A 1 V The points of W lying over a point a 2 V are the pairs.a; b/ 2 V A 1 such that b is a root of P ci.a/X i. I claim that the map W! V is ´etale at.a; b/ if and only if b is a simple root of P ci.a/X i. To see this, write A D kŒX1; : : : ; Xn=a, a D.f1; : : : ; fr /, so that
AŒX=.f / D kŒX1; : : : ; Xn=.f1; : : : ; fr ; f /: The tangent spaces to W and V at.a; b/ and a respectively are the null spaces of the matrices 0 B B B B @.a/ : : : @f1 @X1 ::: @f1 @Xn :::.a/ 0 @fr @X1 @f @X1.a/.a/ : : : : : : @fr @Xn @f @Xn.a/.a/ 0 @f @X.a; bf1 @X1 ::: @fr @X1.a/ : : :.a/ : : : 1 C C A.a/ @f1 @Xn ::: @fr @Xn.a/ and the map T.a;b/.W /! Ta.V / is induced by the projection map knC1! kn omitting the last coordinate. This map is an isomorphism if and only if @f @X.a; b/¤ 0, because then every solution of the smaller set of equations extends uniquely to a solution of the larger set. But @f @X.a; b/ D d.P i ci.a/X i / dX which is zero if and only if b is a multiple root of P i ci.a/X i. The intuitive picture is that W! V is a finite covering with deg.f / sheets, which is ramified exactly at the points where two or more sheets cross..b/; 5.47. Consider a dominant map 'W W! V of smooth affine varieties, corresponding to a map A! B of rings. Suppose B can be written B D AŒY1; : : : ; Yn=.P1; : : : ; Pn/ (same number of polynomials as variables). A similar argument to the above shows that'is ´etale if and only if det is never zero..a/ @Pi @Xj 5.48. The example in 5.46 is typical; in fact every ´etale map is locally of this form, provided P is a normal domain for all P 2 V. More precisely, let 'W W! V V is normal, i.e., O 118 5.
ALGEBRAIC VARIETIES be ´etale at P 2 W, and assume V to be normal; then there exist a map '0W W 0! V 0 with kŒW 0 D kŒV 0ŒX=.f.X//, and a commutative diagram W'V U1 ´etale U2 U 0 1 ´etale U 0 2 W 0 '0 V 0 with all the U open subvarieties and P 2 U1. The failure of the inverse function theorem for the Zariski topology 5.49. In advanced calculus (or differential topology, or complex analysis), the inverse function theorem says that a map'that is ´etale at a point a is a local isomorphism there, i.e., there exist open neighbourhoods U and U 0 of a and '.a/ such that'induces an isomorphism U! U 0. This is not true in algebraic geometry, at least not for the Zariski topology: a map can be ´etale at a point without being a local isomorphism. Consider for example the map 'W A 1 X f0g! A 1 X f0g; a 7! a2: This is ´etale if the characteristic is ¤ 2, because the Jacobian matrix is.2X/, which has rank one for all X ¤ 0 (alternatively, it is of the form 5.46 with f.X/ D X 2 T, where T is the 1, and X 2 c has distinct roots for c ¤ 0). Nevertheless, I claim coordinate function on A that there do not exist nonempty open subsets U and U 0 of A 1 f0g such that'defines an isomorphism U! U 0. If there did, then'would define an isomorphism kŒU 0! kŒU 1/. But on the fields of and hence an isomorphism on the fields of fractions k.A fractions,'defines the map k.X/! k.X/, X 7! X 2, which is not an isomorphism. 1/! k.A 5.50. Let V be the plane curve Y 2 D X and'the map V! A 2 W 1 except over 0, and so we may view it schematically as 1,.x; y/ 7! x. Then'is However, when
viewed as a Riemann surface, V.C/ consists of two sheets joined at a single point O. As a point on the surface moves around O, it shifts from one sheet to the other. Thus the true picture is more complicated. To get a section to ', it is necessary to remove a line in C from 0 to infinity, which is not closed for the Zariski topology. \|0VA1ϕ n. ´Etale maps 119 It is not possible to fit the graph of the complex curve Y 2 D X into 3-space, but the picture at right is an early depiction of it (from Neumann, Carl, Vorlesungen ¨uber Riemann’s theorie der Abel’schen integrale, Leipzig: Teubner, 1865). ´Etale maps of singular varieties Using tangent cones, we can extend the notion of an ´etale morphism to singular varieties. Obviously, a regular map ˛W V! W induces a homomorphism gr. P /. We say that ˛ is ´etale at P if this is an isomorphism. Note that then there is an isomorphism of the geometric tangent cones CP.V /! C˛.P /.W /, but this map may be an isomorphism without ˛ being ´etale at P. Roughly speaking, to be ´etale at P, we need the map on geometric tangent cones to be an isomorphism and to preserve the “multiplicities” of the components. O˛.P //! gr. O O O˛.P /! O P! kŒŒX1; : : : ; Xd. It is a fairly elementary result that a local homomorphism of local rings ˛W A! B induces an isomorphism on the graded rings if and only if it induces an isomorphism on the completions (Atiyah-MacDonald 1969, 10.23).7 Thus ˛W V! W is ´etale at P if and only if the map O P is an isomorphism. Hence 5.53 shows that the choice of a local system of parameters f1; : : : ; fd at a nonsingular point P determines an isomorphism O O nonsingular point P ; then there is a canonical isomorph
this becomes the statement: n! A n is an isomorphism. If we.a/ is never zero (for a 2 kn), then'has an inverse. never zero, implies that det @Pi @Xj @Pi @Xj is a nonzero constant (by the Null- stellensatz 2.11 applied to the ideal generated by det ). This conjecture, which is known as the Jacobian conjecture, has not been settled even for k D C and n D 2, despite the existence of several published proofs and innumerable announced proofs. It has caused many mathematicians a good deal of grief. It is probably harder than it is interesting. See the Wikipedia: JACOBIAN CONJECTURE. if det @Pi @Xj.a/ @Pi @Xj The condition, det o. ´Etale neighbourhoods Recall that a regular map ˛W W! V is said to be ´etale at a nonsingular point P of W if the map.d˛/P W TP.W /! T˛.P /.V / is an isomorphism. O Let P be a nonsingular point on a variety V of dimension d. A local system of parameters at P is a family ff1; : : : ; fd g of germs of regular functions at P generating the maximal ideal nP P. Equivalent conditions: the images of f1; : : : ; fd in nP =n2 P generate it as a k-vector space (see 1.4); or.df1/P ; : : : ;.dfd /P is a basis for the dual space to TP.V /. PROPOSITION 5.53. Let ff1; : : : ; fd g be a local system of parameters at a nonsingular point P of V. Then there is a nonsingular open neighbourhood U of P such that f1; f2; : : : ; fd are represented by pairs. Qf1; U /; : : : ;. Qfd ; U / and the map. Qf1; : : : ; Qfd /W U! A PROOF. Obviously, the fi are represented by regular functions Qfi defined on a single open neighbourhood U 0 of P, which, because of 4.37, we can choose to be nonsingular. The map d is ´etale at P, because the dual
map to.d˛/a is.dXi /o 7!.d Qfi /a. ˛ D. Qf1; : : : ; Qfd /W U 0! A The next lemma then shows that ˛ is ´etale on an open neighbourhood U of P. d is ´etale. LEMMA 5.54. Let W and V be nonsingular varieties. If ˛W W! V is ´etale at P, then it is ´etale at all points in an open neighbourhood of P. PROOF. The hypotheses imply that W and V have the same dimension d, and that their tangent spaces all have dimension d. We may assume W and V to be affine, say W A m and V A n, and that ˛ is given by polynomials P1.X1; : : : ; Xm/; : : : ; Pn.X1; : : : ; Xm/. Then m/! T˛.a/.A.d˛/aW Ta.A, and ˛ is not ´etale at a if and only if the kernel of this map contains a nonzero vector in the subspace Ta.V / of n/. Let f1; : : : ; fr generate I.W /. Then ˛ is not ´etale at a if and only if the matrix Ta.A n/ is a linear map with matrix @Pi @Xj.afi @Xj @Pi @Xj.a/.a/ has rank less than m. This is a polynomial condition on a, and so it fails on a closed subset of W, which doesn’t contain P. Let V be a nonsingular variety, and let P 2 V. An ´etale neighbourhood of a point P of V is a pair.Q; W U! V / with an ´etale map from a nonsingular variety U to V and Q a point of U such that.Q/ D P. o. ´Etale neighbourhoods 121 COROLLARY 5.55. Let V be a nonsingular variety of dimension d, and let P 2 V. There is an open Zariski neighbourhood U of P and a map W U! A d realizing.P; U / as an ´etale neighbourhood of.0
; : : : ; 0/ 2 A d. PROOF. This is a restatement of the Proposition. ASIDE 5.56. Note the similarity to the definition of a differentiable manifold: every point P on a nonsingular variety of dimension d has an open neighbourhood that is also a “neighbourhood” of d. There is a “topology” on algebraic varieties for which the “open neighbourhoods” the origin in A of a point are the ´etale neighbourhoods. Relative to this “topology”, any two nonsingular varieties are locally isomorphic (this is not true for the Zariski topology). The “topology” is called the ´etale topology — see my notes Lectures on ´Etale Cohomology. The inverse function theorem (for the ´etale topology) THEOREM 5.57 (INVERSE FUNCTION THEOREM). If a regular map of nonsingular varieties 'W V! W is ´etale at P 2 V, then there exists a commutative diagram V'W open UP '0 ´etale U'.P / with UP an open neighbourhood of P, Uf.P / an ´etale neighbourhood '.P /, and '0 an isomorphism. PROOF. According to 5.54, there exists an open neighbourhood U of P such that the restriction 'jU of'to U is ´etale. To get the above diagram, we can take UP D U, U'.P / to be the ´etale neighbourhood 'jU W U! W of '.P /, and '0 to be the identity map. The rank theorem For vector spaces, the rank theorem says the following: let ˛W V! W be a linear map of k-vector spaces of rank r; then there exist bases for V and W relative to which ˛ has matrix Ir 0. In other words, there is a commutative diagram 0 0 ˛.x1;:::;xm/7!.x1;:::;xr ;0;:::/ V km W kn: A similar result holds locally for differentiable manifolds. In algebraic geometry, there is the following weaker analogue. THEOREM 5.58 (RANK THEOREM). Let 'W V! W be a regular map
of nonsingular varieties of dimensions m and n respectively, and let P 2 V. If rank.TP.'// D n, then there exists a commutative diagram 'jUP.x1;:::;xm/7!.x1;:::;xn/ UP ´etale m A U'.P / ´etale n A 122 5. ALGEBRAIC VARIETIES in which UP and U'.P / are open neighbourhoods of P and '.P / respectively and the vertical maps are ´etale. PROOF. Choose a local system of parameters g1; : : : ; gn at '.P /, and let f1 D g1 ı'; : : : ; fn D gn ı '. Then df1; : : : ; dfn are linearly independent forms on TP.V /, and there exist fnC1; : : : ; fm such df1; : : : ; dfm is a basis for TP.V /_. Then f1; : : : ; fm is a local system of parameters at P. According to 5.54, there exist open neighbourhoods UP of P and U'.P / of '.P / such that the maps.f1; : : : ; fm/W UP! A.g1; : : : ; gn/W U'.P /! A m n are ´etale. They give the vertical maps in the above diagram. ASIDE 5.59. Tangent vectors at a point P on a smooth manifold V can be defined to be certain equivalence classes of curves through P (Wikipedia: TANGENT SPACE). For V D A n, there is a similar description with a curve taken to be a regular map from an open neighbourhood U of 0 1 to V. In the general case there is a map from an open neighbourhood of the point P in X in A onto affine space sending P to 0 and inducing an isomorphism from tangent space at P to that at 0 (5.53). Unfortunately, the maps from U A n need not lift to X, and so it is necessary to allow maps from smooth curves into X (pull-backs of the covering X! A n). There is a description of the tangent vectors at a point P on a smooth algebraic variety V as certain equivalence classes of regular maps from an ´etale neighbourhood U of 0 in A n by the maps from
U into A 1 to A 1 to V. p. Smooth maps DEFINITION 5.60. A regular map 'W V! W of nonsingular varieties is smooth at a point P of V if.d'/P W TP.V /! T'.P /.W / is surjective;'is smooth if it is smooth at all points of V. THEOREM 5.61. A map 'W V! W is smooth at P 2 V if and only if there exist open neighbourhoods UP and U'.P / of P and '.P / respectively such that 'jUP factors into UP ´etale!A dim V dim W U'.P / q! U'.P /: PROOF. Certainly, if 'jUP factors in this way, it is smooth. Conversely, if'is smooth at P, then we get a diagram as in the rank theorem. From it we get maps UP! A m An U'.P /! U'.P /: The first is ´etale, and the second is the projection of A mn U'.P / onto U'.P /. COROLLARY 5.62. Let V and W be nonsingular varieties. If 'W V! W is smooth at P, then it is smooth on an open neighbourhood of V. PROOF. In fact, it is smooth on the neighbourhood UP in the theorem. q. Algebraic varieties as functors 123 Separable maps A transcendence basis S of an extension E F of fields is separating if the algebraic extension E F.S/ is separable. A finitely generated extension E F of fields is separable if it admits a separating transcendence basis. DEFINITION 5.63. A dominant map 'W W! V of irreducible algebraic varieties is separable if k.W / is a separable extension of k.V /. THEOREM 5.64. Let 'W W! V be a map of irreducible varieties. (a) If there exists a nonsingular point P of W such that 'P is nonsingular and.d'/P is surjective, then'is dominant and separable. (b) Conversely if'is dominant and separable, then the set of P 2 W satisfying (a) is open and dense. PROOF. Replace W and V with their open subsets of nonsingular points. Then apply the
rank theorem. q. Algebraic varieties as functors Let R be an affine k-algebra, and let V be an algebraic variety. We define a point of V with coordinates in R (or an R-point of V ) to be a regular map Spm.R/! V. For example, if V D V.a/ A n, then V.R/ D f.a1; : : : ; an/ 2 Rn j f.a1; : : : ; an/ D 0 all f 2 ag; which is what you should expect. In particular V.k/ D V (as a set), i.e., V (as a set) can be identified with the set of points of V with coordinates in k. Note that.V W /.R/ D V.R/ W.R/ (property of a product). CAUTION 5.65. If V is the union of two subvarieties, V D V1 [ V2, then it need not be true that V.R/ D V1.R/ [ V2.R/. For example, for any polynomial f.X1; : : : ; Xn/, n D Df [ V.f /; A where Df'Spm.kŒX1; : : : ; Xn; T =.1 Tf // and V.f / is the zero set of f, but Rn ¤ fa 2 Rn j f.a/ 2 Rg [ fa 2 Rn j f.a/ D 0g in general. In fact, it need not be true even when V1 and V2 are open in V. Indeed, this would say that every regular map U! V with U affine must factor through V1 or V2, which 2 n f.0; 0/g is the union of the open subsets is nonsense. For example, the variety V D A V1W X ¤ 0 and V2W Y ¤ 0, but the affine subvariety U W X C Y D 1 of V is not contained in V1 or V2. 124 5. ALGEBRAIC VARIETIES THEOREM 5.66. A regular map 'W V! W of algebraic varieties defines a family of maps of sets, '.R/W V.R/! W.R/, one for each affine k-
algebra R, such that for every homomorphism ˛W R! S of affine k-algebras, rhe diagram '.R/ V.R/ W.R/ V.˛/ V.ˇ / '.S/ V.S/ W.S/ (*) commutes. Every family of maps with this property arises from a unique morphism of algebraic varieties. Let Vark (resp. Affk) denote the category of algebraic varieties over k (resp. affine V denote the functor sending an affine algebraic varieties over k). For a variety V, let haff variety T D Spm.R/ to V.R/ D Hom.T; V /. We can restate Theorem 5.66 as follows. THEOREM 5.67. The functor V haff V W Vark! Fun.Affk; Sets/ if fully faithful. PROOF. For an algebraic variety V over k, let hV denote the functor T Hom.T; V /W Vark! Set: According to the Yoneda lemma (q.v. Wikipedia) the functor V hV W Vark! Fun.Vark; Sets/ is fully faithful. Let'be a morphism of functors haff V 0, and let T be an algebraic V variety. Let.Ui /i 2I be a finite affine covering of T. Each intersection Ui \ Uj is affine (5.29), and so'gives rise to a commutative diagram! haff 0 0 hV.T / hV 0.T / Y i hV.Ui / '.Ui / Y i hV 0.Ui / Y i;j hV.Ui \ Uj // '.Ui \Uj / Y i;j hV 0.Ui \ Uj // in which the pairs of maps are defined by the inclusions Ui \ Uj,! Ui ; Uj. As the rows are exact (5.15, last sentence), this shows that 'V extends uniquely to a functor hV! hV 0, which (by the Yoneda lemma) arises from a unique regular map V! V 0. COROLLARY 5.68. To give an affine group variety is the same as giving a functor GW Affk! Grp such that for some
n and some finite set S of polynomials in kŒX1; X2; : : : ; Xn, G is isomorphic to the functor sending R to the set of zeros of S in Rn. PROOF. Certainly an affine group variety defines such a functor. Conversely, the conditions imply that G D hV for an affine algebraic variety V (unique up to a unique isomorphism). The multiplication maps G.R/G.R/! G.R/ give a morphism of functors hV hV! hV. As hV hV'hV V (by definition of V V ), we see that they arise from a regular map V V! V. Similarly, the inverse map and the identity-element map are regular. q. Algebraic varieties as functors 125 It is not unusual for a variety to be most naturally defined in terms of its points functor. For example: SLnW R fM 2 Mn.R/ j det.M / D 1g GLnW R fM 2 Mn.R/ j det.M / 2 Rg GaW R.R; C/: We now describe the essential image of h 7! hV W Vark! Fun.Affk; Sets/. The fibred product of two maps ˛1W F1! F3, ˛2W F2! F3 of sets is the set F1 F3 F2 D f.x1; x2/ j ˛1.x1/ D ˛2.x2/g: When F1; F2; F3 are functors and ˛1; ˛2; ˛3 are morphisms of functors, there is a functor F D F1 F3 F2 such that.F1 F3 F2/.R/ D F1.R/ F3.R/ F2.R/ for all affine k-algebras R. To simplify the statement of the next proposition, we write U for hU when U is an affine variety. PROPOSITION 5.69. A functor F W Affk! Sets is in the essential image of Vark if and only if there exists an affine variety U and a morphism U! F such that (a) the functor R defD U F U is a closed aff
ine subvariety of U U and the maps R U defined by the projections are open immersions; (b) the set R.k/ is an equivalence relation on U.k/, and the map U.k/! F.k/ realizes F.k/ as the quotient of U.k/ by R.k/. PROOF. Let F D hV for V an algebraic variety. Choose a finite open affine covering V D S Ui of V, and let U D F Ui. It is again an affine variety (Exercise 5-2). The functor R is hU 0, where U 0 is the disjoint union of the varieties Ui \ Uj. These are affine (5.29), and so U 0 is affine. As U 0 is the inverse image of V in U U, it is closed (5.26). This proves (a), and (b) is obvious. The converse is omitted for the present. ASIDE 5.70. A variety V defines a functor R V.R/ from the category of all k-algebras to Sets. Again, we call the elements of V.R/ the points of V with coordinates in R. For example, if V is affine, V.R/ D Homk-algebra.kŒV ; R/: More explicitly, if V kn and I.V / D.f1; : : : ; fm/, then V.R/ is the set of solutions in Rn of the system equations fi.X1; : : : ; Xn/ D 0; i D 1; : : : ; m: Note that, when we allow R to have nilpotent elements, it is important to choose the fi to generate I.V / (i.e., a radical ideal) and not just an ideal a such that V.a/ D V.8 For a general variety V, we write V as a finite union of open affines V D S i Vi, and we define V.R/ to be the set of families.˛i /i 2I 2 Q i 2I Vi.R/ such that ˛i agrees with ˛j on Vi \ Vj for all i; j 2 I. This is independent of the choice of the covering, and agrees with the previous definition when V is affine. 8Let a
be an ideal in kŒX1; : : :. If A has no nonzero nilpotent elements, then every k-algebra homomorphism kŒX1; : : :! A that is zero on a is also zero on rad.a/, and so Homk.kŒX1; : : :=a; A/'Homk.kŒX1; : : :=rad.a/; A/: This is not true if A has nonzero nilpotents. 126 5. ALGEBRAIC VARIETIES The functor defined by A.E/ (see p. 72) is R R ˝k E. A criterion for a functor to arise from an algebraic prevariety 5.71. By a functor we mean a functor from the category of affine k-algebras to sets. A subfunctor U of a functor X is open if, for all maps 'W hA! X, the subfunctor '1.U / of hA is defined by an open subvariety of Spm.A/. A family.Ui /i 2I of open subfunctors of X is an open covering of X if each Ui is open in X and X.K/ D S Ui.K/ for every field K. A functor X is local if, for all k-algebras R and all finite families.fi /i of elements of A generating A as an ideal, the sequence of sets X.R/! Y i X.Rfi / Y X.Rfi fj / i;j is exact. Let A.U / D Hom.U; A 1 denote the functor sending a k-algebra R to its underlying set. For a functor U,.U / is an affine let k-algebra and the canonical map U! hO.U / is an isomorphism. A local functor admitting a finite covering by open affines is representable by an algebraic variety over k. 1/ — it is a k-algebra.9 A functor U is affine if O O In the functorial approach to algebraic geometry, an algebraic prevariety over k is defined to be a functor satisfying this criterion. See, for example, I, 1, 3.11, p. 13, of Demazure and
Gabriel, Groupes alg´ebriques: g´eom´etrie alg´ebrique, g´en´eralit´es, groupes commutatifs. 1970. r. Rational and unirational varieties DEFINITION 5.72. Let V be an algebraic variety over k. (a) V is unirational if there exists a dominant rational map P (b) V is rational if there exists a birational map P n Ü V: n Ü V. In more down-to-earth terms, V is rational if k.V / is a pure transcendental extension of k, and it is unirational if k.V / is contained in such an extension of k. In 1876 (over C), L¨uroth proved that every unirational curve is rational. For a proof over any field, see FT 9.19. The L¨uroth problem asks whether every unirational variety is rational. Already for surfaces, this is a difficult problem. In characteristic zero, Castelnuovo and Severi proved that all unirational surfaces are rational, but in characteristic p ¤ 0, Zariski showed that some surfaces of the form Zp D f.X; Y /; while obviously unirational, are not rational. Surfaces of this form are now called Zariski surfaces. Fano attempted to find counter-examples to the L¨uroth problem in dimension 3 among the so-called Fano varieties, but none of his attempted proofs satisfies modern standards. In 1971-72, three examples of nonrational unirational three-folds were found. For a description of them, and more discussion of the L¨uroth problem in characteristic zero, see: Arnaud Beauville, The L¨uroth problem, arXiv:1507.02476. 9Actually, one needs to be more careful to ensure that O.U / is a set; for example, restrict U and A category of k-algebras of the form kŒX0; X1; : : :=a for a fixed family of symbols.Xi / indexed by N. 1 to the r. Rational and unirational varieties 127 A little history In his first proof of the Riemann hypothesis for curves over finite fields, Weil made use of the Jacobian variety of the curve, but initially he was not able to construct this as a projective variety
. This led him to introduce “abstract” algebraic varieties, neither affine nor projective (in 1946). Weil first made use of the Zariski topology when he introduced fibre spaces into algebraic geometry (in 1949). For more on this, see my article: The Riemann hypothesis over finite fields: from Weil to the present day. Exercises 1 is 5-1. Show that the only regular functions on P not affine. When k D C, P 1 is the Riemann sphere (as a set), and one knows from complex analysis that the only holomorphic functions on the Riemann sphere are constant. Since regular functions are holomorphic, this proves the statement in this case. The general case is easier.] 1 are the constant functions. [Thus P 5-2. Let V be the disjoint union of algebraic varieties V1; : : : ; Vn. This set has an obvious topology and ringed space structure for which it is an algebraic variety. Show that V is affine if and only if each Vi is affine. 5-3. Show that an algebraic variety G equipped with a group structure is an algebraic group if the map.x; y/ 7! x1yW G G! G is regular. 5-4. Let G be an algebraic group. Show: (a) The neutral element e of G is contained in a unique irreducible component Gı of G, which is also the unique connected component of G containing e. (b) The subvariety Gı is a normal subgroup of G of finite index, and every algebraic subgroup of G of finite index contains Gı. 5-5. Show that every subgroup variety of a group variety is closed. 5-6. Show that a prevariety V is separated if and only if it satisfies the following condition: a regular map U X fP g! V with U a curve and P a nonsingular point on U extends in at most one way to a regular map U! V. 5-7. Prove the final statement in 5.71. CHAPTER 6 Projective Varieties Recall (5.3) that we defined P the relation n to be the set of equivalence classes in knC1 X foriging for.a0; : : : ; an/.b0; : : : ; bn/ ”.a0
; : : : ; an/ D c.b0; : : : ; bn/ for some c 2 k: Let.a0 W : : : W an/ denote the equivalence class of.a0; : : : ; an/, and let denote the map knC1 X f.0; : : : ; 0/g! P n: Let Ui be the set of.a0 W : : : W an/ 2 P n such that ai ¤ 0, and let ui be the bijection.a0W : : : W an/ 7! a0 ai ; : : : ; bai ai ; : : : ; an ai W Ui ui! A n ( ai ai omitted). n has a unique structure of an algebraic variety for which In this chapter, we show that P these maps become isomorphisms of affine algebraic varieties. A variety isomorphic to n is called a projective variety, and a variety isomorphic to a a closed subvariety of P n is called a quasiprojective variety. Every affine variety is locally closed subvariety of P quasiprojective, but not all algebraic varieties are quasiprojective. We study morphisms between quasiprojective varieties. Projective varieties are important for the same reason compact manifolds are important: results are often simpler when stated for projective varieties, and the “part at infinity” often plays a role, even when we would like to ignore it. For example, a famous theorem of Bezout (see 6.37 below) says that a curve of degree m in the projective plane intersects a curve of degree n in exactly mn points (counting multiplicities). For affine curves, one has only an inequality. a. Algebraic subsets of Pn A polynomial F.X0; : : : ; Xn/ is said to be homogeneous of degree d if it is a sum of terms ai0;:::;inX i0 0 n with i0 C C in D d ; equivalently, X in F.tX0; : : : ; tXn/ D t d F.X0; : : : ; Xn/ for all t 2 k. The polynomials homogeneous of degree d form a subspace kŒX0;
: : : ; Xnd of kŒX0; : : : ; Xn, and kŒX0; : : : ; Xn D M d 0 kŒX0; : : : ; Xnd I 129 130 6. PROJECTIVE VARIETIES in other words, every polynomial F can be written uniquely as a sum F D P Fd with Fd homogeneous of degree d. Let P D.a0 W : : : W an/ 2 P n. Then P also equals.ca0 W : : : W can/ for any c 2 k, and so we can’t speak of the value of a polynomial F.X0; : : : ; Xn/ at P. However, if F is homogeneous, then F.ca0; : : : ; can/ D cd F.a0; : : : ; an/, and so it does make sense to say n (or projective algebraic set) is the set that F is zero or not zero at P. An algebraic set in P of common zeros in P n of some set of homogeneous polynomials. EXAMPLE 6.1. Consider the projective algebraic subset of P equation 2 defined by the homogeneous E W Y 2Z D X 3 C aXZ2 C bZ3. It consists of the points.x W y W 1/ on the affine curve E \ U2 (26) Y 2 D X 3 C aX C b (see 2.2) together with the point “at infinity”.0 W 1 W 0/. Note that E \ U1 is the affine curve Z D X 3 C aXZ2 C bZ3; and that.0W 1W 0/ corresponds to the point.0; 0/ on E \ U1: Z D X 3 C XZ2 C Z3 As.0; 0/ is nonsingular on E \ U1, we deduce from (4.5) that E is nonsingular unless X 3 C aX C b has a multiple root. A nonsingular curve of the form (26) is called an elliptic curve. An elliptic curve has a unique structure of a group variety for which the point at infinity n a. Algebraic subsets of P 131 is the zero: Q P P C Q When a; b 2 Q,
we can speak of the zeros of (26) with coordinates in Q. They also form a group E.Q/, which Mordell showed to be finitely generated. It is easy to compute the torsion subgroup of E.Q/, but there is at present no known algorithm for computing the rank of E.Q/. More precisely, there is an “algorithm” which works in practice, but which has not been proved to always terminate after a finite amount of time. There is a very beautiful theory surrounding elliptic curves over Q and other number fields, whose origins can be traced back almost 1,800 years to Diophantus. (See my book on Elliptic Curves for all of this.) An ideal a kŒX0; : : : ; Xn is said to be graded or homogeneous if it contains with any polynomial F all the homogeneous components of F, i.e., if F 2 a H) Fd 2 a, all d: It is straightforward to check that ˘ an ideal is graded if and only if it is generated by (a finite set of) homogeneous polynomials; ˘ the radical of a graded ideal is graded; ˘ an intersection, product, or sum of graded ideals is graded. For a graded ideal a, we let V.a/ denote the set of common zeros of the homogeneous polynomials in a. Clearly a b H) V.a/ V.b/. If F1; : : : ; Fr are homogeneous generators for a, then V.a/ is also the set of common zeros of the Fi. Clearly every polynomial in a is zero on every representative of a point in V.a/. We write V aff.a/ for the set of common zeros of a in knC1. It is a cone in knC1, i.e., together with any point P it contains the line through P and the origin, and V.a/ D V aff.a/ X f.0; : : : ; 0/g : The sets V.a/ in P n have similar properties to their namesakes in A n. 132 6. PROJECTIVE VARIETIES PROPOSITION 6.2. There are the following relations: nI V.a/ D ; ” rad.a/.X0; : : : ; Xn/I (a
) V.0/ D P (b) V.ab/ D V.a \ b/ D V.a/ [ V.b/I (c) V.P ai / D T V.ai /. PROOF. For the second statement in (a), note that V.a/ D ; ” V aff.a/ f.0; : : : ; 0/g ” rad.a/.X0; : : : ; Xn/ (strong Nullstellensatz 2.16). The remaining statements can be proved directly, as in (2.10), or by using the relation between V.a/ and V aff.a/. on P Proposition 6.2 shows that the projective algebraic sets are the closed sets for a topology n. This topology is called the Zariski topology on P If C is a cone in knC1, then I.C / is a graded ideal in kŒX0; : : : ; Xn: if F.ca0; : : : ; can/ D n. 0 for all c 2 k, then Fd.a0; : : : ; an/ cd D F.ca0; : : : ; can/ D 0; X d for infinitely many c, and so P Fd.a0; : : : ; an/X d is the zero polynomial. For a subset S of n, we define the affine cone over S in knC1 to be P C D 1.S / [ foriging and we set I.S/ D I.C /. Note that if S is nonempty and closed, then C is the closure of 1.S/ ¤ ;, and that I.S/ is spanned by the homogeneous polynomials in kŒX0; : : : ; Xn that are zero on S. PROPOSITION 6.3. The maps V and I define inverse bijections between the set of algebraic n and the set of proper graded radical ideals of kŒX0; : : : ; Xn. An algebraic set subsets of P V in P n is irreducible if and only if I.V / is prime; in particular, P n is irreducible. PROOF. Note that we have bijections falgebraic subsets of P ng S 7! C
fnonempty closed cones in knC1g V I fproper graded radical ideals in kŒX0; : : : ; Xng Here the top map sends S to the affine cone over S, and the maps V and I are in the sense of projective geometry and affine geometry respectively. The composite of any three of these maps is the identity map, which proves the first statement because the composite of the top map with I is I in the sense of projective geometry. Obviously, V is irreducible if and only if the closure of 1.V / is irreducible, which is true if and only if I.V / is a prime ideal. b. The Zariski topology on P n 133 Note that the graded ideals.X0; : : : ; Xn/ and kŒX0; : : : ; Xn are both radical, but V.X0; : : : ; Xn/ D ; D V.kŒX0; : : : ; Xn/ and so the correspondence between irreducible subsets of P quite one-to-one. n and radical graded ideals is not ASIDE 6.4. In English “homogeneous ideal” is more common than “graded ideal”, but we follow Bourbaki, Alg, II, 11. A graded ring is a pair.S;.Sd /d 2N/ consisting of a ring S and a family of additive subgroups Sd such that ( S D M Sd Sd Se Sd Ce, all d; e 2 N: d 2N An ideal a in S is graded if and only if a D M d 2N.a \ Sd /; this means that it is a graded submodule of.S;.Sd //. The quotient of a graded ring S by a graded ideal a is a graded ring S=a D L d Sd =.a \ Sd /. b. The Zariski topology on Pn For a graded polynomial F, let D.F / D fP 2 P n j F.P / ¤ 0g: Then, just as in the affine case, D.F / is open and the sets of this type form a base for the topology of P n. As in the opening paragraph of this chapter, we let Ui D D.Xi /. To
each polynomial f.X1; : : : ; Xn/, we attach the homogeneous polynomial of the same degree f.X0; : : : ; Xn/ D X deg.f / 0 X1 X0 f ; : : : ; Xn X0 ; and to each homogeneous polynomial F.X0; : : : ; Xn/, we attach the polynomial F.X1; : : : ; Xn/ D F.1; X1; : : : ; Xn/: PROPOSITION 6.5. Each subset Ui of P we endow it with the induced topology, the bijection n is open in the Zariski topology on P n, and when Ui $ A n,.a0 an/ $.a0; : : : ; ai1; ai C1; : : : ; an/ becomes a homeomorphism. PROOF. It suffices to prove this with i D 0. The set U0 D D.X0/, and so it is a basic open subset in P n. Clearly, for any homogeneous polynomial F 2 kŒX0; : : : ; Xn, D.F.X0; : : : ; Xn// \ U0 D D.F.1; X1; : : : ; Xn// D D.F/ and, for any polynomial f 2 kŒX1; : : : ; Xn, D.f / D D.f / \ U0: Thus, under the bijection U0 $ A tersections with Ui of the basic open subsets of P homeomorphism. n correspond to the inn, which proves that the bijection is a n, the basic open subsets of A 134 6. PROJECTIVE VARIETIES n is irreducible. We REMARK 6.6. It is possible to use this to give a different proof that P apply the criterion that a space is irreducible if and only if every nonempty open subset is dense (see p. 46). Note that each Ui is irreducible, and that Ui \ Uj is open and dense in each of Ui and Uj (as a subset of Ui, it is the set of points.a0 aj W : : : W an/ with aj ¤
0/. Let U be a nonempty open subset of P n; then U \ Ui is open in Ui. For some i, U \ Ui is nonempty, and so must meet Ui \ Uj. Therefore U meets every Uj, and so is dense in every Uj. It follows that its closure is all of P n. c. Closed subsets of An and Pn We identify A n with U0, and examine the closures in P n of closed subsets of A n. Note that n D A P n t H1; H1 D V.X0/: generated by ff j f 2 ag. For a closed subset V of A With each ideal a in kŒX1; : : : ; Xn, we associate the graded ideal a in kŒX0; : : : ; Xn n, set V D V.a/ with a D I.V /. With each graded ideal a in kŒX0; X1; : : : ; Xn], we associate the ideal a in kŒX1; : : : ; Xn n, we set V D V.a/ with generated by fF j F 2 ag. When V is a closed subset of P a D I.V /. n, PROPOSITION 6.7. (a) Let V be a closed subset of A and.V / D V. If V D S Vi is the decomposition of V into its irreducible components, then V D S V i is the decomposition of V into its irreducible components. n. Then V D V \ A n. Then V is the closure of V in P (b) Let V be a closed subset of P n, and if no irreducible component n, and.V/ D V. of V lies in H1 or contains H1, then V is a proper subset of A PROOF. Straightforward. Examples 6.8. For we have and. aX C b; V W Y 2Z D X 3 C aXZ2 C bZ3; 6.9. Let V D V.f1; : : : ; fm/; then the closure of V in P components of V.f m/ not contained in H1. For example, let 1 ; : : : ; f n is the union of the irreducible V D V.
X1; X 2 1 C X2/ D f.0; 0/gI then V.X0X1; X 2 1 (which is contained in H1).1 C X0X2/ consists of the two points.1W 0W 0/ (the closure of V ) and.0W 0W 1/ 6.10. For V D H1 D V.X0/, we have V D ; D V.1/ and.V/ D ; ¤ V. 1Of course, in this case a D.X1; X2/, a D.X1; X2/, and V D f.1W 0W 0/g, and so this example doesn’t contradict the proposition. d. The hyperplane at infinity 135 d. The hyperplane at infinity n as being A It is often convenient to think of P More precisely, we identify the set U0 with A n D U0 with a hyperplane added “at infinity”. n; the complement of U0 in P n is H1 D f.0 W a1 W : : : W an/ 2 P ng; which can be identified with P For example, P 1 D A n1. 2 D A P 2 [ H1 with H1 a projective line. Consider the line 1 t H1 (disjoint union), with H1 consisting of a single point, and in A 2. Its closure in P 2 is the line 1 C aX1 C bX2 D 0 X0 C aX1 C bX2 D 0: This line intersects the line H1 D V.X0/ at the point.0 W b W a/, which equals.0 W 1 W a=b/ when b ¤ 0. Note that a=b is the slope of the line 1 C aX1 C bX2 D 0, and so the point at which a line intersects H1 depends only on the slope of the line: parallel lines meet in one 2 with point at infinity. We can think of the projective plane P one point added at infinity for each “direction” in A 2 as being the affine plane A 2. Similarly, we can think of P n as being A direction in A one point at infinity for each equivalence class of lines. n — being parallel is an equivalence relation on the lines in A n with one point added at infinity for each n
, and there is We can replace U0 with Un in the above discussion, and write P n D Un t H1 with H1 D f.a0W : : : W an1W 0/g, as in Example 6.1. Note that in this example the point at infinity on the elliptic curve Y 2 D X 3 C aX C b is the intersection of the closure of any vertical line with H1. e. Pn is an algebraic variety O For each i, write i for the sheaf on Ui P n defined by the homeomorphism ui W Ui! A j jUij. When endowed with this sheaf; LEMMA 6.11. Let Uij D Ui \ Uj ; then i jUij D O O Uij is an affine algebraic variety; moreover,.Uij ; i / is generated as a k-algebra by the i /, g 2.Uj ; functions.f jUij /.gjUij / with f 2.Ui ; j /. O n. O O PROOF. It suffices to prove this for.i; j / D.0; 1/. All rings occurring in the proof will be identified with subrings of the field k.X0; X1; : : : ; Xn/. Recall that U0 D f.a0 W a1 W : : : W an/ j a0 ¤ 0g;.a0 W a1 W : : : W an/ $. a1 a0 ; a2 a0 ; : : : ; an a0 / 2 A n: ; X2 X0 ; : : : ; Xn X0 Let kŒ X1 X0 — it is the polynomial ring in the n symbols X1 X0 kŒ X1 X0 be the subring of k.X0; X1; : : : ; Xn/ generated by the quotients Xi X0 ; : : : ; Xn / 2 X0. An element f. X1 X0 defines a map ; : : : ; Xn X0 ; : : : ; Xn X0.a0 W a1 W : : : W an/ 7! f. a1 a0 ; : : : ; an a0 /W U0! k; 136 6. PROJECTIVE VARIETIES and in this way k
Œ X1 X0 U0; and U0 with Spm ; X2 X0 kŒ X1 X0 ; : : : ; Xn X0 ; : : : ; Xn X0. becomes identified with the ring of regular functions on Next consider the open subset of U0; U01 D f.a0 W : : : W an/ j a0 ¤ 0, a1 ¤ 0g: /, and is therefore an affine subvariety of.U0; It is D. X1 X0 corresponds to the inclusion of rings kŒ X1 X0 f. X1 / of kŒ X1 X0 X0 on U01. ; : : : ; Xn X0 ; : : : ; Xn X0 ; X0 X1 ; X0 X1 O 0/. The inclusion U01,! U0. An element ; : : : ; Xn X0,! kŒ X1 X0 defines the function.a0 W : : : W an/ 7! f. a1 a0 ; : : : ; Xn X0 ; X0 X1 Similarly, U1 D f.a0 W a1 W : : : W an/ j a1 ¤ 0g;.a0 W a1 W : : : W an/ $. a0 a1 ; X2 and we identify U1 with Spm X0 defines the map.a0 W : : : W an/ 7! f. a0 a1 ; : : : ; Xn X1 ; : : : ; an a1 kŒ X0 X1 /W U1! k.. A polynomial f. X0 X1 / in kŒ X0 X1 ; : : : ; Xn X1 / 2 A ; : : : ; an a1 ; : : : ; Xn X1 ; : : : ; an a0 ; a0 a1 / n; ; X1 X0 ; X1 X0 ; X0 X1 ; : : : ; Xn X1 ; : : : ; Xn X1 ; : : : ; Xn X0 / of kŒ X0 X1 ; a1 a0 ; : : : ; Xn X1 The two subrings kŒ X1 X0 When regarded as an open subset of U1; U01 D D. X
0 X1 /, and is therefore an affine 1/, and the inclusion U01,! U1 corresponds to the inclusion of rings ; X1 X0 subvariety of.U1; O kŒ X0,! kŒ X0. An element f. X0 ; : : : ; Xn ; : : : ; Xn X1 X1 X1 X1 X1 defines the function.a0 W : : : W an/ 7! f. a0 ; : : : ; an / on U01. a1 a1 ; X1 and kŒ X0 of k.X0; X1; : : : ; Xn/ are X0 X1 equal, and an element of this ring defines the same function on U01 regardless of which of the two rings it is considered an element. Therefore, whether we regard U01 as a subvariety of U0 or of U1 it inherits the same structure as an affine algebraic variety (3.15). This proves the first two assertions, and the third is obvious: kŒ X1 is generated by X0 and kŒ X0 its subrings kŒ X1 X1 X0 ; X2 X1 PROPOSITION 6.12. There is a unique structure of an algebraic variety on P each Ui is an open affine subvariety of P varieties. Moreover, P n for which n and each map ui is an isomorphism of algebraic n is separated. ; : : : ; Xn X0 ; : : : ; Xn X1 ; : : : ; Xn X0 ; X0 X1. PROOF. Endow each Ui with the structure of an affine algebraic variety for which ui is an n D S Ui, and the lemma shows that this covering satisfies the patching isomorphism. Then P n has a unique structure of a ringed space for which Ui,! P n is a condition 5.15, and so P OPnjUi D n and homeomorphism onto an open subset of P Ui. Moreover, because each n into an algebraic prevariety. Finally, the Ui is an algebraic variety, this structure makes P lemma shows that P n satisfies the condition 5.29(c) to be separated. O EXAMPLE 6.13. Let C be the plane projective
curve C W Y 2Z D X 3 and assume that char.k/ ¤ 2. For each a 2 k, there is an automorphism 'a! C:.x W y W z/ 7!.ax W y W a3z/W C Patch two copies of C A 1 together along C.A 1 f0g/ by identifying.P; a/ with.'a.P /; a1/, P 2 C, a 2 A 1 X f0g. One obtains in this way a singular surface that is not quasiprojective (see Hartshorne 1977, Exercise 7.13). It is even complete — see below — and so if it were quasiprojective, it would be projective. In Shafarevich 1994, VI 2.3, there is an example of a nonsingular complete variety of dimension 3 that is not projective. It is known that every irreducible separated curve is quasiprojective, and every nonsingular complete surface is projective, and so these examples are minimal. f. The homogeneous coordinate ring of a projective variety 137 f. The homogeneous coordinate ring of a projective variety Recall (p. 114) that attached to each irreducible variety V, there is a field k.V / with the property that k.V / is the field of fractions of kŒU for any open affine U V. We now describe this field in the case that V D P. We regard this as a subring of k.X0; : : : ; Xn/, and wish to identify the field of fractions of kŒU0 as a subfield of k.X0; : : : ; Xn/. Every nonzero F 2 kŒU0 can be written n. Recall that kŒU0 D kŒ X1 X0 ; : : : ; Xn X0 F. X1 X0 ; : : : ; Xn X0 / D F.X0; : : : ; Xn/ X deg.F / 0 with F homogeneous of degree deg.F /, and it follows that the field of fractions of kŒU0 is k.U0/ D G.X0; : : : ; Xn/ H.X0; : : : ; Xn/ ˇ ˇ ˇ �
� G, H homogeneous of the same degree [ f0g: Write k.X0; : : : ; Xn/0 for this field (the subscript 0 is short for “subfield of elements of n/ D k.X0; : : : ; Xn/0. Note that for F D G degree 0”), so that k.P H in k.X0; : : : ; Xn/0;.a0 W : : : W an/ 7! G.a0; : : : ; an/ H.a0; : : : ; an/ W D.H /! k, is a well-defined function, which is obviously regular (look at its restriction to Ui /. We now extend this discussion to any irreducible projective variety V. Such a V can be written V D V.p/ with p a graded radical ideal in kŒX0; : : : ; Xn, and we define the homogeneous coordinate ring of V (with its given embedding) to be khomŒV D kŒX0; : : : ; Xn=p. Note that khomŒV is the ring of regular functions on the affine cone over V ; therefore its dimension is dim.V / C 1: It depends, not only on V, but on the embedding of V into P n, i.e., it is not intrinsic to V. For example,.a0 W a1/ 7!.a2 0 W a0a1 W a2 2 1! P 1/W P 1/W X0X2 D X 2 1 D kŒX0; X1, which is the affine coordinate ring of the smooth variety A 1 onto its image.P is an isomorphism from P khomŒP 1/ D kŒX0; X1; X2=.X0X2 X 2 khomŒ.P gular variety X0X2 X 2 1. 1 (see 6.23 below), but 2, whereas 1 /, which is the affine coordinate ring of the sin- We say that a nonzero f 2 khomŒV is homogeneous of degree d if it can be represented by a homogeneous polynomial F of degree d in kŒX0; : : : ; Xn, and we say that 0
is homogeneous of degree 0. LEMMA 6.14. Each element of khomŒV can be written uniquely in the form with fi homogeneous of degree i. f D f0 C C fd PROOF. Let F represent f ; then F can be written F D F0 C C Fd with Fi homogeneous of degree i ; when read modulo p, this gives a decomposition of f of the required type. Suppose f also has a decomposition f D P gi, with gi represented by the homogeneous polynomial Gi of degree i. Then F G 2 p, and the homogeneity of p implies that Fi Gi D.F G/i 2 p. Therefore fi D gi. 138 6. PROJECTIVE VARIETIES It therefore makes sense to speak of homogeneous elements of kŒV. For such an element h, we define D.h/ D fP 2 V j h.P / ¤ 0g. Since khomŒV is an integral domain, we can form its field of fractions khom.V /. Define khom.V /0 D n g h 2 khom.V / ˇ o ˇ ˇ g and h homogeneous of the same degree [ f0g: PROPOSITION 6.15. The field of rational functions on V is k.V / defD khom.V /0. PROOF. Consider V0 of khomŒV, and then the field of fractions of kŒV0 becomes identified with khom.V /0. n, we can identify kŒV0 with a subring defD U0 \ V. As in the case of P g. Regular functions on a projective variety Let V be an irreducible projective variety, and let f 2 k.V /. By definition, we can write f D g h with g and h homogeneous of the same degree in khomŒV and h ¤ 0. For any P D.a0 W : : : W an/ with h.P / ¤ 0, f.P / defD g.a0; : : : ; an/ h.a0; : : : ; an/ is well-defined: if.a0; : : : ; an/ is replaced by.ca0; : : : ; can/, then both the numerator
and denominator are multiplied by cdeg.g/ D cdeg.h/. We can write f in the form g h in many different ways,2 but if then f D g h D g0 h0 (in k.V /0), gh0 D g0h (in khomŒV ) and so g.a0; : : : ; an/ h0.a0; : : : ; an/ D g0.a0; : : : ; an/ h.a0; : : : ; an/: Thus, if h0.P / ¤ 0, the two representations give the same value for f.P /. PROPOSITION 6.16. For each f 2 k.V / defD khom.V /0, there is an open subset U of V, where f.P / is defined, and P 7! f.P / is a regular function on U ; every regular function on an open subset of V arises from a unique element of k.V /. PROOF. From the above discussion, we see that f defines a regular function on U D S D.h/, where h runs over the denominators of expressions f D g h with g and h homogeneous of the same degree in khomŒV. Conversely, let f be a regular function on an open subset U of V, and let P 2 U. Then P lies in the open affine subvariety V \ Ui for some i, and so f coincides with the function defined by some fP 2 k.V \ Ui / D k.V / on an open neighbourhood of P. If f coincides with the function defined by fQ 2 k.V / in a neighbourhood of a second point Q of U, then fP and fQ define the same function on some open affine U 0, and so fP D fQ as elements of kŒU 0 k.V /. This shows that f is the function defined by fP on the whole of U. 2Unless khomŒV is a unique factorization domain, there will be no preferred representation f D g h. h. Maps from projective varieties 139 REMARK 6.17. (a) The elements of k.V / D khom.V /0 should be regarded as the algebraic analogues of meromorphic functions on a complex manifold; the regular functions on an open subset U of V
let U 0 be the union of all the lines through the origin that meet U, that is, U 0 D 1.U /. Then U 0 is again open in knC1 X foriging, because U 0 D S cU, c 2 k, and x 7! cx is an automorphism of knC1 X foriging. The complement Z of U 0 in knC1 X foriging is a closed cone, and the proof of (6.3) shows n; but.U / is the complement of.Z/. Thus sends open sets that its image is closed in P to open sets. The rest of the proof is straightforward. Thus, the regular maps P n (as maps of sets). through P n! V are just the regular maps A nC1 X foriging! V factoring 140 6. PROJECTIVE VARIETIES REMARK 6.19. Consider polynomials F0.X0; : : : ; Xm/; : : : ; Fn.X0; : : : ; Xm/ of the same degree. The map.a0 W : : : W am/ 7!.F0.a0; : : : ; am/ W : : : W Fn.a0; : : : ; am// obviously defines a regular map to P that is, on the set S D.Fi / D P will also be regular. It may be possible to extend the map to a larger set by representing it by different polynomials. Conversely, every such map arises in this way, at least locally. More precisely, there is the following result. n X V.F1; : : : ; Fn/. Its restriction to any subvariety V of P m, where not all Fi vanish, m n on the open subset of P PROPOSITION 6.20. Let V D V.a/ P regular if and only if, for every P 2 V, there exist polynomials m and W D V.b/ P n. A map 'W V! W is F0.X0; : : : ; Xm/; : : : ; Fn.X0; : : : ; Xm/; homogeneous of the same degree, such that '..b0 W : : : W bn// D.F0.b0; : : : ; bm/ W : : : W Fn.b0
; : : : ; bm// for all points.b0 W : : : W bm/ in some neighbourhood of P in V.a/. PROOF. Straightforward. EXAMPLE 6.21. We prove that the circle X 2 CY 2 D Z2 is isomorphic to P 1. This equation can be rewritten.X C iY /.X iY / D Z2, and so, after a change of variables, the equation of the circle becomes C W XZ D Y 2. Define 'W P 1! C,.a W b/ 7!.a2 W ab W b2/: For the inverse, define W C! P 1 by.a W b W c/ 7!.a W b/.a W b W c/ 7!.b W c/ if a ¤ 0 if b ¤ 0 : Note that, a ¤ 0 ¤ b; ac D b2 H) c b D b a and so the two maps agree on the set where they are both defined. Clearly, both'and are regular, and one checks directly that they are inverse. i. Some classical maps of projective varieties We list some of the classic maps. HYPERPLANE SECTIONS AND COMPLEMENTS 6.22. Let L D P ci Xi be a nonzero linear form in n C 1 variables. Then the map.a0 W : : : W an/ 7! a0 L.a/ ; : : : ; an L.a/ i. Some classical maps of projective varieties 141 is a bijection of D.L/ P inverse n onto the hyperplane L.X0; X1; : : : ; Xn/ D 1 of A nC1, with.a0; : : : ; an/ 7!.a0 W : : : W an/: Both maps are regular — for example, the components of the first map are the regular functions Xj. As V.L 1/ is affine, so also is D.L/, and its ring of regular functions is kŒ X0 P ci Xi symbol, and P cj is to be thought of as a single D 1; thus it is a polynomial ring in n symbols; any one symbol : In this ring, each quotient Xj P ci Xi ; : : : ; Xn P ci Xi P ci Xi Xj P c
i Xi Xj P ci Xi for which cj ¤ 0 can be omitted. For a fixed P D.a0W : : : W an/ 2 P n, the set of c D.c0W : : : W cn/ such that Lc.P / defD X ci ai ¤ 0 is a nonempty open subset of P n (n > 0). Therefore, for any finite set S of points of P n, fc 2 P n j S D.Lc/g n is irreducible). In particular, S is contained in is a nonempty open subset of P n. Moreover, if S V, where V is a closed subvariety of an open affine subset D.Lc/ of P n, then S V \ D.Lc/: any finite set of points of a projective variety is contained in an P open affine subvariety. n (because P THE VERONESE MAP; HYPERSURFACE SECTIONS 6.23. Let I D f.i0; : : : ; in/ 2 N nC1 j X ij D mg: elements3. Note that I indexes the monomials of degree m in n C 1 variables. It has mCn m Write n;m D mCn n;m whose coordinates are W : : :/. The Veronese mapping indexed by I ; thus a point of P is defined to be 1, and consider the projective space P n;m can be written.: : : W bi0:::in m vW P n! P n;m,.a0 W : : : W an/ 7!.: : : W bi0:::in W : : :/; bi0:::in D ai0 0 : : : ain n : In other words, the Veronese mapping sends an n C 1-tuple.a0W : : : W an/ to the set of monomials in the ai of degree m. For example, when n D 1 and m D 2, the Veronese map is 3This can be proved by induction on m C n. If m D 0 D n, then 0 0 homogeneous polynomial of degree m can be written uniquely as 1! P P 2,.a0 W a1/ 7!.a2 0 W a0a1 W a2 1
/: D 1, which is correct. A general F.X0; X1; : : : ; Xn/ D F1.X1; : : : ; Xn/ C X0F2.X0; X1; : : : ; Xn/ with F1 homogeneous of degree m and F2 homogeneous of degree m 1. But mCn m D mCn1 m C mCn1 m1 because they are the coefficients of X m in.X C 1/mCn D.X C 1/.X C 1/mCn1; and this proves the induction. 142 6. PROJECTIVE VARIETIES Its image is the curve.P 1/ W X0X2 D X 2.b2;0 W b1;1 W b0;2/ 7! 1, and the map.b2;0 W b1;1/ if b2;0 ¤ 1.b1;1 W b0;2/ if b0;2 ¤ 0 is an inverse.P 1/! P 1. (Cf. Example 6.22.) When n D 1 and m is general, the Veronese map is 1! P m,.a0 W a1/ 7!.am 0 P W am1 0 a1 W : : : W am 1 /: I claim that, in the general case, the image of is a closed subset of P defines an isomorphism of projective varieties W P n!.P n/. n;m and that First note that the map has the following interpretation: if we regard the coordinates ai n as being the coefficients of a linear form L D P ai Xi (well-defined up of a point P of P to multiplication by nonzero scalar), then the coordinates of.P / are the coefficients of the homogeneous polynomial Lm with the binomial coefficients omitted. As L ¤ 0 ) Lm ¤ 0, the map is defined on the whole of P n, that is,.a0; : : : ; an/ ¤.0; : : : ; 0/ ).: : : ; bi0:::in; : : :/ ¤.0; : : : ; 0/: Moreover, L1 ¤ cL2 ) Lm ¤ cLm 1 and so is injective. It is clear from its definition
that is regular. 2, because kŒX0; : : : ; Xn is a unique factorization domain, We shall see in the next chapter that the image of any projective variety under a regular n/ is defined by the system of map is closed, but in this case we can prove directly that.P equations: bi0:::inbj0:::jn D bk0:::knb`0:::`n; ih C jh D kh C `h, all h: (*) Obviously P n maps into the algebraic set defined by these equations. Conversely, let Vi D f.: : : : W bi0:::in W : : :/ j b0:::0m0:::0 ¤ 0g: Then.Ui / Vi and 1.Vi / D Ui. It is possible to write down a regular map Vi! Ui inverse to jUi : for example, define V0! P n to be.: : : W bi0:::in W : : :/ 7!.bm;0;:::;0 W bm1;1;0;:::;0 W bm1;0;1;0;:::;0 W : : : W bm1;0;:::;0;1/: Finally, one checks that.P For any closed variety W P.W / of.P n/ P n;m. n/ S Vi. n, jW is an isomorphism of W onto a closed subvariety 6.24. The Veronese mapping has a very important property. If F is a nonzero homogeneous form of degree m 1, then V.F / P n is called a hypersurface of degree m and V.F / \ W is called a hypersurface section of the projective variety W. When m D 1, “surface” is replaced by “plane”. Now let H be the hypersurface in P n of degree m X ai0:::inX i0 0 X in n D 0, and let L be the hyperplane in P n;m defined by X ai0:::inXi0:::in: i. Some classical maps of projective varieties 143 Then.H / D.P n/ \ L, i.e., H.a/ D 0 ” L..a// D 0
: n, defines an isomorphism of the hypersurface Thus for any closed subvariety W of P section W \ H of V onto the hyperplane section.W / \ L of.W /. This observation often allows one to reduce questions about hypersurface sections to questions about hyperplane sections. As one example of this, note that maps the complement of a hypersurface section of W isomorphically onto the complement of a hyperplane section of.W /, which we know to be affine. Thus the complement of any hypersurface section of a projective variety is an affine variety. AUTOMORPHISMS OF P 6.25. An element A D.aij / of GLnC1 defines an automorphism of P n n:.x0 W : : : W xn/ 7!.: : : W P aij xj W : : :/I clearly it is a regular map, and the inverse matrix gives the inverse map. Scalar matrices act as the identity map. Let PGLnC1 D GLnC1 =kI, where I is the identity matrix, that is, PGLnC1 is the.nC1/21 of the quotient of GLnC1 by its centre. Then PGLnC1 is the complement in P hypersurface det.Xij / D 0, and so it is an affine variety with ring of regular functions kŒPGLnC1 D fF.: : : ; Xij ; : : :/= det.Xij /m j deg.F / D m.n C 1/g [ f0g: It is an affine group variety. The homomorphism PGLnC1! Aut.P it is surjective.4 Consider a hypersurface n/ is obviously injective. We sketch a proof that in P n and a line H W F.X0; : : : ; Xn/ D 0 L D f.ta0 W : : : W tan/ j t 2 kg in P n. The points of H \ L are given by the solutions of F.ta0; : : : ; tan/ D 0, which is a polynomial of degree deg.F / in t unless L H. Therefore, H \ L contains deg.F / points, and it is not hard to show that for a fixed H and most L it
will contain n exactly deg.F / points. Thus, the hyperplanes are exactly the closed subvarieties H of P such that (a) dim.H / D n 1; (b) #.H \ L/ D 1 for all lines L not contained in H. These are geometric conditions, and so any automorphism of P hyperplanes. But on an open subset of P n, such an automorphism takes the form n must map hyperplanes to.b0 W : : : W bn/ 7!.F0.b0; : : : ; bn/ W : : : W Fn.b0; : : : ; bn//; where the Fi are homogeneous of the same degree d (see 6.20). Such a map will take hyperplanes to hyperplanes if and only if d D 1. 4This is related to the fundamental theorem of projective geometry — see E. Artin, Geometric Algebra, Interscience, 1957, Theorem 2.26. 144 6. PROJECTIVE VARIETIES THE SEGRE MAP 6.26. This is the mapping..a0 W : : : W am/;.b0 W : : : W bn// 7!..: : : W ai bj W : : ://W P m P n! P mnCmCn: mnCmCn is f.i; j / j 0 i m; 0 j ng. Note that if we interpret The index set for P the tuples on the left as being the coefficients of two linear forms L1 D P ai Xi and L2 D P bj Yj, then the image of the pair is the set of coefficients of the homogeneous form of degree 2, L1L2. From this observation, it is obvious that the map is defined on n.L1 ¤ 0 ¤ L2 ) L1L2 ¤ 0/ and is injective. On any subset of the the whole of P form Ui Uj it is defined by polynomials, and so it is regular. Again one can show that it mnCmCn defined by the is an isomorphism onto its image, which is the closed subset of P equations m P – see Shafarevich 1994, I 5.1. For example, the map wij wkl wil wkj D 0..a0 W a1/
7!.x0W W xn). Somewhat surprisingly, there are surjective regular n! P n. However, there is a surjective regular map A n. Consider the map n! A.x0W : : : W xn/ 7!.x2 0 W W x2 n/W P n! P n: It is mW 1 with m > 1 except over the points.0W W 1W W 0/. If H is a general hyperplane avoiding these points, then P n. For example, when we take n still maps onto P n X H A we obtain the surjective map H W x0 C C xn D 0,.x1; : : : ; xn/ 7!.x2 1 W W x2 n W.1 x1 xn/2/W A n! P n: k. Projective space without coordinates Let E be a vector space over k of dimension n. The set P.E/ of lines through zero in E has a natural structure of an algebraic variety: the choice of a basis for E defines a bijection P.E/! P n, and the inherited structure of an algebraic variety on P.E/ is independent of 5A nonsingular curve of degree d in P 2 has genus.d 1/.d 2/ 2 genus g can’t be realized as a nonsingular curve in P 2.. Thus, if g is not of this form, a curve of 146 6. PROJECTIVE VARIETIES the choice of the basis (because the bijections defined by two different bases differ by an n, which has n C 1 distinguished hyperplanes, automorphism of P namely, X0 D 0; : : : ; Xn D 0, no hyperplane in P.E/ is distinguished. n). Note that in contrast to P l. The functor defined by projective space Let R be a k-algebra. A submodule M of an R-module N is said to be a direct summand of N if there exists another submodule M 0 of M (a complement of M ) such that N D M ˚ M 0. Let M be a direct summand of a finitely generated projective R-module N. Then M is also finitely generated and projective, and so Mm is a free Rm-module of finite rank for every maximal ideal m in R
. If Mm is of constant rank r, then we say that M has rank r. See CA 12. Let P n.R/ D fdirect summands of rank 1 of RnC1g. Then P n is a functor from k-algebras to sets. When K is a field, every K-subspace of KnC1 is a direct summand, and so P n.K/ consists of the lines through the origin in KnC1. Let Hi be the hyperplane Xi D 0 in knC1, and let Pi.R/ D fL 2 P n.R/ j L ˚ HiR D RnC1g: Let L 2 Pi.R/; then Now ei D ` CX j ¤i aj ej. L 7!.aj /j ¤i W Pi.R/! Ui.R/'Rn is a bijection. These combine to give an isomorphism P n.R/! P Y Y n.R/: P n.R/ Pi.R/ Pi.R/ \ Pj.R/ 0i n 0i;j n n.R/ P Y 0i n Ui.R/ Y 0i;j n Ui.R/ \ Uj.R/: More generally, to give a regular map from a variety V to P n is the same as giving an isomorphism class of pairs.L;.s0; : : : ; sn// where L is an invertible sheaf on V and s0; : : : ; sn are sections of L that generate it. m. Grassmann varieties Let E be a vector space over k of dimension n, and let Gd.E/ be the set of d -dimensional subspaces of E. When d D 0 or n, Gd.E/ has a single element, and so from now on we assume that 0 < d < n. Fix a basis for E, and let S 2 Gd.E/. The choice of a basis for S then determines a d n matrix A.S/ whose rows are the coordinates of the basis elements. Changing the basis for S multiplies A.S/ on the left by an invertible d d matrix. Thus, the family of d d minors of A.S/ is determined up to multiplication by a nonzero constant, and so defines a point P.
S/ in P n d 1. m. Grassmann varieties 147 PROPOSITION 6.29. The map S 7! P.S/W Gd.E/! P closed subset of P 1 n d. n d 1 is injective, with image a n d 1 We give the proof below. The maps P defined by different bases of E differ by an automorphism of P, and so the statement is independent of the choice of the basis — later (6.34) we shall give a “coordinate-free description” of the map. The map realizes Gd.E/ as a projective algebraic variety called the Grassmann variety of d -dimensional subspaces of E. EXAMPLE 6.30. The affine cone over a line in P Thus, G2.k4/ can be identified with the set of lines in P x D.x0 W x1 W x2 W x3/ and y D.y0 W y1 W y2 W y3/ be distinct points on L. Then 3 is a two-dimensional subspace of k4. 3, and let 3. Let L be a line in P P.L/ D.p01 W p02 W p03 W p12 W p13 W p23/ 2 P 5; pij defD ˇ ˇ ˇ ˇ xi xj yi yj ˇ ˇ ˇ ˇ ; depends only on L. The map L 7! P.L/ is a bijection from G2.k4/ onto the quadric ˘ W X01X23 X02X13 C X03X12 D 0 in P 5. For a direct elementary proof of this, see (9.41, 9.42) below. REMARK 6.31. Let S 0 be a subspace of E of complementary dimension n d, and let Gd.E/S 0 be the set of S 2 Gd.V / such that S \ S 0 D f0g. Fix an S0 2 Gd.E/S 0, so that E D S0 ˚ S 0. For any S 2 Gd.V /S 0, the projection S! S0 given by this decomposition is an isomorphism, and so S is the graph of a homomorphism S0! S 0: s 7! s
0 ”.s; s0/ 2 S: Conversely, the graph of any homomorphism S0! S 0 lies in Gd.V /S 0. Thus, Gd.V /S 0 Hom.S0; S 0/ Hom.E=S 0; S 0/: (27) The isomorphism Gd.V /S 0 Hom.E=S 0; S 0/ depends on the choice of S0 — it is the element of Gd.V /S 0 corresponding to 0 2 Hom.E=S 0; S 0/. The decomposition E D S0 ˚ S 0 gives a decomposition End.E/ D End.S0/ Hom.S0; S 0/ Hom.S 0; S0/ End.S 0/ ; and the bijections (27) show that the group Gd.E/S 0. 0 1 Hom.S0;S 0/ 1 acts simply transitively on REMARK 6.32. The bijection (27) identifies Gd.E/S 0 with the affine variety A.Hom.S0; S 0// defined by the vector space Hom.S0; S 0/ (cf. p. 72). Therefore, the tangent space to Gd.E/ at S0, TS0.Gd.E//'Hom.S0; S 0/'Hom.S0; E=S0/: Since the dimension of this space doesn’t depend on the choice of S0, this shows that Gd.E/ is nonsingular (4.39). (28) 148 6. PROJECTIVE VARIETIES REMARK 6.33. Let B be the set of all bases of E. The choice of a basis for E identifies B with GLn, which is the principal open subset of A In particular, B has a natural structure as an irreducible algebraic variety. The map.e1; : : : ; en/ 7! he1; : : : ; ed iW B! Gd.E/ is a surjective regular map, and so Gd.E/ is also irreducible. where det ¤ 0. n2 REMARK 6.34. The exterior algebra V E D L Vd E of E is the quotient of the tensor algebra by the ideal generated by all vectors e ˝ e
, e 2 E. The elements of Vd E are called (exterior) d -vectors:The exterior algebra of E is a finite-dimensional graded algebra over k wedge with V0 E D k, V1 E D E; if e1; : : : ; en form an ordered basis for V, then the n products d 0 d ei1 ^ : : : ^ eid.i1 < < id / form an ordered basis for Vd E. In particular, Vn E has dimension 1. For a subspace S of E of dimension d, Vd S is the one-dimensional subspace of Vd E spanned by e1 ^ : : : ^ ed for any basis e1; : : : ; ed of S. Thus, there is a well-defined map S 7! ^d S W Gd.E/! P. ^d E/ (29) which the choice of a basis for E identifies with S 7! P.S/. Note that the subspace spanned by e1; : : : ; en can be recovered from the line through e1 ^ : : : ^ ed as the space of vectors v such that v ^ e1 ^ : : : ^ ed D 0 (cf. 6.35 below). FIRST PROOF OF PROPOSITION 6.29. Fix a basis e1; : : : ; en of E, and let S0 D he1; : : : ; ed i and S 0 D hed C1; : : : ; eni. Order the coordinates in P so that n d 1 P.S/ D.a0W : : : W aij W : : : W : : :/; where a0 is the left-most d d minor of A.S/, and aij, 1 i d, d < j n, is the minor obtained from the left-most d d minor by replacing the i th column with the j th column. Let U0 be the (“typical”) standard open subset of P consisting of the points with nonzero zeroth coordinate. Clearly,6 P.S/ 2 U0 if and only if S 2 Gd.E/S 0. We shall prove the proposition by showing that P W Gd.E/S 0! U0 is injective with closed image. For S 2 Gd.E/S 0, the projection S! S0 is biject
ive. For each i, 1 i d, let n d 1 e0 i D ei C P d <j n aij ej (30) denote the unique element of S projecting to ei. Then e0 d is a basis for S. Conversely, for any.aij / 2 kd.nd /, the e0 i defined by (30) span an S 2 Gd.E/S 0 and project to the ei. Therefore, S $.aij / gives a one-to-one correspondence Gd.E/S 0 $ kd.nd / (this is a restatement of (27) in terms of matrices). 1; : : : ; e0 Now, if S $.aij /, then P.S/ D.1 W : : : W aij W : : : W : : : W fk.aij /W : : :/: 6If e 2 S 0 \ S is nonzero, we may choose it to be part of the basis for S, and then the left-most d d submatrix of A.S/ has a row of zeros. Conversely, if the left-most d d submatrix is singular, we can change the basis for S so that it has a row of zeros; then the basis element corresponding to the zero row lies in S 0 \ S. m. Grassmann varieties 149 where fk.aij / is a polynomial in the aij whose coefficients are independent of S. Thus, P.S/ determines.aij / and hence also S. Moreover, the image of P W Gd.E/S 0! U0 is the graph of the regular map.: : : ; aij ; : : :/ 7!.: : : ; fk.aij /; : : :/W A d.nd /! A n d d.nd /1 ; which is closed (5.28). SECOND PROOF OF PROPOSITION 6.29. An exterior d -vector v is said to be pure (or decomposable) if there exist vectors e1; : : : ; ed 2 V such that v D e1 ^ : : : ^ ed. According to 6.34, the image of Gd.E/ in P.Vd E/ consists of the lines through the pure d -vectors. LEMMA 6.
35. Let w be a nonzero d -vector and let M.w/ D fv 2 E j v ^ w D 0gI then dimk M.w/ d, with equality if and only if w is pure. PROOF. Let e1; : : : ; em be a basis of M.w/, and extend it to a basis e1; : : : ; em; : : : ; en of V. Write w D X ai1:::id ei1 ^ : : : ^ eid ; ai1:::id 2 k. 1i1<:::<id If there is a nonzero term in this sum in which ej does not occur, then ej ^ w ¤ 0. Therefore, each nonzero term in the sum is of the form ae1 ^ : : : ^ em ^ : : :. It follows that m d, and m D d if and only if w D ae1 ^ : : : ^ ed with a ¤ 0. For a nonzero d -vector w, let Œw denote the line through w. The lemma shows that Œw 2 Gd.E/ if and only if the linear map v 7! v ^ wW E 7! Vd C1 E has rank n d (in which case the rank is n d ). Thus Gd.E/ is defined by the vanishing of the minors of order n d C 1 of this map. 7 Flag varieties The discussion in the last subsection extends easily to chains of subspaces. Let d D.d1; : : : ; dr / be a sequence of integers with 0 < d1 < < dr < n, and let Gd.E/ be the set of flags F W E E1 Er 0 (31) 7In more detail, the map w 7!.v 7! v ^ w/W^d E! Homk.E; ^d C1 E/ is injective and linear, and so defines an injective regular map ^d P. E/,! P.Homk.E; ^d C1 E//: The condition rank n d defines a closed subset W of P.Homk.E; Vd C1 E// (once a basis has been chosen for E, the condition becomes the vanishing of the minors of order n d C 1 of a linear map E! Vd C1 E), and G
d.E/ D P.Vd E/ \ W: 150 6. PROJECTIVE VARIETIES with Ei a subspace of E of dimension di. The map Gd.E/ F 7!.E i /! Q i Gdi.E/ Q i P.Vdi E/ realizes Gd.E/ as a closed subset8 Q i Gdi.E/, and so it is a projective variety, called a flag variety. The tangent space to Gd.E/ at the flag F consists of the families of homomorphisms 'i W Ei! E=Ei ; 1 i r; (32) that are compatible in the sense that 'i jEi C1 'i C1 mod EiC1: ASIDE 6.36. A basis e1; : : : ; en for E is adapted to the flag F if it contains a basis e1; : : : ; eji for each E i. Clearly, every flag admits such a basis, and the basis then determines the flag. As in (6.33), this implies that Gd.E/ is irreducible. Because GL.E/ acts transitively on the set of bases for E, it acts transitively on Gd.E/. For a flag F, the subgroup P.F / stabilizing F is an algebraic subgroup of GL.E/, and the map g 7! gF0W GL.E/=P.F0/! Gd.E/ is an isomorphism of algebraic varieties. Because Gd.E/ is projective, this shows that P.F0/ is a parabolic subgroup of GL.E/. n. Bezout’s theorem n (that is, a closed subvariety of dimension n 1). For such a Let V be a hypersurface in P variety, I.V / D.F.X0; : : : ; Xn// with F a homogenous polynomial without repeated factors. We define the degree of V to be the degree of F. The next theorem is one of the oldest, and most famous, in algebraic geometry. 2 of degrees m and n respectively. If C and D THEOREM 6.37. Let C and D be curves in P have no irreducible component in common, then they intersect in exactly mn points, counted with appropriate multiplic
ities. PROOF. Decompose C and D into their irreducible components. Clearly it suffices to prove the theorem for each irreducible component of C and each irreducible component of D. We can therefore assume that C and D are themselves irreducible. We know from 2.62 that C \ D is of dimension zero, and so is finite. After a change of variables, we can assume that a ¤ 0 for all points.a W b W c/ 2 C \ D. Let F.X; Y; Z/ and G.X; Y; Z/ be the polynomials defining C and D, and write F D s0Zm C s1Zm1 C C sm; G D t0Zn C t1Zn1 C C tn with si and tj polynomials in X and Y of degrees i and j respectively. Clearly sm ¤ 0 ¤ tn, for otherwise F and G would have Z as a common factor. Let R be the resultant of F and G, regarded as polynomials in Z. It is a homogeneous polynomial of degree mn in X and 8For example, if ui is a pure di -vector and ui C1 is a pure diC1-vector, then it follows from (6.35) that M.ui / M.ui C1/ if and only if the map v 7!.v ^ ui ; v ^ ui C1/W E! ^di C1 E ˚ ^diC1C1 E has rank n di (in which case it has rank n di ). Thus, Gd.E/ is defined by the vanishing of many minors. o. Hilbert polynomials (sketch) 151 Y, or else it is identically zero. If the latter occurs, then for every.a; b/ 2 k2, F.a; b; Z/ and G.a; b; Z/ have a common zero, which contradicts the finiteness of C \ D. Thus R is a nonzero polynomial of degree mn. Write R.X; Y / D X mnR. Y X /, where R.T / is a polynomial of degree mn in T D Y X. Suppose first that deg R D mn, and let ˛1; : : :
; ˛mn be the roots of R (some of them may be multiple). Each such root can be written ˛i D bi, and R.ai ; bi / D 0. According to ai 7.28 this means that the polynomials F.ai ; bi ; Z/ and G.ai ; bi ; Z/ have a common root ci. Thus.ai W bi W ci / is a point on C \ D, and conversely, if.a W b W c/ is a point on C \ D (so a is a root of R.T /. Thus we see in this case, that C \ D has precisely mn a ¤ 0/, then b points, provided we take the multiplicity of.a W b W c/ to be the multiplicity of b a as a root of R. Now suppose that R has degree r < mn. Then R.X; Y / D X mnr P.X; Y /, where P.X; Y / is a homogeneous polynomial of degree r not divisible by X. Obviously R.0; 1/ D 0, and so there is a point.0 W 1 W c/ in C \ D, in contradiction with our assumption. REMARK 6.38. The above proof has the defect that the notion of multiplicity has been too obviously chosen to make the theorem come out right. It is possible to show that the theorem holds with the following more natural definition of multiplicity. Let P be an isolated point of C \ D. There will be an affine neighbourhood U of P and regular functions f and g on U such that C \ U D V.f / and D \ U D V.g/. We can regard f and g as elements P, and clearly rad.f; g/ D m, the maximal ideal in P. It follows that of the local ring P =.f; g/ is finite-dimensional over k, and we define the multiplicity of P in C \ D to be P =.f; g//. For example, if C and D cross transversely at P, then f and g will form O dimk. a system of local parameters at P —.f; g/ D m — and so the multiplicity is one. O O O The attempt to find good notions of multiplicities in very general situations motivated much of the most interesting work in commutative algebra in
the second half of the twentieth century. o. Hilbert polynomials (sketch) Recall that for a projective variety V P n, khomŒV D kŒX0; : : : ; Xn=b D kŒx0; : : : ; xn; where b D I.V /. We observed that b is graded, and therefore khomŒV is a graded ring: khomŒV D M m0 khomŒV m; where khomŒV m is the subspace generated by the monomials in the xi of degree m. Clearly khomŒV m is a finite-dimensional k-vector space. THEOREM 6.39. There is a unique polynomial P.V; T / such that P.V; m/ D dimk kŒV m for all m sufficiently large. PROOF. Omitted. EXAMPLE 6.40. For V D P 141), dim khomŒV m D mCn n n, khomŒV D kŒX0; : : : ; Xn, and (see the footnote on page D.mCn/.mC1/, and so nŠ P.P n; T / D T Cn n D.T C n/.T C 1/ nŠ : 152 6. PROJECTIVE VARIETIES The polynomial P.V; T / in the theorem is called the Hilbert polynomial of V. Despite the notation, it depends not just on V but also on its embedding in projective space. THEOREM 6.41. Let V be a projective variety of dimension d and degree ı; then P.V terms of lower degree. PROOF. Omitted. The degree of a projective variety is the number of points in the intersection of the variety and of a general linear variety of complementary dimension (see later). EXAMPLE 6.42. Let V be the image of the Veronese map.a0 W a1/ 7!.ad 0 W ad 1 0 a1 W : : : W ad 1 /W P 1! P d : Then khomŒV m can be identified with the set of homogeneous polynomials of degree m d d C1 given by the same equations), which is a in two variables (look at
the map A space of dimension d m C 1, and so 2! A P.V; T / D d T C 1: Thus V has dimension 1 (which we certainly knew) and degree d. Macaulay knows how to compute Hilbert polynomials. REFERENCES: Hartshorne 1977, I.7; Harris 1992, Lecture 13. p. Dimensions The results for affine varieties extend to projective varieties with one important simplification: n and r C s n, then if V and W are projective varieties of dimensions r and s in P V \ W ¤ ;. n be a projective variety of dimension 1, and let THEOREM 6.43. Let V D V.a/ P f 2 kŒX0; : : : ; Xn be homogeneous, nonconstant, and … a; then V \ V.f / is nonempty and of pure codimension 1. PROOF. Since the dimension of a variety is equal to the dimension of any dense open affine subset, the only part that doesn’t follow immediately from 3.42 is the fact that V \ V.f / nC1 (that is, the affine cone over V /. is nonempty. Let V aff.a/ be the zero set of a in A Then V aff.a/ \ V aff.f / is nonempty (it contains.0; : : : ; 0/), and so it has codimension 1 in V aff.a/. Clearly V aff.a/ has dimension 2, and so V aff.a/ \ V aff.f / has dimension 1. This implies that the polynomials in a have a zero in common with f other than the origin, and so V.a/ \ V.f / ¤ ;. COROLLARY 6.44. Let f1; : : : ; fr be homogeneous nonconstant elements of kŒX0; : : : ; Xn; and let Z be an irreducible component of V \ V.f1; : : : fr /. Then codim.Z/ r, and if dim.V / r, then V \ V.f1; : : : fr / is nonempty. PROOF. Induction on r, as before. p. Dimensions 153 PROPOSITION 6.45. Let Z be an irreducible
closed subvariety of V ; if codim.Z/ D r, then there exist homogeneous polynomials f1; : : : ; fr in kŒX0; : : : ; Xn such that Z is an irreducible component of V \ V.f1; : : : ; fr /. PROOF. Use the same argument as in the proof 3.47. PROPOSITION 6.46. Every pure closed subvariety Z of P i.e., I.Z/ D.f / for some f homogeneous element of kŒX0; : : : ; Xn. n of codimension one is principal, PROOF. Follows from the affine case. COROLLARY 6.47. Let V and W be closed subvarieties of P then V \W ¤ ;, and every irreducible component of it has codim.Z/ codim.V /Ccodim.W /. PROOF. Write V D V.a/ and W D V.b/, and consider the affine cones V 0 D V.a/ and W 0 D V.b/ over them. Then n; if dim.V / C dim.W / n, dim.V 0/ C dim.W 0/ D dim.V / C 1 C dim.W / C 1 n C 2: As V 0 \ W 0 ¤ ;, V 0 \ W 0 has dimension 1, and so it contains a point other than the origin. Therefore V \ W ¤ ;. The rest of the statement follows from the affine case. n of dimension r < n; then there is a PROPOSITION 6.48. Let V be a closed subvariety of P linear projective variety E of dimension n r 1 (that is, E is defined by r C 1 independent linear forms) such that E \ V D ;. PROOF. Induction on r. If r D 0, then V is a finite set, and the lemma below shows that there is a hyperplane in knC1 not meeting V. Suppose r > 0, and let V1; : : : ; Vs be the irreducible components of V. By assumption, they all have dimension r. The intersection Ei of all the linear projective varieties containing Vi is the smallest such variety. The lemma below shows that there is
a hyperplane H containing none of the nonzero Ei ; consequently, H contains none of the irreducible components Vi of V, and so each Vi \ H is a pure variety of dimension r 1 (or is empty). By induction, there is an linear subvariety E0 not meeting V \ H. Take E D E0 \ H. LEMMA 6.49. Let W be a vector space of dimension d over an infinite field k, and let E1; : : : ; Er be a finite set of nonzero subspaces of W. Then there is a hyperplane H in W containing none of the Ei. PROOF. Pass to the dual space V of W. The problem becomes that of showing V is not a finite union of proper subspaces E_ i by a hyperplane Hi containing it. Then Hi is defined by a nonzero linear form Li. We have to show that Q Lj is not identically zero on V. But this follows from the statement that a polynomial in n variables, with coefficients not all zero, cannot be identically zero on kn (Exercise 1-1). i. Replace each E_ Let V and E be as in Proposition 6.48. If E is defined by the linear forms L0; : : : ; Lr then the projection a 7!.L0.a/ W W Lr.a// defines a map V! P r. We shall see later that this map is finite, and so it can be regarded as a projective version of the Noether normalization theorem. In general, a regular map from a variety V to P n corresponds to a line bundle on V n X foriging are trivial and a set of global sections of the line bundle. All line bundles on A (see, for example, Hartshorne II 7.1 and II 6.2), from which it follows that all regular maps nC1 X foriging! P m are given by a family of homogeneous polynomials. Assuming this, A it is possible to prove the following result. 154 6. PROJECTIVE VARIETIES n! P nC1 foriging! P COROLLARY 6.50. Let ˛W P m be regular; if m < n, then ˛ is constant. n be the map.a0; : : : ; an/ 7!.a0 W : : : W
an/. Then ˛ ı PROOF. Let W A is regular, and there exist polynomials F0; : : : ; Fm 2 kŒX0; : : : ; Xn such that ˛ ı is the map.a0; : : : ; an/ 7!.F0.a/ W : : : W Fm.a//: As ˛ ı factors through P n, the Fi must be homogeneous of the same degree. Note that ˛.a0 W : : : W an/ D.F0.a/ W : : : W Fm.a//: If m < n and the Fi are nonconstant, then 6.43 shows they have a common zero and so ˛ is not defined on all of P n. Hence the Fi must be constant. q. Products It is useful to have an explicit description of the topology on some product varieties. The topology on Pm Pn. Suppose we have a collection of polynomials Fi.X0; : : : ; XmI Y0; : : : ; Yn/, i 2 I, each of which is separately homogeneous in the Xi and Yj. Then the equations Fi.X0; : : : ; XmI Y0; : : : ; Yn/ D 0; i 2 I; m P define a closed subset of P from a (finite) set of polynomials. n, and every closed subset of P m P n arises in this way The topology on Am Pn m P The closed subsets of A n are exactly those defined by sets of equations Fi.X1; : : : ; XmI Y0; : : : ; Yn/ D 0; i 2 I; with each Fi homogeneous in the Yj. The topology on V Pn Let V be an irreducible affine algebraic variety. We look more closely at the topology n in terms of ideals. Let A D kŒV, and let B D AŒX0; : : : ; Xn. Note that B D on V P nC1: for A ˝k kŒX0; : : : ; Xn, and so we can view it as the ring of regular functions on V A f 2 A and g 2 kŒX0;
: : : ; Xn, f ˝ g is the function.v; a/ 7! f.v/ g.a/W V A nC1! k: n, a 2 A, has degree P ij — The ring B has an obvious grading — a monomial aX i0 and so we have the notion of a graded ideal b B. It makes sense to speak of the zero set V.b/ V P n of such an ideal. For any ideal a A, aB is graded, and V.aB/ D V.a/ P n. LEMMA 6.51. (a) For each graded ideal b B, the set V.b/ is closed, and every closed subset of V P n is of this form. 0 : : : X in (b) The set V.b/ is empty if and only if rad.b/.X0; : : : ; Xn/. (c) If V is irreducible, then V D V.b/ for some graded prime ideal b. q. Products 155 PROOF. (a) In the case that A D k, we proved this in 6.1 and 6.2, and similar arguments apply in the present more general situation. For example, to see that V.b/ is closed, cover n with the standard open affines Ui and show that V.b/ \ Ui is closed for all i. P The set V.b/ is empty if and only if the cone V aff.b/ V A nC1 defined by b is contained in V foriging. But X ai0:::inX i0 0 : : : X in n ; ai0:::in 2 kŒV ; is zero on V foriging if and only if its constant term is zero, and so I aff.V foriging/ D.X0; X1; : : : ; Xn/: Thus, the Nullstellensatz shows that V.b/ D ; ) rad.b/ D.X0; : : : ; Xn/. Conversely, if X N i 2 b for all i, then obviously V.b/ is empty. For (c), note that if V.b/ is irreducible, then the closure of its inverse image in V A nC1 is also irreducible, and so I V
.b/ is prime. Exercises 6-1. Show that a point P on a projective curve F.X; Y; Z/ D 0 is singular if and only if @F=@X, @F=@Y, and @F=@Z are all zero at P. If P is nonsingular, show that the tangent line at P has the (homogeneous) equation.@F=@X/P X C.@F=@Y /P Y C.@F=@Z/P Z D 0. Verify that Y 2Z D X 3 C aXZ2 C bZ3 is nonsingular if X 3 C aX C b has no repeated root, and find the tangent line at the point at infinity on the curve. 6-2. Let L be a line in P by a homogeneous polynomial of degree 2). Show that either 2 and let C be a nonsingular conic in P 2 (i.e., a curve in P 2 defined (a) L intersects C in exactly 2 points, or (b) L intersects C in exactly 1 point, and it is the tangent at that point. 3. Prove 6-3. Let V D V.Y X 2; Z X 3/ A (a) I.V / D.Y X 2; Z X 3/; (b) ZW X Y 2 I.V / kŒW; X; Y; Z, but ZW XY …..Y X 2/;.Z X 3//. r generate a, even (Thus, if F1; : : : ; Fr generate a, it does not follow that F if ais radical.) 1 ; : : : ; F 6-4. Let P0; : : : ; Pr be points in P through P0 but not passing through any of P1; : : : ; Pr. n. Show that there is a hyperplane H in P n passing 6-5. Is the subset of P 2 locally closed? f.a W b W c/ j a ¤ 0; b ¤ 0g [ f.1 W 0 W 0/g 6-6. Show that the image of the Segre map P in any hyperplane of P mnCmCn. m P n! P mnCmCn (see 6.26) is not contained 156 6. PRO
JECTIVE VARIETIES 6-7. Write 0, 1, 1 for the points.0W 1/,.1W 1/, and.1W 0/ on P (a) Let ˛ be an automorphism of P 1 such that 1. ˛.0/ D 0; ˛.1/ D 1; ˛.1/ D 1: Show that ˛ is the identity map. (b) Let P0, P1, P2 be distinct points on P 1. Show that there exists an ˛ 2 PGL2.k/ such that ˛.0/ D P0; ˛.1/ D P1; ˛.1/ D P2: (c) Deduce that Aut.P 1/'PGL2.k/. 6-8. Show that the functor R P n.R/ D fdirect summands of rank 1 of RnC1g satisfies the criterion 5.71 to arise from an algebraic prevariety. (This gives an alternative definition of P n.) 6-9. (a) Let V A m be algebraic varieties and 'W V! W a map. Show that'is regular if and only if every point in V has an open neighbourhood U on which there are regular functions f0; : : : ; fm such that n and W P '.a1; : : : ; an/ D.f0.a1; : : : ; an/W : : : W fm.a1; : : : ; an// for all.a1; : : : ; an/ 2 U. (b) Show that, for a regular map'as in (a), it may not be possible to take U D V. Hint: Let V A 4 be the complement of.0; 0; 0; 0/ in XY ZW D 0; and let 'W V! P.wW y/. See sx4626969 (Mohan). 1 send.w; x; y; z/ to.xW z/ if one of x or z is nonzero and.w; 0; y; 0/ to CHAPTER 7 Complete Varieties Complete varieties are the analogues in the category of algebraic varieties of compact topological spaces in the category of Hausdorff topological spaces. Recall that the image of a compact space under a continuous
map is compact, and hence is closed if the image space is Hausdorff. Moreover, a Hausdorff space V is compact if and only if, for all topological spaces T, the projection map qW V T! T is closed, i.e., maps closed sets to closed sets (see Bourbaki, N., General Topology, I, 10.2, Corollary 1 to Theorem 1). a. Definition and basic properties Definition DEFINITION 7.1. An algebraic variety V is complete if for all algebraic varieties T, the projection map qW V T! T is closed. Note that a complete variety is required to be separated — we really mean it to be a variety and not a prevariety. We shall see 7.22 that projective varieties are complete. EXAMPLE 7.2. Consider the projection map.x; y/ 7! yW A 1 A 1! A 1: This is not closed; for example, the variety V W XY D 1 is closed in A omits the origin. However, when we replace V with its closure in P 1. To see this, note that becomes the whole of A NV defD f..xW z/; y/ 2 P 1 j xy D z2g 1 A 2 but its image in A 1 1 A 1, its projection contains V as an open dense subset, and so must be its closure in P..xW 0/; 0/ of NV maps to 0. 1 A 1. The point Properties 7.3. Closed subvarieties of complete varieties are complete. Let Z be a closed subvariety of a complete variety V. For any variety T, Z T is closed in V T, and so the restriction of the closed map qW V T! T to Z T is also closed. 7.4. A variety is complete if and only if its irreducible components are complete. 157 158 7. COMPLETE VARIETIES Each irreducible component is closed, and hence complete if the variety is complete (7.3). Conversely, suppose that the irreducible components Vi of a variety V are complete. If Z is defD Z \.Vi T / is closed in Vi T. Therefore, q.Zi / is closed in closed in V T, then Zi T, and so q.Z/ D S q.Zi / is also closed. 7.5. Products
of complete varieties are complete. Let V1; : : : ; Vn be complete varieties, and let T be a variety. The projection Q is the composite of the projections i Vi T! T V1 Vn T! V2 Vn T!! Vn T! T; all of which are closed. 7.6. If 'W W! V is surjective and W is complete, then V is complete. Let T be a variety, and let Z be a closed subset of V T. Let Z0 be the inverse image of Z in W T. Then Z0 is closed, and its image in T equals that of Z. 7.7. Let 'W W! V be a regular map of varieties. If W is complete, then '.W / is a complete closed subvariety of V. In particular, every complete subvariety of a variety is closed. defD f.w; '.w//g W V be the graph of '. It is a closed subset of W V (because Let'V is a variety, see 5.28), and '.W / is the projection of'into V. Therefore '.W / is closed, and 7.6 shows that it is complete. The second statement follows from the first applied to the identity map. 7.8. A regular map V! P surjective. 1 from a complete connected variety V is either constant or 1 are the finite sets, and such a set is connected if and The only proper closed subsets of P only if it consists of a single point. Because '.V / is connected and closed, it must either be 1 (and'is onto). a single point (and'is constant) or P 7.9. The only regular functions on a complete connected variety are the constant functions. A regular function on a variety V is a regular map f W V! A 7.8. 1 P 1, to which we can apply 7.10. A regular map 'W V! W from a complete connected variety to an affine variety has image equal to a point. In particular, every complete connected affine variety is a point. n, and write'D.'1; : : : ; 'n/, where 'i is the composite Embed W as a closed subvariety of A of'with the coordinate function xi W A 1. Each 'i is a regular function on V, and hence is constant. (Alternatively, apply 5.12
.) This proves the first statement, and the second follows from the first applied to the identity map. n! A 7.11. In order to show that a variety V is complete, it suffices to check that qW V T! T is a closed mapping when T is affine (or even an affine space A Every variety T can be written as a finite union of open affine subvarieties T D S Ti. If Z is closed in V T, then Zi defD Z \.V Ti / is closed in V Ti. Therefore, q.Zi / is closed in Ti for all i. As q.Zi / D q.Z/ \ Ti, this shows that q.Z/ is closed. This shows that it suffices to check that V T! T is closed for all affine varieties T. But T can be realized as a closed subvariety of A n, and then V T! T is closed if V A n is closed. n! A n). b. Proper maps 159 Remarks 7.12. The statement that a complete variety V is closed in every larger variety W perhaps explains the name: if V is complete, W is connected, and dim V D dim W, then V D W. Contrast A n P n. 7.13. Here is another criterion: a variety V is complete if and only if every regular map C X fP g! V extends uniquely to a regular map C! V ; here P is a nonsingular point on a curve C. Intuitively, this says that all Cauchy sequences have limits in V and that the limits are unique. b. Proper maps DEFINITION 7.14. A regular map 'W V! S of varieties is said to be proper if it is “universally closed”, that is, if for all regular maps T! S, the base change '0W V S T! T of'is closed. 7.15. For example, a variety V is complete if and only if the map V! fpointg is proper. 7.16. From its very definition, it is clear that the base change of a proper map is proper. In particular, (a) if V is complete, then V S! S is proper, (b) if 'W V! S is proper, then the fibre '1.P / over a point P of S is complete. 7.17. If 'W V
! S is proper, and W is a closed subvariety of V, then W '! S is proper. PROPOSITION 7.18. A composite of proper maps is proper. PROOF. Let V3! V2! V1 be proper maps, and let T be a variety. Consider the diagram V3 V2 V1 V3 V2.V2 V1 T /'V3 V1 T closed V2 V1 T closed T: Both smaller squares are cartesian, and hence so also is the outer square. The statement is now obvious from the fact that a composite of closed maps is closed. COROLLARY 7.19. If V! S is proper and S is complete, then V is complete. PROOF. Special case of the proposition. COROLLARY 7.20. The inverse image of a complete variety under a proper map is complete. PROOF. Let 'W V! S be proper, and let Z be a complete subvariety of S. Then V S Z! Z is proper, and V S Z''1.Z/. 160 7. COMPLETE VARIETIES EXAMPLE 7.21. Let f 2 kŒT1; : : : ; Tn; X; Y be homogeneous of degree m in X and Y, and let H be the subvariety of A 1 defined by n P f.T1; : : : ; Tn; X; Y / D 0. The projection map A 7.15). The fibre over a point.t1; : : : ; tn/ 2 A mial 1! A n P n defines a regular map H! A n is the subvariety of P n, which is proper (7.22, 1 defined by the polyno- f.t1; : : : ; tn; X; Y / D a0X m C a1X m1Y C C amY m; ai 2 k: Assume that not all ai are zero. Then this is a homogeneous of degree m and so the fibre always has m points counting multiplicities. The points that “disappeared off to infinity” when P 1 (see p. 51) have literally become the point at infinity on P 1 was taken to be A 1. c. Projective varieties are complete The reader may skip this section since the main theorem is given a more explicit proof
in Theorem 7.31 below. THEOREM 7.22. A projective variety is complete. n itself; thus we n W! W is a closed mapping in the case that W PROOF. After 7.3, it suffices to prove the Theorem for projective space P have to prove that the projection map P is an irreducible affine variety (7.11). Write p for the projection W P n! W. We have to show that Z closed in W P implies that p.Z/ closed in W. If Z is empty, this is true, and so we can assume it to be nonempty. Then Z is a finite union of irreducible closed subsets Zi of W P n, and it suffices to show that each p.Zi / is closed. Thus we may assume that Z is irreducible, and hence that Z D V.b/ with b a graded prime ideal in B D AŒX0; : : : ; Xn (6.51). If p.Z/ is contained in some closed subvariety W 0 of W, then Z is contained in W 0 P n, and we can replace W with W 0. This allows us to assume that p.Z/ is dense in W, and we now have to show that p.Z/ D W. n Because p.Z/ is dense in W, the image of the cone V aff.b/ under the projection W nC1! W is also dense in W, and so (see 3.34a) the map A! B=b is injective. Let w 2 W : we shall show that if w … p.Z/, i.e., if there does not exist a P 2 P n such A that.w; P / 2 Z, then p.Z/ is empty, which is a contradiction. Let m A be the maximal ideal corresponding to w. Then mB C b is a graded ideal, n/ \ V.b/, and so w will be in the image of Z and V.mB C b/ D V.mB/ \ V.b/ D.w P unless V.mB C b/ ¤ ;. But if V.mB C b/ D ;, then mB C b.X0; : : : ; Xn/N for some N (by 6.51b), and so mB C

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