# On the number of generators of an algebra over a commutative ring

Uriya A. First  Department of Mathematics
Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa, 3498838
Israel<EMAIL_ADDRESS>,  Zinovy Reichstein  Department of
Mathematics
University of British Columbia
Vancouver BC V6T 1Z2
Canada<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Ben Williams

###### Abstract.

A theorem of O. Forster says that if $R$ is a noetherian ring of Krull
dimension $d$, then every projective $R$-module of rank $n$ can be generated
by $d+n$ elements. S. Chase and R. Swan subsequently showed that this bound is
sharp: there exist examples that cannot be generated by fewer than $d+n$
elements. We view rank-$n$ projective $R$-modules as _$R$ -forms_ of the non-
unital $R$-algebra $R^{n}$ where the product of any two elements is $0$. The
first two authors generalized Forster’s theorem to forms of other algebras
(not necessarily commutative, associative or unital); A. Shukla and the third
author then showed that this generalized Forster bound is optimal for étale
algebras.

In this paper, we prove new upper and lower bounds on the number of generators
of an $R$-form of a $k$-algebra, where $k$ is an infinite field and $R$ is a
finitely generated $k$-ring of Krull dimension $d$. In particular, we show
that, contrary to expectations, for most types of algebras, the generalized
Forster bound is far from optimal. Our results are particularly detailed in
the case of Azumaya algebras. Our proofs are based on reinterpreting the
problem as a question about approximating the classifying stack $BG$, where
$G$ is the automorphism group of the algebra in question, by algebraic spaces
of a certain type.

###### Key words and phrases:

Algebra over a ring, multialgebra, Azumaya algebra, number of generators,
versal torsor, classifying space, equivariant cohomology

###### 2020 Mathematics Subject Classification:

Primary 14L30, 16H05, 16S15, 14F25, 55R40. Secondary 17A36, 13E15, 20G10,
14M17.

The second and third authors have been partially supported by Discovery Grants
from the Natural Sciences and Engineering Research Council of Canada.

##  1\. Introduction

Throughout this paper $k$ will denote a base field, $A$ a finite-dimensional
$k$-algebra, $R$ a $k$-ring, and $B$ a finitely generated $R$-algebra. By a
ring we will always mean a commutative associative unital ring. In contrast,
an algebra will not be required to be commutative, associative, or unital. We
write $\operatorname{gen}_{R}(B)$ for the smallest cardinality of a set of
generators of $B$ as an $R$-algebra.

Recall that an $R$-form of $A$ is an $R$-algebra $B$ such that there exists a
faithfully flat $R$-ring $S$ for which $A\otimes_{k}S\cong B\otimes_{R}S$ as
$S$-algebras. Many important classes of algebras arise as forms of particular
$k$-algebras. For example:

  1. (i)

$R$-forms of $A=k^{n}$ with the zero multiplication are projective $R$-modules
of rank $n$.

  2. (ii)

$R$-forms of $A=k^{n}$ with componentwise multiplication are finite étale
$R$-algebras of rank $n$.

  3. (iii)

$R$-forms of $A=\operatorname{M}_{s}(k)$ are Azumaya algebras of degree $s$
over $R$.

  4. (iv)

$R$-forms of the split octonion $k$-algebra are octonion algebras over $R$.

O. Forster [For64] showed that a projective module of rank $n$ over a
noetherian ring $R$ can be generated by $\operatorname{Kdim}R+n$ elements,
where $\operatorname{Kdim}R$ denotes the Krull dimension of $R$; see also
[Swa67]. In [FR17, Theorem 1.2 and Proposition 4.1], the first two authors
extended this result to algebras as follows.

######  Theorem 1.1 (Generalized Forster bound).

Assume $k$ is infinite. Let $A$ be a finite-dimensional $k$-algebra. If $R$ is
a noetherian $k$-ring and $B$ is an $R$-form of $A$, then
$\operatorname{gen}_{R}(B)\leqslant\operatorname{Kdim}R+\operatorname{gen}_{k}(A)$.

In case (i) above, Theorem 1.1 specializes to Forster’s bound, since in this
case $R$-forms of $A$ are projective $R$-modules of rank $n$. Forster’s bound
was shown to be optimal by S. Chase and R. Swan [Swa62, Theorem 4]. More
precisely, they showed that for every $n\geqslant 1$ and $d\geqslant 0$ there
exist a $d$-dimensional ring $R$ of finite type (i.e., generated by finitely
many elements) over the field of real numbers and a projective $R$-module $B$
of rank $n$ such that $\operatorname{gen}_{R}(B)=d+n$.

In case (ii), Theorem 1.1 shows that
$\operatorname{gen}(B)\leqslant\operatorname{Kdim}R+1$ for every finite étale
algebra $B$ over $R$. This upper bound was recently shown to be optimal by A.
Shukla and the third author [SW20]. Specifically, for every pair of integers
$n\geqslant 2$ and $d\geqslant 0$, they constructed a finitely generated ring
$R$ over the field of real numbers, of dimension $d$, and a finite étale
$R$-algebra $B$ of rank $n$ such that $\operatorname{gen}_{R}(B)=d+1$.111A
similar example was independently found by M. Ojanguren (2017, unpublished).

In this paper we will focus on the case where $R$ is of finite type over an
infinite field $k$ and will address the following question:

######  Question 1.2.

Is the generalized Forster bound of Theorem 1.1 optimal for every finite-
dimensional $k$-algebra $A$?

We will show that the answer to this question is “no”, contrary to what one
might expect based on [Swa62] and [SW20]. In particular, we will show that the
following bounds given by Theorem 1.1,

  1. (iii)

$\operatorname{gen}_{R}(B)\leqslant\operatorname{Kdim}R+2$ for every Azumaya
algebra $B$ over $R$ and

  2. (iv)

$\operatorname{gen}_{R}(B)\leqslant\operatorname{Kdim}R+3$ for every octonion
algebra $B$ over $R$;

(see [FR17, Corollary 4.2]), are far from optimal.

Our first main result is:

######  Theorem 1.3.

Assume that $k$ is an infinite field, $A$ is an $n$-dimensional $k$-algebra,
and $R$ is a finite type $k$-ring of Krull dimension $d$. Let $\overline{k}$
be an algebraic closure of $k$ and set
$\overline{A}=A\otimes_{k}\overline{k}$. If the maximal dimension of a proper
$\overline{k}$-subalgebra of $\overline{A}$ is $n_{\rm max}$, then

$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\frac{d}{n-n_{\rm
max}}}\right\rfloor+n_{\rm max}+1$

for any $R$-form $B$ of $A$.

If $n_{\rm max}\leqslant n-2$, as in examples (iii) and (iv) above, then
Theorem 1.3 yields an asymptotically stronger upper bound on
$\operatorname{gen}_{R}(B)$ than Theorem 1.1 (for large $d$).

Our second main result is a lower bound on $\operatorname{gen}_{R}(B)$ for
certain rings $R$ and certain $R$-forms of $A$. While this lower bound does
not quite match the upper bound of Theorem 1.3, it is not far off in the sense
that both are linear in $d=\operatorname{Kdim}R$.

######  Theorem 1.4.

Let $k$ be a field of characteristic $0$ and let $A$ be an $n$-dimensional
$k$-algebra such that the group $G=\operatorname{Aut}_{k}(A)$ is not
unipotent. Then there exists a positive integer $\rho_{G}$ with the following
property: For any positive integer $d$, there exist a finite type $k$-ring $R$
of Krull dimension $d$ and an $R$-form $B$ of $A$ such that

$\operatorname{gen}_{R}(B)\geqslant\left\lfloor{\frac{d+2\dim
G-\rho_{G}}{2n}}\right\rfloor+1.$

Here $\rho_{G}$ depends only on $G$ and not on $A$. Moreover, writing $H$ for
the quotient of the identity connected component of $G$ by its unipotent
radical, we may take $\rho_{G}=4$ if $\dim H>0$ and $\rho_{G}=2$ if $\dim
Z(H)>0$.

Theorems 1.3 and 1.4 remain valid in the more general setting where we allow
$A$ and $B$ to be _multialgebras_ , rather than algebras. The same is true for
most of the other results of this paper where the structure of the algebra is
not specified. Here by a _multialgebra_ over a ring $R$ we mean a module $B$
together with a (not necessarily finite) collection of $R$-multilinear maps
$\\{m_{i}:B^{n_{i}}\to B\\}_{i\in I}$, where each $n_{i}$ is a non-negative
integer.222This definition of multialgebra is more restrictive than the one
used in [FR17]. For example, a usual binary (algebra) product on $B$ can be
specified by an $R$-bilinear map $B^{2}\to B$, an involution by an $R$-linear
map $B^{1}\to B$, and a unit element in $B$ by a map $B^{0}\to B$ (which is
always multilinear). A multialgebra structure on $B$ naturally induces one on
the tensor product $B\otimes_{R}S$ for any $R$-ring $S$. Subalgebras, forms
and generating sets of multialgebras are defined in the obvious manner.
Viewing $B$ as a multialgebra is useful even in the case where $B$ is an
$R$-algebra in the usual sense, with a unit element. Here we may view $B$ as a
unital algebra or a non-unital algebra, and the multialgebra structure of $B$
allows us to distinguish between these two settings. (See also Remark 1.6
below.) Aside from usual algebras, unital and non-unital, the multialgebras of
greatest interest to us will be algebras with involution.

Our third main result sharpens the upper and lower bounds of Theorems 1.3 and
1.4 in the case of Azumaya algebras.

######  Theorem 1.5.

Let $k$ be an infinite field.

  1. (a)

Suppose $R$ is a finite type $k$-ring of Krull dimension $d$, and $B$ is an
Azumaya $R$-algebra of degree $s$. Then
$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\dfrac{d}{s-1}}\right\rfloor+2$.

  2. (b)

On the other hand, if $\operatorname{char}(k)=0$, then for any $s\geqslant 3$
and $d\geqslant 0$ there exists a finite type $k$-ring $R$ of Krull dimension
$d$ and an Azumaya $R$-algebra $B$ of degree $s$ such that
$\displaystyle\operatorname{gen}_{R}(B)\geqslant\left\lfloor{\dfrac{d}{2(s-1)}}\right\rfloor+2$.

An important role in this paper will be played by the automorphism group
$G=\operatorname{Aut}_{k}(A)$ (viewed as a linear algebraic group defined over
$k$) and by the $k$-schemes $U_{r}$ and $Z_{r}$ of $r$-tuples
$(a_{1},\ldots,a_{r})$ of elements of $A$ that do and do not generate $A$,
respectively. It is well known that the category of $R$-forms of $A$ is
equivalent to the category of $G$-torsors over $\operatorname{Spec}R$, or
equivalently, to the category of $R$-points of the classifying stack $BG$. On
the other hand, the pairs $(B,(b_{1},\ldots,b_{r}))$, consisting of an
$R$-form $B$ of $A$ and generators $b_{1},\ldots,b_{r}$ of the $R$-algebra
$B$, are in a natural bijective correspondence with $R$-points of the quotient
stack $U_{r}/G$; see Proposition 4.1. In fact, $U_{r}/G$ is an algebraic
space; see Remark 3.3. There is an evident morphism $U_{r}/G\to BG$ sending
the pair $(B,(b_{1},\ldots,b_{r}))$ to $B$. An $R$-algebra $B$ can be
generated by $r$ elements if and only if the $R$-point of $BG$ corresponding
to $B$ can be lifted to $U_{r}/G$. Informally speaking, much of this paper
revolves around the following question: how well does $U_{r}/G$ approximate
$BG$? The upper bounds on $\operatorname{gen}_{R}(B)$ in Theorems 1.3 and
1.5(a) are proved by constructing explicit liftings when
$\operatorname{Kdim}R$ is suitably small. On the other hand, the lower bounds
of Theorems 1.4 and 1.5(b) are proved by exhibiting topological obstructions
to the existence of a such liftings.

The remainder of this paper is structured as follows. In Section 2 we collect
notational conventions used throughout the paper. Sections 3-9 are devoted to
upper bounds on the number of generators. In Sections 3 and 4 we study the
spaces $U_{r}$ and $Z_{r}$ of generators and non-generators of a
(multi)algebra $A$. In Section 5 we introduce the notion of a _$d$ -versal
torsor_. This is a variant of the usual notion of versality from [Ser03,
Section 5] or [DR15], where instead of torsors over fields we consider torsors
over rings of bounded Krull dimension. In Section 6 we prove an upper bound on
$\operatorname{gen}_{R}(B)$ for any $R$-form $B$ of $A$ in terms of the
codimension of $Z_{r}$ inside $A^{r}$ (Theorem 6.1) and use it to deduce
Theorem 1.3. In Sections 7 and 8 we use Theorem 6.1 to prove new upper bounds
on the number of generators of an Azumaya algebra (Theorem 1.5(a)), an Azumaya
algebra with involution (Proposition 7.3) and an octonion algebra (Proposition
8.1). These bounds are of the same form,

$\operatorname{gen}_{R}(B)\leqslant\dfrac{d}{n-n_{\rm max}}+O(1),$

as the bound given by Theorem 1.3 but they are stronger, because the additive
constant is smaller. In Section 9 we use Theorem 6.1 to give a new short proof
of the generalized Forster bound (Theorem 1.1) in the case where $R$ is a
$k$-ring of finite type. The second half of the paper, starting from Section
10, is devoted to lower bounds on the number of generators. In Section 10, we
prove a version of the Lefschetz principle that will allow us to assume that
$k$ is a subfield of $\mathbb{C}$ in these proofs, thus opening the door to
the use of non-algebraic techniques such as singular cohomology. In Section
11, we prove Theorem 11.4, which is a general device for constructing examples
of $R$-forms $B$ of a $k$-algebra $A$ requiring many generators. We use this
device to prove Theorem 1.4 in Section 12 and Theorem 1.5(b) in Sections 13
and 14. In Appendix A we give proofs of some results on equivariant cohomology
of nonsingular varieties that are used in the second half of this paper. These
results are largely known but are not always documented in the literature in
the form we need.

######  Remark 1.6.

Let $R$ be a $k$-ring and $B$ be an $R$-algebra (in the usual sense) with a
unit element. Unless otherwise specified, we will view $B$ as a unital
algebra, and similarly for algebras with involution. Note that if $B_{0}$ is
$B$ viewed as a non-unital algebra and $B_{1}$ is $B$ viewed as a unital
algebra, then clearly
$\operatorname{gen}_{R}(B_{1})\leqslant\operatorname{gen}_{R}(B_{0})\leqslant\operatorname{gen}_{R}(B_{1})+1$,
and we often have

(1.1) $\operatorname{gen}_{R}(B_{1})=\operatorname{gen}_{R}(B_{0});$

see Lemma 2.2. In particular, equality (1.1) holds for the algebras we
consider in Sections 7 and 8: Azumaya algebras of degree $\geqslant 2$ with
and without involution and octonion algebras. On the other hand, if $B=R$
(viewed as a rank $1$ étale algebra over $R$), then (1.1) fails:
$\operatorname{gen}_{R}(B_{0})=1$, whereas $\operatorname{gen}_{R}(B_{1})=0$.

######  Remark 1.7.

We prove the lower bounds of Theorems 1.3 and 1.5(b) in characteristic $0$
only. The main reason is that our arguments rely on the Affine Lefschetz
Hyperplane Theorem (Theorem 11.1) which requires the characteristic-$0$
assumption. Since we are forced to work in characteristic $0$, we liberally
use non-algebraic techniques in our proofs of Theorems 1.4 and 1.5(b). In
particular, we work with singular cohomology of complex algebraic varieties
and treat the complex points of the algebraic space $U_{r}/G$ as a
differentiable manifold. It is likely that the use of these non-algebraic
techniques can be avoided by replacing singular cohomology with another
oriented (as axiomatized in [Pan03] and [Pan09]), bounded-below,
${{\mathbb{A}}^{1}}$-invariant cohomology theory of smooth varieties, for
instance $\ell$-adic étale cohomology. If one can prove an Affine Lefschetz
Hyperplane Theorem for such a cohomology theory, then one should be able to
establish lower bounds similar to Theorems 1.4 and 1.5(b) in positive
characteristic.

######  Remark 1.8.

The results of Forster [For64] and the first two authors [FR17] assume only
that $R$ is noetherian, whereas Theorem 1.3 and 1.5(a) assume that $R$ is a
finite type $k$-ring for some infinite field $k$. We do not know whether the
upper bounds of Theorem 1.3 and Theorem 1.5(a) remain valid for an arbitrary
noetherian ring $R$.

######  Remark 1.9.

The argument we use to establish the lower bounds of Theorem 1.5(b) does not
extend to the case of quaternion algebras, i.e., when $s=2$. Proposition 13.2
does not hold when $s=2$ which means that our proof of Lemma 14.5 breaks down.
In fact, Lemma 14.5 fails for $s=2$; see Remark 14.6. Nonetheless, the thesis
of W. S. Gant [Gan20, Theorem 5.2] shows that when $s=2$ and $r$ is odd, Lemma
14.5 continues to hold. Arguing as in the proof of Theorem 1.5(b), we obtain
the following:

> If $\operatorname{char}(k)=0$ and $d\geqslant 0$, then there exists a
> finitely generated $k$-ring $R$ of Krull dimension $d$ and an Azumaya
> $R$-algebra $B$ of degree $2$ such that
>
> $\displaystyle\operatorname{gen}_{R}(B)\geqslant
> 2\left\lfloor{\frac{d+2}{4}}\right\rfloor+2.$

The lower bound on $\operatorname{gen}_{R}(B)$ here is the same as that of
Theorem 1.5(b) when $d$ is congruent to $2$ or $3$ modulo $4$, and is lower by
$1$ when $d$ is congruent to $0$ or $1$.

##  2\. Notational conventions

Throughout this paper:

  * •

$k$ will denote a field. All schemes, varieties, algebraic groups, groups
actions and morphisms are assumed to be defined over $k$.

By a variety (or equivalently, a $k$-variety) we shall mean a finite-type
separated and reduced scheme over $k$. An algebraic group is a group scheme of
finite type over $k$.

  * •

$A$ will denote a finite-dimensional $k$-algebra, or more generally, a finite-
dimensional $k$-multialgebra.

  * •

$n$ will denote the dimension of $A$ as a $k$-vector space, unless otherwise
stated.

  * •

$G$ will denote $\operatorname{Aut}(A)$, the automorphism group scheme of $A$
defined over $k$.

By definition, $G$ is a closed subgroup of
$\operatorname{GL}(A)\cong\operatorname{GL}_{n}$. In positive characteristic
$G$ need not be smooth. If $k$ contains at least $8$ elements, then every
affine group scheme of finite type arises as the automorphism group scheme of
some $k$-algebra; see [GP03] in the case $k$ is of characteristic $0$, and
[Mil20] for the general case. The algebras in question may not be associative,
but they are algebras in the usual sense (and not just multialgebras).

  * •

$R$ will denote a $k$-ring, i.e., a commutative associative unital
$k$-algebra.

  * •

$B$ will denote an $R$-form of $A$. This means that there exists a faithfully
flat $R$-ring $S$ for which $A\otimes_{k}S\cong B\otimes_{R}S$ as
$S$-algebras.

  * •

$V_{1}$ will denote the affine space $\mathbb{A}(A)$ underlying our finite-
dimensional $k$-algebra $A$. More generally, $V_{r}$ will denote the $r$-fold
product $V_{1}\times_{k}\ldots\times_{k}V_{1}$ for any $r\geqslant 1$. If $R$
is a $k$-ring, then $R$-points of $V_{r}$ are in a natural bijection with
$r$-tuples of elements in $A_{R}=A\otimes_{k}R$.

  * •

$Z_{r}$ will denote the closed subscheme of $V_{r}$ consisting of $r$-tuples
$a_{1},\ldots,a_{r}\in A$ that do not generate $A$.

More precisely, $Z_{r}$ is defined as follows. Denote the multialgebra
operations on $A$ by $\\{m_{i}:A^{r_{i}}\to A\\}_{i\in I}$ and recall that
$n=\dim_{k}A$. Let $W_{r}$ denote the collection of $\\{m_{i}\\}_{i\in
I}$-monomials in the letters $x_{1},\dots,x_{r}$. That is,
$W_{r}=\bigcup_{t=1}^{\infty}W_{r,t}$, where $W_{r,1}=\\{x_{1},\dots,x_{r}\\}$
and for any $t>1$, $W_{r,t}$ is the set of formal expressions obtained by
substituting elements from $W_{r,t-1}$ in the $\\{m_{i}\\}_{i\in I}$ in every
possible way. Each $w\in W_{r}$ determines a morphism $w:V_{r}\to V_{1}$,
given by substitution. By definition, $\overline{a}=(a_{1},\dots,a_{r})\in
V_{r}(k)=A^{r}$ generates $A$ if and only if there exists
$\overline{w}=(w_{1},\dots,w_{n})\in(W_{r})^{n}$ such that
$w_{1}(\overline{a}),\dots,w_{n}(\overline{a})$ form a $k$-basis of $A$. For
every $n$-tuple of monomials $\overline{w}=(w_{1},\dots,w_{n})\in(W_{r})^{n}$,
denote by $f_{\overline{w}}$ the composition of $(w_{1},\dots,w_{n}):V_{r}\to
V_{n}\cong\operatorname{M}_{n\times n}$ with the determinant morphism
$\operatorname{M}_{n\times n}\to\mathbb{A}^{1}_{k}$. The scheme of non-
generators $Z_{r}$ is defined as the closed subscheme of $V_{r}$ cut by the
equations $f_{\overline{w}}=0$ as $\overline{w}$ ranges over $(W_{r})^{n}$.

######  Remark 2.1.

Note that $Z_{r}$ may not be reduced, and the natural $G$-action on $Z_{r}$
may not restrict to the associated reduced scheme $Z_{r}^{\mathrm{red}}$. For
an example, take $r=1$ and let $A$ be the non-unital algebra
$k[\varepsilon]/(\varepsilon^{2})$ when $k$ is a field of characteristic $2$.

  * •

$U_{r}=V_{r}-Z_{r}$ will denote the open subscheme of $V_{r}$ consisting of
$r$-tuples $a_{1},\ldots,a_{r}\in A$ that generate $A$.

The group scheme $G$ acts on $V_{r}$ from the left by
$g\cdot(a_{1},\dots,a_{r})=(g(a_{1}),\dots,g(a_{r}))$, and the action
restricts to $Z_{r}$ and $U_{r}$.

  * •

$\operatorname{gen}_{R}(B)$ will denote the minimal cardinality of all sets of
elements of $B$ that generate $B$ as an $R$-algebra (or multialgebra).

######  Lemma 2.2.

Let $A$ be a finite-dimensional unital $k$-algebra (in the usual sense, here
we do not allow $A$ to be a general multialgebra), $R$ be a $k$-ring, and $B$
be an $R$-form of $A$. Let $\overline{k}$ be an algebraic closure of $k$ and
suppose $A\otimes_{k}\overline{k}$ does not have an augmentation, i.e., there
does not exist a morphism $A\otimes_{k}\overline{k}\to\overline{k}$ of unital
$\overline{k}$-algebras. Write $B_{1}$, resp. $B_{0}$, to denote $B$
considered as a unital, resp. non-unital, algebra. Then
$\operatorname{gen}_{R}(B_{1})=\operatorname{gen}_{R}(B_{0})$.

###### Proof.

The inequality
$\operatorname{gen}_{R}(B_{1})\leqslant\operatorname{gen}_{R}(B_{0})$ is
obvious, since any generating set for $B_{0}$ also generates $B_{1}$. To prove
the opposite inequality, we argue by contradiction. Assume the contrary: there
exist elements $b_{1},\dots,b_{r}\in B$ that generate $B_{1}$ but not $B_{0}$.
Let $B^{\prime}$ be the non-unital subalgebra generated by
$b_{1},\ldots,b_{r}$. Then $B^{\prime}$ is a proper two-sided ideal of $B_{1}$
and the $R$-algebra $B/B^{\prime}$ is generated as an $R$-module by the
identity element $1_{B}$ of $B$. In other words, there a is unital $R$-algebra
homomorphism $\epsilon:B\to B/B^{\prime}\cong R/I$ for some proper ideal $I$
of $R$.

Since we are assuming that $B$ is an $R$-form of $A$, there exists a
faithfully flat $R$-ring $S$ such that $B\otimes_{R}S$ is isomorphic to
$A_{S}=A\otimes_{k}S$ as an $S$-algebra. Tensoring the map $\epsilon:B\to R/I$
with $S$, we obtain a map $\epsilon_{S}\colon A_{S}\to S/IS$. Tensoring
further with a field $K=S/M$, where $M$ is a maximal ideal of $S$ containing
$IS$, we obtain an augmentation $\epsilon_{K}\colon A_{K}\to K$, where $K$ is
a field containing $k$.

Now let $X$ be the $k$-scheme of unital $k$-algebra homomorphisms from $A$ to
$k$; it is a closed subscheme of the affine space associated to the dual space
of $A$. By what we have shown, $X$ has a $K$-point, namely, the augmentation
$\epsilon_{K}:A_{K}\to K$. We conclude that $X\neq\emptyset$ and hence, $X$
has a $\overline{k}$-point. Equivalently $A_{\overline{k}}$, has an
augmentation $A_{\overline{k}}\to\overline{k}$, contradicting our assumption.
This contradiction shows that
$\operatorname{gen}_{R}(B_{0})=\operatorname{gen}_{R}(B_{1})$. ∎

All $G$-torsors in this paper are _left_ $G$-torsors $\pi:T\to X$ such that
$\pi$ admits a section after base-changing along some faithfully flat quasi-
compact morphism $X^{\prime}\to X$ (that is, we consider left $G$-torsors for
the fpqc topology).

  * •

For a $G$-torsor $T\to\operatorname{Spec}R$ (with
$G=\operatorname{Aut}_{k}(A)$ and $R$ a $k$-ring), we let ${}^{T}\\!A$ denote
the twist of $A$ by $T$.

Recall that ${\,}^{T}\\!A$ is an $R$-form of $A$ defined as follows. Elements
of ${\,}^{T}\\!A$ are $R$-points of the twisted variety ${\,}^{T}\\!V_{1}$
over $\operatorname{Spec}R$, which is the quotient of $T\times V_{1}$ by the
diagonal action of $G$. When $A$ is an algebra (in the usual sense), the
$R$-algebra operations on ${\,}^{T}\\!A$ are given by twisting each of the
following operations on $A$ by $T$:

  * $\circ$

addition, $+\colon V_{1}\times V_{1}\to V_{1}$, given by $(a_{1},a_{2})\mapsto
a_{1}+a_{2}$;

  * $\circ$

scalar multiplication $\mathbb{A}^{1}\times V_{1}\to V_{1}$ given by
$(c,a)\mapsto ca$; and

  * $\circ$

algebra multiplication $\cdot\,\colon V_{1}\times V_{1}\to V_{1}$ given by
$(a_{1},a_{2})\mapsto a_{1}\cdot a_{2}$.

Here $G$ acts trivially on $\mathbb{A}^{1}$. For details, see [Gir71, Section
III.2.3]. The same construction works for multialgebras.

  * •

$\mathrm{H}^{i}(X)$ will denote the singular cohomology of a topological space
$X$ with coefficients in the ring $\mathbb{Z}$.

  * •

$\mathrm{H}^{*}(X)$ will denote the graded ring
$\bigoplus_{i=0}^{\infty}\mathrm{H}^{i}(X)$.

If different coefficients are required, they will be specified, e.g.,
$\mathrm{H}^{i}(X;\mathbb{Q})$.

  * •

If $Y$ is a variety over a field $k$ and $k$ is equipped with an embedding
$k\hookrightarrow\mathbb{C}$, then $Y(\mathbb{C})$ will denote the topological
space underlying the associated analytic space of $Y_{\mathbb{C}}$, [Ser55,
Sec. 2]. If $Y$ is such a variety, then notation $\mathrm{H}^{i}(Y)$ will be
taken to mean $\mathrm{H}^{i}(Y(\mathbb{C}))$.

  * •

If $\Gamma$ is a topological group, then $E\Gamma\to B\Gamma$ will denote a
universal principal $\Gamma$-bundle, and if $\Gamma$ acts on a space $X$, then
$\mathrm{H}^{i}_{\Gamma}(X)$ will denote the Borel equivariant cohomology
group $\mathrm{H}^{i}((X\times E\Gamma)/\Gamma)$, [Hsi75, Chapter III].

The notation $\mathrm{H}^{i}_{G}(Y)$ will be used in place of
$\mathrm{H}^{i}_{G(\mathbb{C})}(Y(\mathbb{C}))$ when $G$ is a linear algebraic
group acting on a variety $Y$ over a field $k\hookrightarrow\mathbb{C}$.

  * •

The notation ${\mathbb{G}_{m}}$ is reserved for the multiplicative group-
scheme $\operatorname{GL}_{1}$.

##  3\. Preliminaries on schemes of generators and non-generators

Let $A$ be a finite-dimensional (multi)algebra over a field $k$. Recall from
Section 2 that we denote the affine space of ordered $r$-tuples of elements in
$A$ by $V_{r}$, the closed subscheme of $r$-tuples not generating $A$ by
$Z_{r}\subset V_{r}$ and the open subscheme of $r$-tuples of generators by
$U_{r}=V_{r}-Z_{r}$. In this section, we will discuss some basic properties of
these schemes. We write $A_{R}$ for $A\otimes_{k}R$.

######  Lemma 3.1.

Let $R$ be a $k$-ring (not necessarily of finite type). Then $U_{r}(R)$ is the
set of $r$-tuples $(b_{1},\dots,b_{r})\in V_{r}(R)=A_{R}^{r}$ that generate
$A_{R}$ as an $R$-algebra.

###### Proof.

Let $\overline{b}=(b_{1},\dots,b_{r})\in V_{r}(R)$ and let
$t:\operatorname{Spec}R\to V_{r}$ be the corresponding morphism.

Suppose that $\overline{b}$ generates $A_{R}$. Let $P\in\operatorname{Spec}R$,
let $A(P)=A\otimes_{k}\mathrm{Frac}(R/P)$ and let $b_{i}(P)$ denote the image
of $b_{i}$ in $A(P)$. Then $\overline{b}(P):=(b_{1}(P),\dots,b_{r}(P))$
generates $A(P)$. By the construction of $Z_{r}$ in Section 2, there is
$\overline{w}\in(W_{r})^{n}$ such that $f_{\overline{w}}(\overline{b}(P))\neq
0$ in $\mathrm{Frac}(R/P)$. This means that the image of $t$ does not meet the
$P$-fibre of $Z_{r}$. As this holds for all $P\in\operatorname{Spec}R$, the
image of $t:\operatorname{Spec}R\to V_{r}$ is contained $V_{r}-Z_{r}=U_{r}$
and $\overline{b}\in U_{r}(R)$.

Conversely, assume $\overline{b}\in U_{r}(R)$. Then for every
$P\in\operatorname{Spec}R$, we have $t(P)\notin Z_{r}$. This means that there
is $\overline{w}\in(W_{r})^{n}$ such that
$f_{\overline{w}}(\overline{b}(P))\neq 0$, hence $\overline{b}(P)$ generates
$A(P)$. This holds for all $P\in\operatorname{Spec}R$, so by [FR17, Lemma
2.1], $\overline{b}$ generates $A_{R}$. ∎

Recall from Section 2 that the group scheme $G=\operatorname{Aut}_{k}(A)$ acts
on $V_{r}$ from the left, and the action restricts to $Z_{r}$ and $U_{r}$.

######  Lemma 3.2.

The action of $G$ on $U_{r}$ is scheme-theoretically free, i.e., the morphism

(3.1) $\Phi:G\times U_{r}\longrightarrow U_{r}\times U_{r},$

given by
$\Phi(g,(a_{1},\ldots,a_{r}))=((g(a_{1}),\ldots,g(a_{r})),(a_{1},\ldots,a_{r}))$
is a closed embedding.

###### Proof.

By [Gro64, Cor. 18.12.6], showing that $\Phi$ is a closed embedding is
equivalent to showing that $\Phi$ is a proper monomorphism. To see that $\Phi$
is a monomorphism, it suffices to test that

(3.2) $\Phi(R):G(R)\times U_{r}(R)\to U_{r}(R)\times U_{r}(R)$

is injective for every $k$-ring $R$. Here, thanks to Lemma 3.1, $U_{r}(R)$ is
the set of all $r$-tuples $(b_{1},\dots,b_{r})$ of generators for $A_{R}$. The
action of $G(R)=\operatorname{Aut}_{R}(A_{R})$ on $U_{r}(R)$ is free because
$g\in G(R)$ fixes $(b_{1},\dots,b_{r})\in U_{r}(R)$ if and only if $g$ fixes
the $R$-algebra generated by the $b_{i}$, i.e., the entire algebra $A_{R}$.
The element $g$ fixes the entire algebra $A_{R}$ if and only if $g$ is the
identity automorphism. It follows that for any $R$, the map $\Phi(R)$ in (3.2)
is an injection, as claimed.

It remains to show that $\Phi$ is a proper map. We use the valuative criterion
for properness. Suppose $R$ is a valuation ring over $k$ with fraction field
$K$, and let $\varphi$, $\psi$ be morphisms as in the following commutative
diagram. We need to show the existence of the dotted arrow.

${\operatorname{Spec}K}$${G\times U_{r}}$${\operatorname{Spec}R}$${U_{r}\times
U_{r}.}$$\scriptstyle{\varphi}$$\scriptstyle{\Phi}$$\scriptstyle{\psi}$

We remark that $A_{R}\subset A_{K}$, and $U_{r}(R)\subset U_{r}(K)$. The arrow
$\varphi$ represents an automorphism $g\in G(K)$ and an $r$-tuple
$(a_{1},\dots,a_{r})\in U_{r}(K)$; the arrow $\psi$ represents a pair of
$r$-tuples in $U_{r}(R)$, and commutativity of the diagram implies that these
$r$-tuples are $(ga_{1},\dots,ga_{r})$ and $(a_{1},\dots,a_{r})$. In
particular, both $(ga_{1},\dots,ga_{r})$ and $(a_{1},\dots,a_{r})$ lie in
$U_{r}(R)$. We must show that $g$ is defined over $R$. If $e_{1},\ldots,e_{n}$
is a $k$-basis of $A$, we claim that $g(e_{i})\in A_{R}$ for every
$i=1,\ldots,n$. This will prove that $g\in\operatorname{GL}_{n}(R)\cap
G(K)=G(R)$.

To prove the claim, note that since $(a_{1},\ldots,a_{r})\in U_{r}(R)$, it is
possible to find polynomials $f_{1},\dots,f_{n}$ in $r$ non-commuting
variables with coefficients in $R$ such that

$f_{i}(a_{1},\dots,a_{r})=e_{i}\quad\forall i\in\\{1,\dots,n\\}.$

(When $A$ is a multialgebra, the $f_{i}$ should be taken to be formal linear
combinations of members of $W_{r}$ from Section 2.) Now consider the result of
applying $g$ to $e_{i}$. Since $g$ is known to lie in $G(K)$, $g(e_{i})$ is a
priori an element of $A_{K}$, but using the polynomials $f_{i}$, we see that

$g(e_{i})=g(f_{i}(a_{1},\dots,a_{r}))=f_{i}(ga_{1},\dots,ga_{r}),$

so $g(e_{i})$ actually lies in $A_{R}$, as claimed. This allows us to produce
a lift, so the valuative criterion for properness is satisfied. ∎

######  Remark 3.3.

Regard $U_{r}/G$ as a sheaf on the large fppf site of $\operatorname{Spec}k$.
Since $G$ is a linear algebraic group, being by definition a closed subgroup
of $\operatorname{GL}(A)$, we may regard $U_{r}/G$ as an Artin stack.
Moreover, since the $G$-action on $U_{r}$ is free, $U_{r}/G$ is an algebraic
space over $k$; see [Sta20, Tag 04SZ].

##  4\. Generators and equivariant maps

Recall from the introduction that an $R$-form of $A$ is an $R$-algebra $B$ for
which there exists a faithfully flat $R$-ring $S$ such that
$A\otimes_{k}S\cong B\otimes_{R}S$ as $S$-algebras. It is well known that the
category of $G$-torsors over $\operatorname{Spec}R$ is equivalent to the
category of $R$-forms of $A$, functorially in $R$. The equivalence is given by
sending an $R$-algebra $B$ to
$\operatorname{Iso}_{R\text{-}\mathrm{alg}}(A_{R},B)$, regarded as an
$R$-scheme and endowed with the _left_ $G_{R}$-action given by
$g\cdot\psi=\psi\circ g^{-1}$ on sections. Conversely, a $G$-torsor
$T\to\operatorname{Spec}R$ corresponds to the twisted algebra ${\,}^{T}\\!A$
defined in Section 2.

######  Proposition 4.1.

Let $R$ be a $k$-ring (not necessarily of finite type), let $B$ be an $R$-form
of $A$, and let $T\to\operatorname{Spec}R$ be its associated $G$-torsor. Then
there is a natural (in $R$) bijective correspondence between the following
sets:

  1. (a)

$G$-equivariant $k$-morphisms from $T$ to $U_{r}$, and

  2. (b)

$r$-tuples $(b_{1},\dots,b_{r})\in B^{r}$ generating $B$ as an $R$-algebra.

Our proof of Proposition 4.1 will rely on the following lemmas.

######  Lemma 4.2.

Let $X$ and $Y$ be $k$-schemes, $G$ a group scheme over $k$, and

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\textstyle{X}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\pi}}$

be a cartesian diagram over $k$, where $\alpha$ and $\beta$ are $G$-torsors
and $\pi$ is $G$-equivariant. Then there is a natural (in the morphism $X\to
Y$) bijection between sections $\overline{f}\colon X\to Y$ of $\overline{\pi}$
and $G$-equivariant sections $f\colon V\to W$ of $\pi$ such that
$\overline{f}\circ\alpha=\beta\circ f$ whenever $f$ corresponds to
$\overline{f}$.

###### Proof.

It is enough to prove the corresponding statement for sheaves over the large
fpqc site of $\operatorname{Spec}k$. To that end, it is enough to show that
for every $k$-scheme $S$, there is a unique (and hence natural in $S$, $X$,
$Y$) bijection between $G(S)$-equivariant $\pi(S)$-sections $f:V(S)\to W(S)$
and $\overline{\pi}(S)$-sections $\overline{f}:X(S)\to Y(S)$ such that
$\overline{f}\circ\alpha(S)=\beta(S)\circ f$ whenever $f$ corresponds to
$\overline{f}$. That is, it is enough to prove the set-theoretic analogue of
the lemma, which is straightforward. ∎

######  Lemma 4.3.

Let $R$ be a $k$-ring and $T\to\operatorname{Spec}R$ be a $G$-torsor. Then
there is a natural isomorphism between the twisted $R$-(multi)algebra
$B={\,}^{T}A$ and the $R$-(multi)algebra $\operatorname{Mor}_{G}(T,V_{1})$ of
$G$-equivariant morphisms $T\to V_{1}$. Here “natural” means that if $S$ is an
$R$-ring, then this isomorphism is compatible with the base change from $R$ to
$S$.

The (multi)algebra structure on $\operatorname{Mor}_{G}(T,V_{1})$ is induced
from the (multi)algebra structure on $A$. For example, to add $G$-equivariant
morphisms $\varphi_{1}\colon T\to V_{1}$ and $\varphi_{2}\colon T\to V_{1}$,
we compose $(\varphi_{1},\varphi_{2})\colon T\to V_{1}\times V_{1}$ with the
addition map $+\colon V_{1}\times V_{1}\to V_{1}$.

###### Proof.

Consider the cartesian diagram

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T\times
V_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{r}}$$\scriptstyle{\pi_{r}}$$\textstyle{\operatorname{Spec}R}$$\textstyle{{\,}^{T}V_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\pi_{r}}}$

where $\pi_{r}\colon T\times V_{r}\to T$ is projection to the first component.
Now set $r=1$. A $G$-equivariant morphism $\varphi\colon T\to V_{1}$ gives
rise to a $G$-equivariant section $(\mathrm{id},\varphi)\colon T\to T\times
V_{1}$ of $\pi_{1}$. The latter descends to a section
$b\colon\operatorname{Spec}R\to{\,}^{T}V_{1}$, which we view as an element of
$B$. This gives us a map $F\colon\operatorname{Mor}_{G}(T,V_{1})\to B$ taking
$\varphi$ to $b$. By Lemma 4.2, $F$ is an isomorphism of sets.

Using the definition of the (multi)algebra structure on $B={\,}^{T}A$, one
verifies that $F$ is in fact a homomorphism, and thus an isomorphism, of
(multi)algebras. For example, to show that $F$ is a homomorphism of additive
groups, we consider the diagram:

$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T\times
V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(\mathrm{id},+)}$$\textstyle{T\times
V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{\operatorname{Spec}R}$$\textstyle{{\,}^{T}V_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\pi_{2}}}$$\scriptstyle{+}$$\textstyle{{\,}^{T}V_{1}}$

Note that by definition, $V_{2}=V_{1}\times V_{1}$, and thus
${\,}^{T}V_{2}={\,}^{T}V_{1}\times{\,}^{T}V_{1}$. Similar arguments with $+$
replaced by scalar multiplication and (multi)algebra operations in $A$ show
that $F$ is a homomorphism of (multi)algebras.

Finally, to show that $F$ is natural in $R$, we consider a $k$-ring morphism
$R\to S$ and examine the diagram

$\textstyle{T_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{S}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{T\times
V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\scriptstyle{\pi_{1}}$$\textstyle{\operatorname{Spec}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Spec}R}$$\textstyle{{\,}^{T}V_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\pi_{1}},}$

where the square on the left is Cartesian. Since
${\,}^{T_{S}}V_{1}\cong({\,}^{T}V_{1})\times_{\operatorname{Spec}R}\operatorname{Spec}S$,
the $S$-points of the $R$-scheme ${\,}^{T}V_{1}$ are in a natural bijection
with the $S$-points of the $S$-scheme ${\,}^{T_{S}}V_{1}$. ∎

###### Proof of Proposition 4.1.

By Lemma 4.3, an $r$-tuple $b=(b_{1},\ldots,b_{r})$ of elements of
$B={\,}^{T}A$ may be viewed as an $r$-tuple of $G$-equivariant maps
$\varphi_{1},\ldots,\varphi_{r}\colon T\to V_{1}$ or equivalently, as a
$G$-equivariant morphism $\varphi=(\varphi_{1},\ldots,\varphi_{r})\colon T\to
V_{r}=(V_{1})^{r}$. It remains to prove the following.

Claim. $b_{1},\ldots,b_{r}$ generate $B$ as an $R$-algebra if and only if
$\varphi(T)$ is contained in $U_{r}$.

To prove this claim, let $S/R$ be a faithfully flat ring extension that splits
$T$. By functoriality ${\,}^{T_{S}}A=({\,}^{T}A)\otimes_{R}S$. Since $S$ is
faithfully flat over $R$, the elements $b_{1},\ldots,b_{r}$ generate
${\,}^{T}A$ over $R$ if and only if they generate ${\,}^{T_{S}}A$ over $S$.
Without loss of generality, we may replace $R$ by $S$ and thus assume that $T$
is split over $R$, i.e., that $T$ has a section
$\psi\colon\operatorname{Spec}R\to T$. This induces an isomorphism
$G\times\operatorname{Spec}R\to T$ given sectionwise by $(g,x)\mapsto
g\cdot\psi(x)$, and an associated algebra isomorphism
$\operatorname{Mor}_{G}(T,V_{1})\cong
B={}^{T}A\cong{}^{G\times\operatorname{Spec}R}A=A_{R}$ which, after unfolding
the definitions, assumes the following simple form: a $G$-equivariant morphism
$\varphi_{i}\colon T\to V_{1}$ corresponds to the element
$b_{i}=\varphi_{i}\circ\psi\colon\operatorname{Spec}R\to V_{1}$ of $A_{R}$. By
Lemma 3.1, $b_{1},\ldots,b_{r}$ generate $B=A_{R}$ if and only if the image of

$b=(b_{1},\ldots,b_{r})=(\varphi_{1}\circ\psi,\ldots,\varphi_{r}\circ\psi)=\varphi\circ\psi\colon\operatorname{Spec}R\to
V_{r}$

lies in $U_{r}$. Since $G\times_{k}\operatorname{Spec}R$ is isomorphic to $T$
via $(g,x)\mapsto g\cdot\psi(x)$, $\varphi\colon T\to V_{r}$ is a
$G$-equivariant map, and $U_{r}$ is a $G$-invariant subvariety of $V_{r}$, we
see that $\varphi(T)$ lies in $U_{r}$ if and only if
$(\varphi\circ\psi)(\operatorname{Spec}R)=b(\operatorname{Spec}R)$ lies in
$U_{r}$. This completes the proof of the claim and thus of Proposition 4.1. ∎

##  5\. Versal group actions

Let $k$ be a field, $G$ be a group scheme over $k$, and ${\mathscr{C}}$ be a
class of $k$-schemes. A $G$-scheme is a $k$-scheme equipped with a $G$-action
$G\times_{k}X\to X$. We will say that a $G$-scheme $X$ is _versal for
${\mathscr{C}}$_ if for every $G$-torsor $\tau\colon T\to Y$ with
$Y\in{\mathscr{C}}$, there exists a $G$-equivariant $k$-morphism $T\to X$. We
will be particularly interested in the the class ${\mathscr{C}}$ of affine
$k$-schemes $Y=\operatorname{Spec}R$, where $R$ is a finite type $k$-algebra
of Krull dimension $\leqslant d$. If $X$ is versal with respect to this class
of $k$-schemes, we will say that $X$ is _$d$ -versal_.

######  Remark 5.1.

The notion of versality has been previously studied in the case where
${\mathscr{C}}$ consists of schemes of the form $\operatorname{Spec}K$ for
some field $K$ containing $k$; see, [Ser03, Section 5] or [DR15]. Our
definition of $d$-versality here is analogous to “weak versality” in [DR15].
For notational simplicity we will use the term “$d$-versal” instead of “weakly
$d$-versal”, even though the latter may be more consistent with the
literature.

######  Proposition 5.2.

Let $G$ be an affine algebraic group over an infinite field $k$,
$\rho:G\to\operatorname{GL}(V)$ be a faithful finite-dimensional
representation, and $Z$ be a closed $G$-subscheme of $V$ of dimension
$\leqslant\dim V-(d+1)$ for some integer $d\geqslant 0$. Then $U=V-Z$ is a
$d$-versal $G$-scheme.

######  Remark 5.3.

It is well known that for any finite type affine group scheme $G$ over $k$ and
any $d\geqslant 0$, there exists a faithful representation
$G\to\operatorname{GL}(V)$ and a closed $G$-invariant subvariety $Z\subseteq
V$ of codimension $>d$ such that $V-Z$ is the total space of a torsor
$(V-Z)\to X$ for some scheme $X$; see [EG98, Lemma 9]. In fact, we may even
assume that $X$ is a quasi-projective variety; see [Tot99, Remark 1.4]. In
particular, Proposition 5.2 shows that $d$-versal $G$-schemes with a free
action of $G$ exist for every $d\geqslant 0$ and every linear algebraic group
$G$.

######  Remark 5.4.

Proposition 5.2 also holds for $Z=\emptyset$ with the convention that
$\dim\emptyset=-\infty$. More generally, if $0\not\in Z$, then $V-Z$ is
$d$-versal for every $d\geqslant 0$ simply because every $G$-torsor
$T\to\operatorname{Spec}(R)$ admits the zero morphism $T\to V$ which is
obviously $G$-equivariant.

Our proof of Proposition 5.2 will rely on the following lemma. Denote the
affine space of $a\times b$-matrices over $k$ by $\operatorname{M}_{a\times
b}$. If $b\geqslant a$, we will denote the Zariski open subvariety of $a\times
b$ matrices of rank $a$ by $\operatorname{M}_{a\times b}^{0}$. We regard
$\operatorname{GL}_{n}$ as an algebraic group over $k$.

######  Lemma 5.5.

Let $G$ be a closed subgroup scheme of $\operatorname{GL}_{n}$, $R$ be a
noetherian $k$-ring, and $\tau\colon T\to\operatorname{Spec}R$ be a
$G$-torsor. Then there exists an integer $m\geqslant n$ and a $G$-equivariant
$k$-morphism $F\colon T\to\operatorname{M}_{n\times m}^{0}$. Here $G$ acts on
$\operatorname{M}_{n\times m}^{0}$ via multiplication on the left.

###### Proof.

We will view the $n$-dimensional $k$-vector space $A=k^{n}$ as a (non-unital)
algebra with trivial multiplication, $a_{1}\cdot a_{2}=0$ for every
$a_{1},a_{2}\in A$. Then $\operatorname{M}_{n\times m}$ is the variety $V_{m}$
of $m$-tuples of elements of $A$ (each column of $\operatorname{M}_{n\times
m}$ is identified with $A$), $\operatorname{M}_{n\times m}^{0}$ is precisely
the variety $U_{m}$ of $m$-tuples of generators of $A$, and
$\operatorname{Aut}_{k}(A)=\operatorname{GL}(A)=\operatorname{GL}_{n}$.
Twisted $R$-forms of $A$ are precisely the projective $R$-modules $B$ of rank
$n$.

It is well known that every projective $R$-module of finite rank is generated
by finitely many elements. In particular, consider the
$\operatorname{GL}_{n}$-torsor
$T^{\prime}=T\times^{G}\operatorname{GL}_{n}\to\operatorname{Spec}R$ and set
$B={\,}^{T^{\prime}}\\!A$ to be the $R$-module corresponding to $T^{\prime}$.
Suppose $B$ is generated by $m$ elements. Then by Proposition 4.1, there exist
a $\operatorname{GL}_{n}$-equivariant morphism $F\colon T^{\prime}\to
U_{m}=\operatorname{M}_{n\times m}^{0}$. Restricting $F$ to $T\subset
T^{\prime}$, we obtain a desired $G$-equivatiant map
$T\to\operatorname{M}_{n\times m}^{0}$. ∎

###### Proof of Proposition 5.2.

Recall that $\rho:G\to\operatorname{GL}(V)$ is a closed embedding [Mil17,
Theorem 5.34]. Let $R$ be a finite type $k$-ring of Krull dimension $\leqslant
d$, and let $\tau\colon T\to\operatorname{Spec}R$ be a $G$-torsor. Our goal is
to establish the existence of a $G$-equivariant morphism $f:T\to U$.

Let $n=\dim_{k}V$. We begin by identifying $V$ with $\operatorname{M}_{n\times
1}$, $\operatorname{GL}(V)$ with $\operatorname{GL}_{n}$ and $G$ with a closed
subgroup of $\operatorname{GL}_{n}$ via $\rho$. Observe that
$\operatorname{GL}(V)=\operatorname{GL}_{n}$ acts on
$\operatorname{M}_{n\times m}$ by multiplication on the left, and so does
$G\subset\operatorname{GL}_{n}$, whereas $\operatorname{GL}_{m}$ acts on
$\operatorname{M}_{n\times m}$ via multiplication on the right. These actions
commute, and both of them restrict to $\operatorname{M}_{n\times m}^{0}$. Note
also that $\operatorname{M}_{n\times m}^{0}$ is a homogeneous space under the
action of $\operatorname{GL}_{m}$ for all $m\geqslant n$.

Lemma 5.5 tells us that there exists a $G$-equivariant morphism $F\colon
T\to\operatorname{M}_{n\times m}^{0}$ for some integer $m\geqslant n$. Let
$Z^{\prime}$ be the preimage of $Z\subset V$ under the projection map
$p\colon\operatorname{M}_{n\times m}^{0}\to V$ to the first column. (Recall
that each column of $\operatorname{M}_{n\times m}$ is $G$-equivariantly
identified with $V$.) In other words, $Z^{\prime}$ consists of $n\times m$
matrices of rank $n$ whose first column lies in $Z$. For
$s\in\operatorname{GL}_{m}({k})$, let $r_{s}:\operatorname{M}_{n\times
m}^{0}\to\operatorname{M}_{n\times m}^{0}$ denote the $G$-equivariant morphism
given by multiplying with $s$ on the right, and let $sT$ denote $T$, regarded
as an $\operatorname{M}_{n\times m}^{0}$-scheme via $r_{s}\circ F$.

Claim. Let $X$ and $Y$ be $G$-schemes and let $f:X\to\operatorname{M}_{n\times
m}^{0}$ and $g:Y\to\operatorname{M}_{n\times m}^{0}$ be $G$-equivariant
morphisms. Suppose that

$\dim X+\dim Y<mn+\dim G.$

Then there exists a non-empty Zariski open subset
$W_{X,Y}\subset\operatorname{GL}_{m}$ such that
$sX\times_{\operatorname{M}^{0}_{n\times m}}Y=\emptyset$ for every $s\in
W_{X,Y}({k})$.

Assume for a moment that this claim is established. Let
$f:X\to\operatorname{M}_{n\times m}^{0}$ be the morphism
$F:T\to\operatorname{M}_{n\times m}^{0}$ and let
$g:Y\to\operatorname{M}_{n\times m}^{0}$ be the closed immersion
$Z^{\prime}\to\operatorname{M}_{n\times m}^{0}$. Then

$\dim X=\dim\,T=\dim\operatorname{Spec}(R)+\dim G\leqslant d+\dim G,$

whereas

$\dim Y=\dim Z+n(m-1)<(\dim V-d)+n(m-1)=n-d+n(m-1)=mn-d.$

Thus, $\dim X+\dim Y<mn+\dim G$, and the claim applies to this choice of $X$
and $Y$. Since $\operatorname{GL}_{m}$ is rational, and $k$ is an infinite
field, $W_{X,Y}$ has a $k$-point $s\in W_{X,Y}(k)$. After replacing $F$ by
$r_{s}\circ F$, we may assume that the image of $T$ in
$\operatorname{M}_{n\times m}^{0}$ is disjoint from $Z^{\prime}$. Composing
$F$ with the projection $p\colon\operatorname{M}_{n\times m}^{0}\to V$ to the
first column, we obtain a desired $G$-equivariant morphism $f=p\circ F\colon
T\to U$.

It thus remains to prove the claim. Consider the natural projection $\pi\colon
I\to\operatorname{GL}_{m}$, where $I\subset\operatorname{GL}_{m}\times X\times
Y$ is the incidence scheme consisting of triples $(u,x,y)$ such that
$f(x)\cdot u=g(y)$. We take $W_{X,Y}$ to be the complement of the Zariski
closure of $\pi(I)$ in $\operatorname{GL}_{m}$. By definition, $W_{X,Y}$ is a
Zariski open subset of $\operatorname{GL}_{mn}$ defined over $k$. We only need
to show that it is non-empty, or equivalently, that $\pi$ is not dominant.
Since $\pi$ is quasi-compact, by [Gro65, Proposition 2.3.7], we may pass to
the algebraic closure of $k$ in order to check this, and thus assume that $k$
is algebraically closed.

Since $k$ is algebraically closed, $G^{\mathrm{red}}$ is a subgroup scheme of
$G$ [Mil17, Corollary 1.39]. It is harmless to replace $G$ with the identity
connected component of $G^{\mathrm{red}}$, so assume $G$ is reduced and
connected. This means that $X^{\mathrm{red}}$ and $Y^{\mathrm{red}}$ are
$G$-varieties [Bri17, Proposition 2.5.1(1)], so we may replace $X$ and $Y$
with their reductions. We may also assume that $X$ and $Y$ are irreducible.
Indeed, if $X_{1},\ldots,X_{\alpha}$ are the irreducible components of $X$,
and $Y_{1},\ldots,Y_{\beta}$ are the irreducible components of $Y$, then

$W_{X,Y}=\bigcap_{i=1}^{\alpha}\bigcap_{j=1}^{\beta}\,W_{X_{i},Y_{j}}.$

As $\operatorname{GL}_{m}$ is irreducible, it is enough to show that each
$W_{X_{i},Y_{j}}$ is nonempty. Note that each $X_{i}$ and each $Y_{j}$ is a
$G$-variety because $G$ is connected.

Write $M=\operatorname{M}^{0}_{n\times m}$. At this point, we may apply the
Kleiman Transversality Theorem, see [Kle74, Corollary 4(i)] or [Har77, Theorem
10.8], which tells us that there exists an open dense subset $W^{\prime}$ of
$\operatorname{GL}_{m}$ such that for every $s\in W^{\prime}(k)$, the fibre
product $sX\times_{M}Y$ is either

  1. (i)

empty, or

  2. (ii)

nonempty and equidimensional of dimension $\dim X+\dim Y-\dim M<\dim G$.

Since the $\operatorname{GL}_{m}$-action and the $G$-action on
$\operatorname{M}_{n\times m}^{0}$ commute, the scheme $sX\times_{M}Y$ admits
a $G$-action, and since $f$ and $g$ are $G$-equivariant, the morphism
$sX\times_{M}Y\to\operatorname{M}^{0}_{n\times m}$ is also $G$-equivariant.
The left $G$-action on $\operatorname{M}_{n\times m}^{0}$ has trivial
stabilizers, so the same applies to $sX\times_{M}Y$. Thus, if $sX\times_{M}Y$
is nonempty, then it is of dimension $\geq\dim G$, meaning that (ii) is
impossible. Consequently, $sX\times_{M}Y=\emptyset$ for every $s\in
W^{\prime}(k)$, which means that $W^{\prime}\subseteq W_{X,Y}$. Since
$W^{\prime}$ is dense in $\operatorname{GL}_{m}$, we have
$W_{X,Y}\neq\emptyset$, as desired. ∎

##  6\. Proof of Theorem 1.3

With Propositions 4.1 and 5.2 at hand, we can deduce the following theorem,
from which all of our subsequent upper bounds will follow. The notation is as
in Section 2.

######  Theorem 6.1.

Let $A$ be a finite-dimensional algebra over an infinite field $k$,
$r\in\mathbb{N}\cup\\{0\\}$, and $c_{A}(r)$ be the codimension of $Z_{r}$ in
$V_{r}$, i.e., $c_{A}(r)=r\dim_{k}A-\dim Z_{r}$. Let $R$ be a $k$-ring of
Krull dimension $d$. If $c_{A}(r)>d$, then $\operatorname{gen}_{R}(B)\leqslant
r$ for any $R$-form $B$ of $A$.

###### Proof.

Assume that $c_{A}(r)>d$. Then Proposition 5.2, with $V=V_{r}$ and $Z=Z_{r}$,
tells us that $U_{r}$ is $d$-versal. Let $R$ be a $k$-ring of Krull dimension
$\leqslant d$, let $B$ be an $R$-form of $A$ and $T\to\operatorname{Spec}R$ be
the $G$-torsor associated to $B$. Since $U_{r}$ is $d$-versal, there exists a
$G$-equivariant morphism $T\to U_{r}$, and by Proposition 4.1, this means that
$B$ can be generated by $r$ elements, i.e.,
$\operatorname{gen}_{R}(B)\leqslant r$. ∎

We now proceed with the proof of Theorem 1.3. Let $\overline{k}$ be an
algebraic closure of $k$ and let $n_{\max}$ be the largest possible dimension
of a proper $\overline{k}$-subalgebra of $A\otimes_{k}\overline{k}$. Theorem
1.3 is an immediate consequence of Theorem 6.1 in combination with part (b) of
the following lemma.

######  Lemma 6.2.

(a) $\dim Z_{r}\geqslant n_{\max}r$. (b) If $r>n_{\max}$, then $\dim
Z_{r}\leqslant n_{\max}r+n_{\max}(n-n_{\max})$.

###### Proof.

Passing from $k$ to $\overline{k}$ does not change $\dim Z_{r}$. Hence, we
assume that $k$ is algebraically closed. We will not distinguish between
varieties and their sets of $k$-points.

(a) Let $A_{\max}$ be a proper subalgebra of $A$ of maximal dimension
$n_{\max}$. If $a_{1},\ldots,a_{r}\in A_{\max}$, then the subalgebra generated
by $a_{1},\ldots,a_{r}$ is contained in $A_{\max}$ and hence the $r$-tuple
$(a_{1},\ldots,a_{r})$ lies in $Z_{r}$. Thus $\dim
Z_{r}\geqslant\dim(\underbrace{A_{\max}\times\ldots\times A_{\max}}_{\text{$r$
times}})=n_{\max}r$.

(b) Any $r$-tuple $a_{1},\ldots,a_{r}\in A$ that spans a linear subspace of
dimension $>n_{\max}$ in $A$ automatically generates $A$ as a $k$-algebra.
Thus,

(6.1)
$Z_{r}^{\mathrm{red}}\subset\operatorname{Rank}_{r,0}(A)\cup\operatorname{Rank}_{r,1}(A)\cup\ldots\cup\operatorname{Rank}_{r,n_{\max}}(A),$

where here, if $V$ is an $n$-dimensional $k$-vector space, then
$\operatorname{Rank}_{r,s}(V)$ stands for the subvariety of $V^{r}$ consisting
of $r$-tuples $(v_{1},\ldots,v_{r})$ such that
$\dim\operatorname{Span}_{k}\\{v_{1},\ldots,v_{r}\\}=s$. In our case, $V=A$,
but from this point on the algebra structure of $A$ will play no role in the
proof. To compute the dimension of $\operatorname{Rank}_{r,s}(V)$, let
$\operatorname{Gr}(V,s)$ denote the Grassmannian of $s$-dimensional subspaces
of $V$ and consider the natural morphism

$\operatorname{Rank}_{r,s}(V)\to\operatorname{Gr}(V,s)$

sending $(v_{1},\ldots,v_{r})$ to
$\operatorname{Span}_{k}\\{v_{1},\ldots,v_{r}\\}$. The fibres of this map are
readily seen to be irreducible of dimension $rs$; hence,
$\operatorname{Rank}_{r,s}(V)$ is irreducible of dimension

$\dim\operatorname{Rank}_{r,s}(V)=\dim\operatorname{Gr}(V,s)+rs=sr+(n-s)s.$

In view of (6.1), it remains to show that the maximal value of the function

$f(x)=xr+x(n-x)$

on the interval $[0,n_{\rm max}]$ is attained for $x=n_{\rm max}$. Since
$n>n_{\rm max}$ and $r>n_{\rm max}$, we have

$f^{\prime}(x)=r+n-2x=(n-x)+(r-x)>0$

for any $x\in[0,n_{\rm max}]$, and the desired conclusion follows. This
completes the proof of Lemma 6.2 and thus of Theorem 1.3. ∎

######  Remark 6.3.

If $nr-\dim Z_{r}>d$, then by Lemma 6.2(a), $(n-n_{\max})r>d$ or equivalently,
$r>\dfrac{d}{n-n_{\max}}$. This shows that the best upper bound we can hope to
deduce from Theorem 6.1 is of the form

$\operatorname{gen}(B)<\frac{d}{n-n_{\max}}+c,$

where $B$ is an arbitrary $R$-form of $A$, $R$ is a $k$-ring of Krull
dimension $d$, and $c$ is a constant which depends only on $A$ and not on $d$.
In Theorem 1.3, $c=n_{\max}$. In Sections 7 and 8 we will find a smaller
constant term $c$ for specific types of algebras. The linear term
$\dfrac{d}{n-n_{\max}}$ in the upper bound of Theorem 1.3 cannot be sharpened
by this method.

##  7\. Upper bounds on the number of generators for Azumaya algebras

Throughout this section, $\operatorname{M}_{s}$ denotes the affine $k$-space
of $s\times s$ matrices.

###  7.1. Proof of Theorem 1.5(a)

Our proof will rely on Theorem 6.1 with $A=\operatorname{M}_{s}(k)$, the
algebra of $s\times s$ matrices over $k$. Recall that Azumaya algebras of
degree $s$ over $R$ are precisely the $R$-forms of $\operatorname{M}_{s}(k)$.
By definition,
$Z_{r}\subset\operatorname{M}_{s}\times\ldots\times\operatorname{M}_{s}$ ($r$
times) consists of $r$-tuples $(a_{1},\ldots,a_{r})$ of $s\times s$ matrices
which do not generate $\operatorname{M}_{s}(k)$ as a $k$-algebra. Our goal now
is to compute the dimension of $Z_{r}$. In order to do so, we may pass to the
algebraic closure of $k$ and thus assume that $k$ is algebraically closed.
Under this assumption, we may apply Burnside’s theorem: $Z_{r}^{\mathrm{red}}$
is the union of the $s-1$ subvarieties $X_{1},\ldots,X_{s-1}$ of
$\operatorname{M}_{s}^{r}$ determined by

(7.1)
$X_{i}(k)=\left\\{(a_{1},...,a_{r})\,\bigg{|}\,\begin{array}[]{c}\text{$a_{1},\dots,a_{r}\in\operatorname{M}_{s}(k)$
have a common}\\\ \text{$i$-dimensional invariant
subspace}\end{array}\right\\}.$

For simple proofs of Burnside’s theorem, see [Lam98] or [LR04].

######  Proposition 7.1.

(a) For every $i=1,\ldots,s-1$, the variety $X_{i}$ is a closed irreducible
subvariety of $\operatorname{M}_{s}^{r}$ of dimension $rs^{2}-(r-1)i(s-i)$.

(b) $\dim Z_{r}=rs^{2}-(r-1)(s-1)$. Equivalently,
$c_{\operatorname{M}_{s}(k)}(r)=(r-1)(s-1)$.

Assume for a moment that Proposition 7.1 is established. Then setting
$r=\left\lfloor{\dfrac{d}{s-1}}\right\rfloor+2$, we obtain

$c_{\operatorname{M}_{s}(k)}(r)=(r-1)(s-1)=\left(\left\lfloor{\frac{d}{s-1}}\right\rfloor+1\right)(s-1)>\frac{d}{s-1}\cdot(s-1)=d.$

Theorem 1.5(a) now follows from Theorem 6.1. It remains to prove Proposition
7.1. The case where $r=1$ is obvious, so we will assume that $r\geqslant 2$.

(a) Again, we treat varieties as their sets of $k$-points. Consider the
incidence variety

$Y_{i}=\\{(a_{1},...,a_{r},W)\,|\,a_{1}(W),\ldots,a_{r}(W)\subset
W\\}\subset\operatorname{M}_{s}\times\ldots\times\operatorname{M}_{s}\times\operatorname{Gr}(i,s),$

equipped with the natural projections

$\textstyle{Y_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\textstyle{V_{r}=\operatorname{M}_{s}^{r}}$$\textstyle{\operatorname{Gr}(i,s).}$

Here, $\operatorname{Gr}(i,s)$ denotes the Grassmannian of $i$-dimensional
subspaces in $k^{s}$. Note that $\pi_{2}$ is surjective. Since $Y_{i}$ is
closed in $V_{r}\times\operatorname{Gr}(i,s)$ and $G(i,s)$ is projective, the
image of $Y_{i}$ under $\pi_{1}$ is closed in $\operatorname{M}^{r}_{s}$. This
image is precisely $X_{i}$.

The fibre of $W\in\operatorname{Gr}(i,s)$ under $\pi_{2}$ consists of the
$r$-tuples of matrices $(a_{1},\ldots,a_{r})$ such that $a_{i}(W)\subset W$
for each $a_{i}$. This means that in a suitable basis of $k^{s}$, the matrices
$a_{1},\dots,a_{r}$ are all block upper-triangular of the form

$\begin{bmatrix}A_{i\times i}&B_{i\times(s-i)}\\\ 0_{(s-i)\times
i}&C_{(s-i)\times(s-i)}\end{bmatrix}$

where $A_{i\times i}$ is an $i\times i$-matrix, $B_{i\times(s-i)}$ is an
$i\times(s-i)$-matrix, $C_{(s-i)\times(s-i)}$ is an $(s-i)\times(s-i)$-matrix,
and $0_{(s-i)\times i}$ is the $(s-i)\times i$ zero matrix. In particular,
every fibre of $\pi_{2}$ is an affine space of dimension $r(s^{2}-i(s-i))$. By
the Fibre Dimension Theorem, $Y_{i}$ is irreducible of dimension
$r(s^{2}-i(s-i))+\dim\operatorname{Gr}(i,s)=rs^{2}-(r-1)i(s-i)$.

Now recall that $X_{i}\subset V_{r}$, is the image of $Y_{i}$ under $\pi_{1}$.
Since $Y_{i}$ is irreducible, so is $X_{i}$. It remains to show that $\dim
X_{i}=\dim Y_{i}$, i.e., the fibre of $\pi_{2}$ over $(a_{1},\ldots,a_{r})\in
X_{i}$ in general position is finite. This fibre consists of $i$-dimensional
subspaces, invariant under each of the matrices $a_{1},\ldots,a_{r}$. If
$(a_{1},\ldots,a_{r})\in X_{i}$ is in general position, we may assume that
$a_{1}$ has distinct eigenvalues and hence, $a_{1}$ alone has only finitely
many invariant subspaces in $k^{n}$. This shows that the fibre
$\pi_{2}^{-1}(a_{1},\ldots,a_{r})$ is finite, and $\dim X_{i}=\dim
Y_{i}=rs^{2}-(r-1)i(s-i)$.

(b) It follows from part (a) that the components of $Z_{r}$ of the largest
dimension are $X_{1}$ and $X_{s-1}$. Thus $\dim Z_{r}=\dim X_{1}=\dim
X_{s-1}=rs^{2}-(r-1)(s-1)$ and
$c_{\operatorname{M}_{s}(k)}(r)=r\dim\operatorname{M}_{s}-\dim
Z_{r}=(r-1)(s-1)$, as claimed. This completes the proof of Proposition 7.1 and
thus of Theorem 1.5(a). ∎

######  Corollary 7.2.

Let $R$ be a $k$-ring of finite type and $B$ an Azumaya algebra of degree $s$.

(a) If $\displaystyle s>1+\frac{1}{2}d$, where $d$ is the Krull dimension $R$,
then $\operatorname{gen}_{R}(B)=2$.

(b) In particular, $\operatorname{gen}_{R}(\operatorname{M}_{m}(B))=2$ for all
but finitely many positive integers $m$.

###### Proof.

(a) follows from the inequality
$\operatorname{gen}(B)\leqslant\dfrac{d}{s-1}+1$ of Theorem 1.4(a), and (b) is
an immediate consequence of (a). ∎

###  7.2. Azumaya algebras with involution

In the following proposition we consider $R$-algebras with involution $B$, and
$\operatorname{gen}_{R}(B)$ denotes the minimal cardinality of a generating
set of $B$ when regarded as an algebra with involution.

######  Proposition 7.3.

Assume $k$ is infinite. Let $R$ be a finite type $k$-ring of Krull dimension
$d$.

  1. (a)

If $B$ is an Azumaya algebra of degree $s$ with orthogonal involution over
$R$, then

and,
$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\dfrac{d+(s-2)}{2s-3}}\right\rfloor+1$,
if $s\neq 4$, and
$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\dfrac{d+1}{4}}\right\rfloor+1$,
if $s=4$.

  2. (b)

If $B$ is an Azumaya algebra of even degree $s$ with symplectic involution
over $R$, then

$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\dfrac{d+(s-1)}{2s-3}}\right\rfloor+1$,
if $s\geqslant 8$,
$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\dfrac{d+6}{9}}\right\rfloor+1$,
if $s=6$, and,
$\operatorname{gen}_{R}(B)\leqslant\left\lfloor{\dfrac{d+3}{4}}\right\rfloor+1$,
if $s=4$, and $\operatorname{gen}_{R}(B)\leqslant d+2$, if $s=2$.

###### Proof.

(a) Recall that Azumaya algebras with an orthogonal involution are forms of
$A=(\operatorname{M}_{s}(k),t)$, where $t$ is the matrix transposition
involution. By [NTW19, Proposition 4.10],

$\dim Z_{r}=\begin{cases}r(s^{2}-2s+3)+(s-2),&\text{if $s\neq 4$, and}\\\
12r+1,&\text{if $s=4$.}\end{cases}$

Substituting these formulas into the inequality of Theorem 6.1, we deduce part
(a).

(b) is proved by the same argument with $t$ replaced by the standard
symplectic involution of $\operatorname{M}_{s}$ (where $s$ is even). In this
case the formula for $\dim Z_{r}$ is given by [NTW19, Proposition 4.13]. ∎

##  8\. Upper bounds on the number of generators for octonion algebras

An octonion algebra $B$ over $R$ is an $R$-form of the split octonion algebra
$A$, as defined in e.g., [BS59, Section 2].

######  Proposition 8.1.

Let $k$ be a field and $A$ be the split octonion algebra over $k$. Then

  1. (a)

for any $r\geqslant 3$, the scheme $Z_{r}$ of $r$-tuples not generating $A$
(see Section 2) is irreducible of dimension $6r+5$.

  2. (b)

If $k$ is infinite, $R$ is a $k$-ring of Krull dimension degree $d$, and $B$
is an octonion $R$-algebra, then
$\operatorname{gen}(B)\leqslant\left\lfloor{\dfrac{d+1}{2}}\right\rfloor+3$.

Part (b) is an immediate consequence of part (a) and Theorem 6.1. To prove
part (a), we may assume without loss of generality that our base field $k$ is
algebraically closed. Again, we will freely identify varieties with their sets
of $k$-points. Recall that the automorphism group of $A$ is the exceptional
group $G_{2}$. Since $k$ is algebraically closed, $A$ has no quaternion
division $k$-subalgebras, and thus [Rac74, Theorem 5] tells us that, up to
translations by elements of $G_{2}$, there is a unique maximal subalgebra
$A_{\max}\subset A$. The subalgebra $A_{\max}$ is $6$-dimensional; it is
sometimes called the algebra of _sextonions_ ; see [Wes06]. We will use the
following description of $A_{\max}$ from [Rac74]. Choose a $k$-basis
$x_{i},y_{i}$ for the octonion algebra $A$, where $i=0,1,2,3$ and
$x_{i},y_{i}$ satisfy the relations [Rac74, (14) on p. 165]. Then $A_{\max}$
is spanned by $x_{0},x_{1},x_{2},y_{0},y_{1},y_{3}$.

######  Lemma 8.2.

(a) $A_{\max}$ is generated by three elements,

(b) the stabilizer $H$ of $A_{\max}$ in $G_{2}$ is a $9$-dimensional parabolic
subgroup of $G_{2}$.

###### Proof.

(a) It suffices to show that $A_{\max}$ is generated by $x_{1}$, $x_{2}$ and
$y_{1}$. Indeed, let $A_{0}$ be the subalgebra of $A_{\max}$ generated by
$x_{1}$, $x_{2}$ and $y_{1}$. Then the relations [Rac74, (14) on p. 165] tell
us that $x_{0}=-x_{1}y_{1}$, $y_{0}=1-x_{0}$ and $y_{3}=x_{1}x_{2}$ also lie
in $B_{0}$. Thus $A_{0}=A_{\max}$, as desired.

(b) Recall that $(x_{i},y_{i})$ are, by construction, mutually orthogonal
hyperbolic pairs relative to the norm form in $A$ for $i=0,1,2,3$. This
implies that

(8.1) $A_{\max}^{\perp}=L$, where $L=\operatorname{Span}_{k}(x_{2},y_{3})$,
and conversely, $L^{\perp}=A_{\max}$.

The multiplication table in [Rac74, (14), p. 165] tells us that
$x_{2}^{2}=y_{3}^{2}=x_{2}y_{3}=0$. This tells us that $L$ is
$\gamma$-isotropic in the sense of D. Anderson; see [And11, Lemma 3.5]. In
view of (8.1), the stabilizer $H$ of $A_{0}$ in $G_{2}$ equals the stabilizer
of $L$ in $G_{2}$. By [And11, Proposition A.5], the stabilizer of $L$ is a
$9$-dimensional parabolic subgroup of $G_{2}$. ∎

We are now ready to complete the proof of Proposition 8.1(a). Recall our
standing assumptions: $k$ is algebraically closed and $r\geqslant 3$.

Let $Z_{r}^{\leqslant t}$ be the subvariety of $r$-tuples
$(a_{1},\ldots,a_{r})\in A^{r}$ such that $a_{1},\ldots,a_{r}$ generate a
$k$-subalgebra of $A$ of dimension $\leqslant t$. More precisely, we choose a
$k$-vector space basis $e_{1},\ldots,e_{8}$ in $A$ and identify an element
$x=x_{1}e_{1}+\ldots+x_{8}e_{8}$ of $A$ with the row vector
$(x_{1},\ldots,x_{8})\in k^{8}$. Then $Z_{r}^{\leqslant t}$ is the Zariski
closed subvariety of $A^{r}$ cut out by the conditions

(8.2) $\operatorname{rank}\begin{bmatrix}p_{1}(a_{1},\ldots,a_{r})\\\
\vdots\\\ p_{t+1}(a_{1},\ldots,a_{r})\end{bmatrix}\leqslant t$

for any $t+1$ monomials $p_{1},\ldots,p_{t+1}$ in $r$ variables. Here
$p_{1},\ldots,p_{t+1}$ are non-commutative non-associative monomials,
$p_{i}(a_{1},\ldots,a_{r})\in A$ represents a row of the $(t+1)\times 8$
matrix in (8.2), and each entry of this matrix is a polynomial function in the
coordinates of $a_{1},\ldots,a_{r}$.

Now set $\displaystyle Z_{r}^{(t)}=Z_{r}^{\leqslant t}-Z^{\leqslant t-1}$ to
be the variety of $r$-tuples $(a_{1},\ldots,a_{r})$ such that
$a_{1},\ldots,a_{r}$ generate a $k$-algebra of dimension exactly $t$. Since
every proper subalgebra of $A$ is of dimension $\leqslant 6$, we have
$Z_{r}=Z_{r}^{\leqslant 6}$. Recall that every proper subalgebra of $A$ is
contained in a $6$-dimensional subalgebra $S$, which is a translate of
$A_{\max}$, and that by Lemma 8.2(a), $A_{\max}$ (and thus $S$) is generated
by $3$ (and thus $r$) elements as a $k$-algebra. This tells us that
$Z_{r}^{(6)}$ is dense in $Z_{r}$.

It remains to show that $Z_{r}^{(6)}$ is an irreducible variety of dimension
$6r+5$. Indeed, consider the natural morphism $f\colon
Z_{r}^{(6)}\to\operatorname{Gr}(A,6)$ taking an $r$-tuple
$(a_{1},\dots,a_{r})$ to the $6$-dimensional subalgebra generated by
$a_{1},\ldots,a_{r}$. To see that $f$ is a morphism, note that $Z_{r}^{(6)}$
is covered by open subsets

$U_{p_{1},\ldots,p_{6}}=\\{(a_{1},\ldots,a_{r})\,|\,p_{1}(a_{1},\ldots,a_{r})\wedge\ldots\wedge
p_{6}(a_{1},\ldots,a_{r})\neq 0\;\;\text{in}\;\;\Lambda^{6}(A)\\}$

indexed by $6$-tuples of monomials $p_{1},\ldots,p_{6}$ in $r$ variables. On
any $U_{p_{1},\ldots,p_{6}}\neq\emptyset$, we define

$\displaystyle f_{p_{1},\ldots,p_{6}}\colon U_{p_{1},\ldots,p_{6}}$
$\displaystyle\to\operatorname{Gr}(A,6)$ $\displaystyle(a_{1},\ldots,a_{r})$
$\displaystyle\mapsto p_{1}(a_{1},\ldots,a_{r})\wedge\dots\wedge
p_{6}(a_{1},\ldots,a_{r})\in\mathbb{P}(\Lambda^{6}(A)).$

Here we identify $\operatorname{Gr}(A,6)$ with the variety of decomposable
tensors in $\mathbb{P}(\Lambda^{6}(A))$ via the Plücker embedding. One readily
sees from this definition that every $f_{p_{1},\ldots,p_{6}}$ is a morphism.
These morphisms agree on the overlaps: the image of $(a_{1},\ldots,a_{r})\in
Z_{r}^{(6)}$ under any of them is the $6$-dimensional subalgebra generated by
$a_{1},\ldots,a_{r}$. Hence, they patch together to a morphism $f\colon
Z_{r}^{(6)}\to\operatorname{Gr}(A,6)$. By Lemma 8.2(b), the image of $f$ is
the irreducible projective homogeneous space $G_{2}/H$ of dimension $14-9=5$.
The fibres of $f$ are isomorphic to Zariski open subvarieties of $V_{r}$. By
Lemma 8.2(a), these open subvarieties are not empty, and so they are
irreducible of dimension $6r=r\dim A_{\max}$. Applying the Fibre Dimension
Theorem to $f$, we conclude that $Z_{r}^{(6)}$ is irreducible of dimension
$6r+5$, as claimed. This completes the proof of Proposition 8.1. ∎

##  9\. A new proof of the generalized Forster bound

As yet another application of Theorem 6.1 we will now give a short proof of
the following variant of Theorem 1.1.

######  Theorem 9.1.

Assume that $k$ is infinite field, $A$ is a finite-dimensional
$k$-(multi)algebra, $R$ is a finite type $k$-ring of Krull dimension $d$, and
$B$ is an $R$-form of $A$. Then $\operatorname{gen}_{R}(B)\leqslant
d+\operatorname{gen}_{k}(A)$.

Our proof will rely on the following.

######  Proposition 9.2.

$c_{A}(r+1)\geqslant c_{A}(r)+1$ whenever
$r\geqslant\operatorname{gen}_{k}(A)-1$.

As a consequence of Proposition 9.2, we see that $c_{A}(r)\geqslant
r-\operatorname{gen}_{k}(A)+1$ for any $r\geqslant\operatorname{gen}_{k}(A)$.
Setting $r=d+\operatorname{gen}_{k}(A)$, we can rewrite this as $c_{A}(r)>d$.
Theorem 6.1 now tells us that $\operatorname{gen}_{R}(B)\leqslant
r=d+\operatorname{gen}_{k}(A)$, and Theorem 9.1 follows.

In the course of proving Proposition 9.2 we may assume without loss of
generality that $k$ is algebraically closed. It is then harmless to replace
$V_{r}$ and $Z_{r}$ by their sets of $k$-points endowed with the Zariski
topology, and this will be our convention in this section.

We say that an $r$-tuple $(a_{1},\dots,a_{r})\in A^{r}$ _almost generates_ $A$
if there exists $a_{r+1}\in A$ such that $a_{1},\dots,a_{r},a_{r+1}$ generates
$A$. Denote the set of almost generating $r$-tuples in $A^{r}$ by
$U^{\prime}_{r}$.

######  Lemma 9.3.

Let $\psi:V_{r+1}\to V_{r}$ denote projection to the first $r$ coordinates.

(a) Let $\overline{a}=(a_{1},\dots,a_{r})\in Z_{r}$. Then
$\dim\big{(}\psi^{-1}(\overline{a})\cap Z_{r+1}\big{)}<\dim A$ if and only if
$\overline{a}\in U^{\prime}_{r}$.

(b) $U_{r}^{\prime}$ is Zariski open in $V_{r}$.

(c) If $r\geqslant\operatorname{gen}_{k}(A)-1$, then $Z_{r}\cap
U^{\prime}_{r}$ is dense in $Z_{r}$.

###### Proof.

(a) is obvious from the definition of $U_{r}^{\prime}$. (b) holds because
$\psi$ is flat and finitely presentated, hence open, and
$U^{\prime}_{r}=\psi(U_{r+1})$.

(c) It is enough to show that if there exists
$\overline{a}=(a_{1},\dots,a_{r})\in Z_{r}-\overline{Z_{r}\cap
U^{\prime}_{r}}$, then $r<\operatorname{gen}_{k}(A)-1$, or equivalently, that
$U^{\prime}_{r}=\emptyset$. To prove this assertion, we will show by induction
on $i=0,1,\dots,r$ that the affine space

(9.1)
$L_{i}:=\\{(x_{1},\dots,x_{i},a_{i+1},\dots,a_{r})\,|\,x_{1},\dots,x_{i+1}\in
A\\}$ is contained in $Z_{r}-U^{\prime}_{r}$.

For $i=r$, (9.1) tells us that $\emptyset=L_{r}\cap U_{r}^{\prime}=V_{r}\cap
U_{r}^{\prime}=U_{r}^{\prime}$. In other words,
$r<\operatorname{gen}_{k}(A)-1$, and part (c) follows.

The base case, where $i=0$, is clear. For the induction step, we assume assume
that $L_{i}\subseteq Z_{r}-U^{\prime}_{r}$ for some $i\in\\{0,\dots,r-1\\}$.
This means that for every $x_{1},\ldots,x_{r}\in A$,
$(x_{1},\dots,x_{i},a_{i+1},\dots,a_{r})\notin U^{\prime}_{r}$. Consequently,
for every $x_{i+1}\in A$, the $r+1$ elements
$x_{1},\dots,x_{i+1},a_{i+1},\dots,a_{r}$ do not generate $A$ as an
$k$-algebra, and, in particular, $(x_{1},\dots,x_{i+1},a_{i+2},\dots,a_{r})\in
Z_{r}$. In other words, $L_{i+1}\subseteq Z_{r}$. Our goal is to show that
$L_{i+1}\cap U^{\prime}_{r}=\emptyset$. Assume the contrary. Then by part (b),
$L_{i+1}\cap U^{\prime}_{r}$ is Zariski dense in $L_{i+1}$. In particular,
$\overline{a}$ lies in the Zariski closure of $Z_{r}\cap U_{r}^{\prime}$,
contradiction our assumption that $\overline{a}\notin
Z_{r}-\overline{Z_{r}\cap U^{\prime}_{r}}$. This contradiction tells us that
$L_{i+1}\cap U^{\prime}_{r}=\emptyset$, completing the proof of (9.1) and thus
of part (c). ∎

###### Proof of Proposition 9.2.

By Lemma 9.3 the fibre of $\psi:Z_{r+1}\to Z_{r}$ over every point of a dense
open subset $Z_{r}\cap U_{r}^{\prime}$ of $Z_{r}$ is of dimension $<\dim(A)$.
Hence, $\dim Z_{r+1}<\dim Z_{r}+\dim A$ by the Fibre Dimension Theorem.
Equivalently, $c_{A}(r+1)>c_{A}(r)$. ∎

######  Remark 9.4.

The above argument is conceptually simpler than the proof given in [FR17,
Section 3]. It does not entirely supplant this proof, though, because the main
result of [FR17] is more general than Theorem 9.1.

######  Example 9.5.

Let $A=k^{n}$, viewed as a $k$-module (i.e., as a $k$-algebra with zero
multiplication). Here $R$-forms of $A$ are projective $R$-modules of rank $n$
and $G=\operatorname{Aut}_{k}(A)=\operatorname{GL}_{n}$. Let us write elements
of $V_{1}$ as column vectors of length $n$ and identify $V_{r}$ with the space
$\operatorname{M}_{n\times r}$ of $n\times r$ matrices. We assume that
$r\geqslant n$; indeed, we need at least $n$ elements to generate $B$. The
group $G=\operatorname{GL}_{n}$ acts on $V_{r}=\operatorname{M}_{n\times r}$
via left multiplication. Clearly $U_{r}=\operatorname{M}_{n\times r}^{0}$ is
the open subvariety of $n\times r$ matrices of rank $n$, and $Z_{r}$ is the
closed subvariety of $\operatorname{M}_{n\times r}$ of matrices of rank $<n$,
i.e., of matrices with linearly dependent rows. An easy calculation shows that
$\dim Z_{r}=(n-1)+(n-1)r=(n-1)(r+1)$, and thus $c_{A}(r)=r\dim A-\dim
Z_{r}=r-n+1$. By Theorem 6.1, if $R$ is of finite type over $k$ and
$r-n+1>\operatorname{Kdim}R$, then $B$ is generated by $r$ elements as an
$R$-module. In other words,
$\operatorname{gen}(B)\leqslant\operatorname{Kdim}R+n$, which is Forster’s
original bound.

######  Example 9.6.

Let $A=k\times\ldots\times k$ ($n$ times) with componentwise multiplication.
Here $R$-forms of $A$ are étale algebras of rank $n$ over $R$ and
$G=\operatorname{Aut}_{k}(A)$ is the symmetric group $\operatorname{S}_{n}$
permuting the $n$ factors of $k$. Once again, we will write elements of $A$ as
column vectors of length $n$, and elements of $V_{r}$ as $n\times r$-matrices.
Then $\operatorname{S}_{n}$ acts on $V_{r}=\operatorname{M}_{n\times r}$ by
permuting the rows, $U_{r}$ consists of matrices with distinct rows, and
$Z_{r}$ consists of matrices whose rows are not distinct. From this we readily
see that $\dim Z_{r}=nr-r$ or equivalently, $c_{A}(r)=rn-\dim Z_{r}=r$. If $R$
is a $k$-ring of finite type, Theorem 6.1 tells us that if
$r>\operatorname{Kdim}R$, then every étale $R$-algebra $B$ is generated by $r$
elements. Equivalently,
$\operatorname{gen}_{R}(B)\leqslant\operatorname{Kdim}R+1$, which is the upper
bound of Theorem 1.1.

##  10\. The Lefschetz principle

Our proofs of Theorems 1.4 and 1.5 will rely on topological methods. These
methods work best if the base field $k$ is equipped with an embedding
$k\hookrightarrow\mathbb{C}$. To extend our arguments to an arbitrary base
field $k$ of characteristic $0$, we will repeatedly use the following version
of the Lefschetz principle.

######  Lemma 10.1.

Let $R$ be a $k$-ring, and $B$ be an $R$-algebra. Suppose $B$ is finitely
generated as an $R$-module. For any field extension $K/k$, set
$R_{K}=R\otimes_{k}K$ and $B_{K}=B\otimes_{R}R_{K}$. Then

  1. (a)

$\operatorname{gen}_{R}(B)\geqslant\operatorname{gen}_{R_{K}}(B_{K})$.

  2. (b)

Moreover, there exists a subextension $k\subset F\subset K$ such that $F/k$ is
finitely generated and
$\operatorname{gen}_{R_{F}}(B_{F})=\operatorname{gen}_{R_{K}}(B_{K})$.

  3. (c)

Suppose $k$ is a finitely generated extension of $\mathbb{Q}$. If
$\operatorname{gen}_{R_{K}}(B_{K})<r$ for some field $K$ containing $k$, then

$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})<r.$

###### Proof.

(a) is obvious: if $a_{1},\ldots,a_{r}$ generate $A$ as an $R$-algebra, then
$a_{1},\ldots,a_{r}$ generate $B_{K}$ as an $R_{K}$-algebra.

(b) Suppose $\operatorname{gen}_{R_{K}}(B_{K})=r$. Choose $r$ elements,
$b_{1},\ldots,b_{r}\in B_{K}$ that generate $B_{K}$ as an $R_{K}$-algebra. We
claim that these same elements will generate $B_{F}$ as an $R_{F}$-algebra for
some extension $k\subset F\subset K$ such that $F$ is finitely generated over
$k$. In particular, the claim tells us that
$\operatorname{gen}_{R_{F}}(B_{F})\leqslant r$. The opposite inequality is
given by part (a), so if we can prove this claim, then part (b) will follow.

To prove the claim, choose elements $a_{1},\ldots,a_{d}$ that generate $B$ as
an $R$-module and write each element $b_{i}$ as

$b_{i}=r_{i1}a_{1}+\ldots+r_{id}a_{d}.$

for some coefficients $r_{ij}\in R_{K}$. Each $r_{ij}$ lies in $R_{F_{ij}}$
for some intermediate subfield $k\subset F_{ij}\subset K$ such that $F_{ij}$
is finitely generated over $k$. After replacing $k$ by the compositum of
$F_{ij}$ in $K$ (which is still finitely generated over $k$), we may assume
without loss of generality that $b_{1},\ldots,b_{r}$ lie in $B$.

Since $b_{1},\ldots,b_{r}$ generate $B_{K}$ as an $R_{K}$-algebra, we can
write each $a_{i}$ as

$a_{i}=s_{i1}M_{i1}+\ldots+s_{it_{i}}M_{it_{i}}$

for some monomials $M_{ij}$ in $b_{1},\ldots,b_{r}$ and some coefficients
$s_{ij}\in R_{K}$. Once again, each $s_{ij}$ lies in some intermediate
subfield $E_{ij}$ finitely generated over $k$. Setting $F$ to be the
compositum of $E_{ij}$ in $K$, we see that the $R_{F}$-subalgebra of $B_{F}$
generated by $b_{1},\ldots,b_{r}$ contains $a_{1},\ldots,a_{d}$. This shows
that $b_{1},\ldots,b_{r}$ generate $B_{F}$ as an $R_{F}$-algebra. Hence,
$\operatorname{gen}_{R_{F}}(B_{F})\leqslant r$, as desired.

(c) Choose $F$ as in part (b), so that $\operatorname{gen}_{R_{F}}(B_{F})<r$.
Since $F$ is a finitely generated extension of $\mathbb{Q}$, it is isomorphic
(over $k$) to a subfield of $\mathbb{C}$. Thus we may assume without loss of
generality that $F\subset\mathbb{C}$. By part (a),

$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})\leqslant\operatorname{gen}_{R_{F}}(B_{F})=\operatorname{gen}_{R_{K}}(B_{K})<r,$

as desired. ∎

##  11\. Equivariant cohomology of $U_{r}$

Suppose the field $k$ is embedded in $\mathbb{C}$, so that a $k$-variety $X$
gives rise to a complex analytic space $X(\mathbb{C})$. The purpose of this
section is to calculate some of the
$\operatorname{PGL}_{s}(\mathbb{C})$-equivariant cohomology of the variety
$U_{r}(\mathbb{C})$. We begin with two short subsections that cover
preliminary material.

We write $\mathrm{H}^{*}(X)$ for the singular cohomology
$\mathrm{H}^{*}(X(\mathbb{C});\mathbb{Z})$.

### The Affine Lefschetz Theorem

We use singular cohomology, rather than some other cohomology theory of
varieties, because of the following theorem.

######  Theorem 11.1 (Affine Lefschetz Hyperplane Theorem).

Let $k\subset\mathbb{C}$ be a field with a chosen complex embedding, let
$\mathbb{A}^{N}$ be an affine space and $X\subset\mathbb{A}^{N}$ be a smooth
closed subvariety of dimension $d$. Then there exists a smooth affine
hyperplane section $Y\subseteq X$ defined over $k$ such that $\dim Y=d-1$ and
$\mathrm{H}^{i}(X)\to\mathrm{H}^{i}(Y)$ is an isomorphism if $i<d-1$ and an
injection if $i=d-1$.

Here, a hyperplane section is the scheme-theoretic intersection of $X$ with a
hyperplane in the ambient $\mathbb{A}^{N}$.

When $k=\mathbb{C}$, the theorem is an immediate consequence of Bertini’s
theorem and [HT85, Theorem 1.1.3]. In fact, if $k$ is a field with a chosen
complex embedding $k\subseteq\mathbb{C}$, then we can still apply these
results. There is an open dense set of complex hyperplanes in some projective
space for which the intersection $Y$ is smooth and such that $Y\to X$ has the
required behaviour in cohomology. Any open dense subset of an affine space
contains a point defined over $\mathbb{Q}$, and therefore over $k$.

### Equivariant cohomology of varieties

If $\Gamma$ is a Lie group, then there exists a universal principal
$\Gamma$-bundle $E\Gamma\to B\Gamma$ where $E\Gamma$ is contractible. If
$\Gamma$ acts on $X$, then the Borel equivariant cohomology ([Hsi75, III.1])
of $X$ is defined by

$\mathrm{H}^{*}_{\Gamma}(X):=\mathrm{H}^{*}((X\times
E\Gamma)/\Gamma;\mathbb{Z}).$

As a special case, if $X$ is a point, then
$\mathrm{H}^{\ast}_{\Gamma}(X)=\mathrm{H}^{\ast}(B\Gamma;\mathbb{Z})$.

If $G$ is a linear algebraic group defined over $k\subseteq\mathbb{C}$, then
we may define a Lie group $G(\mathbb{C})$. Suppose $G$ acts on a $k$-variety
$X$. We abuse notation and write $\mathrm{H}^{*}_{G}(X)$ for
$\mathrm{H}^{*}_{G(\mathbb{C})}(X(\mathbb{C}))$. The ideas of B. Totaro
[Tot99] and D. Edidin and W. Graham [EG98] allow us to treat
$\mathrm{H}^{*}_{G}(X)$ as though it were the cohomology of a variety, as we
now recall.

For any $n$, we can find a $G$-representation $V$ and an open $G$-invariant
subvariety $U\subset V$ with the properties that $U\to U/G$ is a $G$-torsor,
$U/G$ is a variety, and $(V-U)\hookrightarrow V$ is a closed subvariety of
codimension exceeding $n/2$; see Remark 5.3. Then

(11.1) $\mathrm{H}^{i}_{G}(X)\cong\mathrm{H}^{i}((X\times U)/G)$

for $i\leqslant n$.

If $G$ acts on a variety $X$ in such a way that $X\to X/G$ is a $G$-torsor,
then the quotient $(X\times U)/G$ appearing in (11.1) above is isomorphic to
$X/G\times U$. We deduce that in this case

$\mathrm{H}^{n}_{G}(X)\cong\mathrm{H}^{n}(X/G)$

for all nonnegative integers $n$.

The equivariant Borel cohomology groups for $G$ are contravariantly functorial
for $G$-equivariant maps. Since there is a unique $G$-equivariant map
$X\to\operatorname{Spec}\mathbb{C}$, there is a natural map
$\mathrm{H}^{*}_{G}(\operatorname{Spec}\mathbb{C})=\mathrm{H}^{*}(BG)\to\mathrm{H}^{*}_{G}(X)$.

### The case of $U_{r}$

The purpose of this section is to prove Theorem 11.4, which is a general
device for constructing examples of forms of an algebra $A$ requiring many
generators.

We continue to write $V_{r}$ for the affine variety whose $k$-points are
$A^{r}$, $Z_{r}$ for the closed subscheme of $V_{r}$ representing $r$-tuples
that fail to generate $A$ as an $k$-algebra, and $c_{A}(r)$ for the
codimension of $Z_{r}$ in $V_{r}$. As before, write $U_{r}=V_{r}-Z_{r}$ and
$c_{A}(r)=\dim V_{r}-\dim Z_{r}$. The automorphism group of the algebra $A$ is
denoted $G$.

######  Lemma 11.2.

The natural map $\mathrm{H}^{n}(BG)\to\mathrm{H}^{n}_{G}(U_{r})$ is an
isomorphism for all values of $n<2c_{A}(r)-1$.

###### Proof.

The inclusion $U_{r}\to V_{r}$ induces an isomorphism

$\mathrm{H}^{n}(BG)\cong\mathrm{H}^{n}_{G}(V_{r})\to\mathrm{H}^{n}_{G}(U_{r})$

for $n<2c_{A}(r)-1$ by virtue of Lemma A.1 and homotopy invariance for
$\mathrm{H}^{n}_{G}(\cdot)$. ∎

Suppose $R$ is a $k$-ring of finite, $B$ is an $R$-form of the algebra $A$,
and $(b_{1},\dots,b_{r})\in B^{r}$ is a generating $r$-tuple of elements. Let
$T\to X$ be the $G$-torsor associated to $B$. There is a $G$-equivariant
classifying map $\varphi_{r}:T\to U_{r}$ by Proposition 4.1, which induces a
morphism

(11.2)
$\varphi_{r}^{*}:\mathrm{H}^{*}_{G}(U_{r})\to\mathrm{H}_{G}^{*}(T)=\mathrm{H}^{*}(\operatorname{Spec}R).$

This morphism is compatible with the natural maps from $\mathrm{H}^{*}(BG)$ to
$\mathrm{H}^{*}_{G}(U_{r})$ and $\mathrm{H}^{*}_{G}(T)$. We have therefore
proved:

######  Lemma 11.3.

Suppose $X=\operatorname{Spec}R$ is an affine variety and $B$ is an $R$-form
of $A$ with associated $G$-torsor $T$. Suppose that $B$ can be generated by
$r$ elements. Then there is a commutative diagram of cohomology rings

$\textstyle{\mathrm{H}^{\ast}(BG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}_{G}(U_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}_{G}(T)=\mathrm{H}^{*}(X)}$

in which the maps with source $\mathrm{H}^{\ast}(BG)$ are the natural maps for
$G$-equivariant cohomology. ∎

If $X$ is a regular $k$-variety, then there exists a vector bundle $E$ on $X$
and an $E$-torsor $p:W\to X$ for which $W$ is an affine scheme. This version
of the Jouanolou construction is due to Thomason, and is proved in greater
generality in [Wei89, Proposition 4.4].

The vector bundle $E$ is to be viewed as a group-scheme on $X$ where the group
structure is given by addition in the vector bundle. When the construction is
applied to a regular variety $X$, the result is an affine $k$-variety $Y$ such
that there is a map $p:Y\to X$ that is Zariski-locally isomorphic to the
projection $U\times\mathbb{A}^{n}_{k}\to U$. In particular, the map
$p:Y(\mathbb{C})\to X(\mathbb{C})$ is a fibre bundle with contractible fibres,
and is therefore a homotopy equivalence, [Jam84, Theorem 7.57].

######  Theorem 11.4.

Let $k$ be a field with a chosen embedding $k\hookrightarrow\mathbb{C}$ and
$A$ be a finite-dimensional algebra over $k$. Suppose the natural map
$\mathrm{H}^{i}(BG)\to\mathrm{H}^{i}_{G}(U_{r})$ is not injective for some
$r\geqslant\operatorname{gen}_{k}(A)$. Then there exists a finite type
$k$-ring $R$ and an $R$-form $B$ of $A$ so that $\operatorname{Kdim}R=i$ and
$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})>r$.

In particular, $\operatorname{gen}_{R}(B)>r$; see Lemma 10.1(a).

###### Proof.

Recall that by Proposition 9.2 $c_{A}(r+t)\to\infty$ as $t\to\infty$. Choose a
sufficiently large positive integer $t$ so that

(11.3) $2c_{A}(r+t)-1>i\quad\text{and}\quad(r+t)\dim(A)\geqslant i+\dim(G).$

The algebra $B$ will be defined as ${\,}^{T}A$, for a suitable $G$-torsor
$T\to\operatorname{Spec}R$. Our goal is to construct this torsor. We will do
so by starting with the $G$-torsor $U_{r+t}\to U_{r+t}/G$ and modifying it in
stages. At each stage, we will produce a $G$-torsor $Y\to Y/G$ so that $Y$ has
the same $G$-equivariant cohomology as $U_{r+t}$ in degrees less than or equal
to $i$.

The first problem with $U_{r+t}$ is that $U_{r+t}/G$ may not be a scheme, only
an algebraic space; see Remark 3.3. To fix this, recall that by Remark 5.3
there exists a linear representation $G\to\operatorname{GL}(\bar{W})$ and a
$G$-invariant open subset $W\subset\bar{W}$ such that $\bar{W}-W$ has
codimension at least $i/2$ and $W$ is the total space of a $G$-torsor $W\to
W/G$ in the category of schemes. There is now a torsor
$q^{\prime}:U_{r+t}\times W\to(U_{r+t}\times W)/G$ in the category of schemes;
see [EG98, Prop. 23(1)] for instance. The quotient scheme $(U_{r+t}\times
W)/G$ is separated [MFK94, Lemma 0.6] and quasi-compact, being the surjective
image of a quasi-compact scheme. It is of finite type by [Gro64, Prop.
1.5.4(v)], since $U_{r}\times W$ is of finite type. The morphism
$U_{r+t}\times W\to(U_{r+t}\times W)/G$ is smooth, since $G$ is smooth, and
because $U_{r+t}\times W$ is a smooth $k$-variety, it follows from [Gro67,
Prop. 17.7.7] that $(U_{r+t}\times W)/G$ is a smooth $k$-variety.

The codimension of $\bar{W}-W$ in $W$ is at least $i/2$, so that there is an
isomorphism $\mathrm{H}^{j}_{G}(U_{r+t})\to\mathrm{H}^{j}_{G}(W\times
U_{r+t})$ for all $j\leqslant i$ (see Lemma A.7).

The next problem is that $(W\times U_{r+t})/G$ is not affine in general. We
will approximate it by an affine scheme as follows. Write
$f:\operatorname{Spec}R\to(W\times U_{r+t})/G$ for an affine vector-bundle
torsor using the Jouanolou construction, and consider the pullback

(11.4) ${T}$${W\times U_{r+t}}$${\operatorname{Spec}R}$${(W\times
U_{r+t})/G.}$$\scriptstyle{F}$$\scriptstyle{q}$${\lrcorner}$$\scriptstyle{f}$

The composite of $F$ with the projection map $W\times U_{r+t}\to U_{r+t}$ is
$G$-equivariant and classifies an $R$-form $B$ of $A$, which is generated by
$r+t$
elements.linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,todo:
linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,Revisit this after
the “associated forms” part has been finalized.—Ben W. The map $f$ is a
vector-bundle torsor, and therefore induces an isomorphism on cohomology:

$\mathrm{H}_{G}^{*}(W\times U_{r+t})\cong\mathrm{H}^{*}((W\times
U_{r+t})/G)\overset{f^{*}}{\longrightarrow}\mathrm{H}^{*}(\operatorname{Spec}R)\cong\mathrm{H}^{*}_{G}(T).$

We now define the $R$-algebra $B={\,}^{T}A$ by twisting $A$ by $T$. We claim
that $B_{\mathbb{C}}$ cannot be generated by $r$ elements.

Suppose for the sake of contradiction that $B_{\mathbb{C}}$ can be generated
by $r$ elements. Then there is a $G_{\mathbb{C}}$-equivariant map
$\varphi\colon T_{\mathbb{C}}\to U_{r,\mathbb{C}}$ representing
$B_{\mathbb{C}}$ and a generating $r$-tuple of elements, by Proposition 4.1.
This leads us to the following commutative diagram

(11.5)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{H}^{i}(BG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$
non-
injective$\textstyle{\mathrm{H}^{i}_{G}(U_{r+t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{i}_{G}(W\times
U_{r+t})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{i}_{G}(T)}$$\textstyle{\mathrm{H}^{i}_{G}(U_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$

in which the first three groups in the bottom row are isomorphic. The
rightmost triangle in (11.5) is the triangle of Lemma 11.3, and the leftmost
diagonal arrow is an isomorphism by Lemma 11.2. The isomorphism
$\mathrm{H}^{i}(BG)\to\mathrm{H}^{i}_{G}(T)$ in (11.5) must factor through the
non-injective map $\mathrm{H}^{i}(BG)\to\mathrm{H}^{i}_{G}(U_{r})$, a
contradiction. Therefore, $B_{\mathbb{C}}$ cannot be generated by $r$
elements. This proves the claim.

Unfortunately, $B={\,}^{T}A$ is not yet the algebra we are looking for.
Indeed,

$\operatorname{Kdim}(R)\geqslant\dim\,(W\times
U_{r+t})/G>\dim(U_{r+t}/G)=(r+t)\dim(A)-\dim(G)\geqslant i;$

see (11.3), whereas we want $\operatorname{Kdim}(R)=i$. To complete the proof,
we modify the torsor $T\to\operatorname{Spec}(R)$ to reduce
$\operatorname{Kdim}(R)$ to $i$. To do this, we embed $\operatorname{Spec}R$
into an affine space $\mathbb{A}^{N}$. By Theorem 11.1, provided
$\operatorname{Kdim}(R)\geqslant i+1$, there is a smooth affine hyperplane
section
$\iota\colon\operatorname{Spec}R^{\prime}\hookrightarrow\operatorname{Spec}R$
such that
$\iota^{*}:\mathrm{H}^{i}(\operatorname{Spec}R)\to\mathrm{H}^{i}(\operatorname{Spec}R^{\prime})$
is injective. If we write $T^{\prime}$ for the pullback of the $G$-torsor $T$
along $\iota$, we deduce that
$\iota^{*}:\mathrm{H}_{G}^{i}(T)\to\mathrm{H}_{G}^{i}(T^{\prime})$ is also
injective. Observe that
$\operatorname{Kdim}R^{\prime}=\operatorname{Kdim}R-1$.

The $R^{\prime}$-algebra $B^{\prime}=({\,}^{T^{\prime}}A)_{\mathbb{C}}$ cannot
be generated by $r$ elements, otherwise there would exist a $G$-equivariant
morphism $\varphi\colon T^{\prime}_{\mathbb{C}}\to U_{r,\mathbb{C}}$ giving
rise to a commutative diagram:

$\textstyle{\mathrm{H}^{i}(BG)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}^{i}_{G}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{i}_{G}(U_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\textstyle{\mathrm{H}^{i}_{G}(T^{\prime}).}$

Here the injective map $\mathrm{H}^{i}(BG)\to\mathrm{H}^{i}_{G}(T^{\prime})$
factors through the non-injective map
$\mathrm{H}^{i}(BG)\to\mathrm{H}^{i}_{G}(U_{r})$, which is a contradiction.
Replacing $R$ by $R^{\prime}$ and $T$ by $T^{\prime}$, we reduce
$\operatorname{Kdim}R$ by $1$ while preserving the property that
$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})>r$. Since this procedure
can be repeated as long as $\operatorname{Kdim}(R)>i$, we eventually arrive at
an example where $\operatorname{Kdim}R=i$. ∎

##  12\. Algebras requiring many generators

Recall that if $X$ is defined over $k\hookrightarrow\mathbb{C}$, then we write
$\mathrm{H}^{*}(X)$ for $\mathrm{H}^{*}(X(\mathbb{C}))$. Similarly, we write
$\mathrm{H}^{*}(BG)$ in place of $\mathrm{H}^{*}(BG(\mathbb{C}))$.

In this section, we establish Theorem 1.4 by applying Theorem 11.4. We must
first prove that $\mathrm{H}^{i}(BG)$ is nonzero for many values of $i$, which
we do in Lemma 12.1. Our main tool, singular cohomology of the complex points,
can be used only when the base field is embedded in $\mathbb{C}$, but by
recourse to the Lefschetz principle in Lemma 10.1, we will be able prove
Theorem 1.4 over any field of characteristic $0$.

######  Lemma 12.1.

Let $k$ be a field with a fixed embedding in $\mathbb{C}$. Let $G$ be an
affine algebraic group over $k$ that is not unipotent. Then there exists a
natural number $\rho_{G}$ such that for all $i\geqslant 1$, there exists
$j\in\\{i+1,i+2,\dots,i+\rho_{G}\\}$ such that

$\mathrm{H}^{j}(BG)\not\cong 0.$

If $H$ denotes the quotient of the identity connected component of $G$ by its
unipotent radical, and $Z$ is the centre of $H$, then we can take $\rho_{G}=4$
if $\dim H>0$, and $\rho_{G}=2$ if $\dim Z>0$.

###### Proof.

Since the cohomology depends only on
$G(\mathbb{C})=G_{\mathbb{C}}(\mathbb{C})$, and since $G_{\mathbb{C}}$ is
unipotent if and only if $G$ is, there is nothing to be lost by assuming
$k=\mathbb{C}$.

The proof proceeds by a sequence of reductions. First, consider the natural
exact sequence

${1}$${N}$${G}$${H}$${1}$

of algebraic groups, where $N$ is the unipotent radical and $H$ is reductive;
see [Hoc81, Theorem 4.3]. Since $N$ is unipotent, it admits a composition
series by subgroups having quotients isomorphic to ${\mathbb{G}_{a}}$ [DG70,
IV Th. de Lazard 4.1]. It follows that $N$ is contractible as a topological
space and so $BG\simeq BH$. Without loss of generality, therefore, we may
assume that $G$ is reductive.

If $G$ is finite, then $\rho_{G}$ exists by [BCR90, Theorem 2.4]. Thus, we may
assume that $G$ is infinite, i.e., $\dim G>0$. We will show that in this case
$\mathrm{H}^{j}(BG;\mathbb{Q})\not\cong 0$ (and hence,
$\mathrm{H}^{j}(BG;\mathbb{Z})\not\cong 0$) for some
$j\in\\{i+1,i+2,i+3,i+4\\}$. Let $G^{0}$ denote the identity component of $G$.
Since the component group $G/G^{0}$ is finite,
$\mathrm{H}^{j}(BG^{0};\mathbb{Q})=\mathrm{H}^{j}(BG;\mathbb{Q})$. We may
therefore assume that $G$ is nontrivial, connected and reductive. Let $Z$ be
the centre of $G$. We will treat the cases where $Z$ is finite and infinite
separately.

If $Z$ is finite, then $G$ is a semisimple linear algebraic group. By [CE48,
Theorem 21.2], we know that $\tilde{\mathrm{H}}^{l}(G;\mathbb{Q})\cong 0$ for
values of $l\in\\{0,1,2\\}$ but $\tilde{\mathrm{H}}^{3}(G;\mathbb{Q})\not\cong
0$. It is an easy consequence of the Serre spectral sequence that
$\tilde{\mathrm{H}}^{l}(BG;\mathbb{Q})\cong 0$ for $l\in\\{0,1,2,3\\}$ but
there exists a nonzero element
$\alpha\in\tilde{\mathrm{H}}^{4}(BG;\mathbb{Q})$. We know, however, that
$\mathrm{H}^{*}(BG;\mathbb{Q})\cong\mathrm{H}^{*}(BT;\mathbb{Q})^{W}$, where
the latter denotes the Weyl invariants of the cohomology of a maximal torus of
a maximal compact subgroup of $G(\mathbb{C})$, by [Mal45, Theorem 11] and
[Hsi75, Reduction 2, p36]. Since $\mathrm{H}^{*}(BT;\mathbb{Q})$ is a
polynomial ring in some number of generators, the element $\alpha$ of the
subring $\mathrm{H}^{*}(BT;\mathbb{Q})^{W}$ cannot be nilpotent. Therefore the
elements $\alpha^{l}$ form a family of nonzero elements in
$\mathrm{H}^{4l}(BG;\mathbb{Q})$. In this case, we may take $\rho_{G}=4$.

If $Z$ is not finite, then $G$ contains a nontrivial torus $Z^{\prime}$ such
that the quotient $G^{\text{ss}}=G/Z^{\prime}$ is semisimple. In particular,
$\tilde{\mathrm{H}}^{l}(BG^{\text{ss}};\mathbb{Q})$ vanishes for values of
$l\in\\{0,1,2,3\\}$. A Serre spectral sequence argument applied to

${BZ^{\prime}}$${BG}$${BG^{\text{ss}}}$

shows that we can find a nontrivial element
$\alpha\in\mathrm{H}^{2}(BG;\mathbb{Q})$ in this case, and we can repeat the
same argument as in the previous case to show that the $\alpha^{l}$ form a
family of nonzero elements in $\mathrm{H}^{2l}(BG;\mathbb{Q})$. In this case,
we may take $\rho_{G}=2$. ∎

######  Remark 12.2.

The construction of $\rho_{G}$ in Lemma 12.1 uses the algebraic group
structure of $G_{\mathbb{C}}$. However, since $\mathrm{H}^{*}(BG(\mathbb{C}))$
depends only on the Lie group $G(\mathbb{C})$, we may assume that $\rho_{G}$
that depends only on $G(\mathbb{C})$ and not on $G$.

###### Proof of Theorem 1.4.

First of all, we consider the case where the field $k$ is a subfield of
$\mathbb{C}$. Set $r$ to be the largest integer satisfying

(12.1) $d\geqslant 2rn-2\dim_{\mathbb{C}}G+\rho_{G}.$

In other words, set

(12.2) $r=\left\lfloor{\dfrac{d+2\dim G-\rho_{G}}{2n}}\right\rfloor.$

By Lemma 12.1, there exists an integer $d+1-\rho_{G}\leqslant i\leqslant d$
such that $\mathrm{H}^{i}(BG)\neq 0$. By (12.1), $i>2rn-2\dim_{\mathbb{C}}G$.

We claim that $\mathrm{H}_{G}^{i}(U_{r})=0$. Consider the quotient
$U_{r}(\mathbb{C})\to U_{r}(\mathbb{C})/G(\mathbb{C})$. This is a principal
$G(\mathbb{C})$-bundle over a manifold of dimension $2rn-2\dim_{\mathbb{C}}G$,
because the Lie group $G(\mathbb{C})$ acts properly on $U_{r}(\mathbb{C})$ by
Lemma 3.2. The quotient map
$EG(\mathbb{C})\times_{G(\mathbb{C})}U_{r}(\mathbb{C})\to
U_{r}(\mathbb{C})/G(\mathbb{C})$ is a fibre bundle with fibre
$EG(\mathbb{C})$, as one can see by using an open cover of the manifold
$U_{r}(\mathbb{C})/G(\mathbb{C})$ trivializing $U_{r}(\mathbb{C})\to
U_{r}(\mathbb{C})/G(\mathbb{C})$. This implies that
$\mathrm{H}^{\ast}_{G}(U_{r})\cong\mathrm{H}^{\ast}(U_{r}(\mathbb{C})/G(\mathbb{C}))$,
see [Jam84, Theorem 7.57]. On the other hand, we know that
$\mathrm{H}^{j}(U_{r}(\mathbb{C})/G(\mathbb{C}))=0$ whenever $j>2rn-2\dim G$,
since $U_{r}(\mathbb{C})/G(\mathbb{C})$ is a manifold of real dimension
$2rn-2\dim_{\mathbb{C}}G$. This proves the claim.

We conclude that the map on equivariant cohomology
$\mathrm{H}^{i}(BG)\to\mathrm{H}_{G}^{i}(U_{r})$ is not injective. By Theorem
11.4, there exists a $k$-ring $R$ of finite type and an $R$-form $B$ of $A$
such that $\dim R=i\leqslant d$ and
$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})>r$. Substituting in the
formula for $r$ from (12.2) and remembering
$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})$ is an integer, we can
rewrite this inequality as

$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})>\dfrac{d+2\dim
G-\rho_{G}}{2n}.$

If $i=\operatorname{Kdim}R<d$, we replace $R$ by the polynomial ring
$R^{\prime}=R[t_{1},\ldots,t_{d-i}]$ and $B$ by
$B^{\prime}=B\otimes_{R}R^{\prime}$; it is easy to see that
$\operatorname{Kdim}R^{\prime}=d$ and

$\operatorname{gen}_{R^{\prime}_{\mathbb{C}}}(B^{\prime}_{\mathbb{C}})=\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})>\frac{d+2\dim
G-\rho_{G}}{2n}.$

By Lemma 10.1(a), this completes the proof of Theorem 1.4 in the case where
$k$ is a subfield of $\mathbb{C}$.

Now we establish the general case, where $k$ is arbitrary field of
characteristic $0$. Since the finite-dimensional $k$-algebra $A$ may be
defined by means of finitely many structure constants, there exists a subfield
$k_{0}\subset k$ and a finite-dimensional $k_{0}$-algebra $A_{0}$, such that
$k_{0}$ is finitely generated over $\mathbb{Q}$ and
$A_{0}\otimes_{k_{0}}k\cong A$. Since $k_{0}$ is finitely generated over
$\mathbb{Q}$, we may embed $k_{0}$ in $\mathbb{C}$. Using the first part of
this proof, we produce a $k_{0}$-ring $R_{0}$ and a form $B_{0}$ of $A_{0}$
over $R_{0}$ such that $\operatorname{Kdim}(R_{0})=d$ and

$\operatorname{gen}_{(R_{0})_{\mathbb{C}}}((B_{0})_{\mathbb{C}})>\frac{d+2\dim
G-\rho_{G}}{2n}.$

By Lemma 10.1(c),

$\operatorname{gen}_{R}(B)>\frac{d+2\dim G-\rho_{G}}{2n},$

where $R=R_{0}\otimes_{k_{0}}k$ is a finite type $k$-ring and
$B=B_{0}\otimes_{k_{0}}k$ is an $R$-form of $A$. Moreover,
$\operatorname{Kdim}R=\operatorname{Kdim}R_{0}=d$, as required. ∎

######  Remark 12.3.

Theorem 1.4 fails if $G$ is unipotent. Indeed, in this case every $G$-torsor
$T\to\operatorname{Spec}R$ splits, see e.g., [AD07, Corollary 3.2].333Here we
are assuming that $\operatorname{char}(k)=0$, as in Theorem 1.4. This is also
a standing assumption in [AD07]. This tells us that the only $R$-form $B$ of
$A$ (up to isomorphism) is $B=A\otimes_{k}R$ and consequently,
$\operatorname{gen}_{R}(B)\leqslant\operatorname{gen}_{k}(A)$ for every $R$.

######  Remark 12.4.

Even among the class of finite-dimensional commutative associative and unital
algebras it is possible for the automorphism group to be nontrivial and
unipotent. In [Pol89, Example 1.7], R. D. Pollack shows that the automorphism
group of the finite-dimensional commutative, associative and unital
$\mathbb{C}$-algebra

$A=\frac{\mathbb{C}[x,y]}{(x,y)^{5}+(x^{2}-y^{3},x^{3}-y^{4})}$

is unipotent. Note that this group is nontrivial, since $x\mapsto
x+y^{4},\,y\mapsto y$ gives a nontrivial automorphism.

##  13\. Geometry of matrix tuples

In this section and the next, we return to the study of the specific case
where $A$ denotes the algebra of $s\times s$ matrices,
$\operatorname{M}_{s}(k)$. We will assume $k$ is field of characteristic $0$
and $s\geqslant 3$ throughout. We continue to use the notation $V_{r}$ to
denote the affine variety whose $k$-points are $A^{r}$, the notation $Z_{r}$
to denote the closed subscheme of $r$-tuples that do not generate $A^{r}$, and
the notation $U_{r}$ for $V_{r}-Z_{r}$. Recall from Section 7.1 that

$Z_{r}^{\mathrm{red}}=X_{1}\cup\ldots\cup X_{s-1},$

where $X_{i}$ is the variety of $r$-tuples $(a_{1},\dots,a_{r})$ of $s\times
s$-matrices having a common $i$-dimensional invariant subspace. Proposition
7.1(a) says that $X_{i}$ is a closed irreducible subvariety of $V_{r}$ of
dimension $rs^{2}-(r-1)i(s-i)$. We will be particularly interested in $X_{1}$,
along with the $\operatorname{PGL}_{s}$-invariant subvariety $T_{2}$ of
$V_{r}$ that we define as the variety of $r$-tuples $(a_{1},\dots,a_{r})$ of
$s\times s$-matrices satisfying the following conditions:

  * •

there exists a subspace $W$ having dimension at least $2$, invariant under
each $a_{i}$, $i=1,\ldots,r$;

  * •

$a_{1},\ldots,a_{r}$ commute pairwise when restricted to $W$.

Our goal is to prove Theorem 1.5(b). We will do this in the next section by
showing that the natural map

$\mathrm{H}^{2(r-1)(s-1)}(B\operatorname{PGL}_{s})\to\mathrm{H}^{2(r-1)(s-1)}_{\operatorname{PGL}_{s}}(U_{r})$

is not injective, then appealing to Theorem 11.4. Since cohomological tools
apply best to smooth varieties and transverse intersections, our focus in this
section is proving the following two preparatory results.

######  Proposition 13.1.

The variety $X_{1}$ is smooth away from $T_{2}$.

######  Proposition 13.2.

Fix an invertible $(s-1)\times(s-1)$ matrix $a$ such that none of the standard
basis vectors of $k^{s-1}$ is an eigenvector for $a$. Fix $(s-1)$ distinct
elements $\lambda_{2},\dots,\lambda_{s}\in k^{\times}$. Let $Y$ be the affine
subspace of $V_{r}=\operatorname{M}_{s}^{r}$ consisting of $r$-tuples of the
form

$\left(\begin{bmatrix}0&0&\dots&0\\\ 0&&&\\\ \vdots&&a&\\\
0&&&\end{bmatrix},\begin{bmatrix}0&0&\dots&0\\\ x_{22}&\lambda_{2}&&0\\\
\vdots&&\ddots&\\\
x_{2s}&0&&\lambda_{s}\end{bmatrix},\begin{bmatrix}0&0&\dots&0\\\
x_{32}&\lambda_{2}&&0\\\ \vdots&&\ddots&\\\
x_{3s}&0&&\lambda_{s}\end{bmatrix},\dots,\begin{bmatrix}0&0&\dots&0\\\
x_{r2}&\lambda_{2}&&0\\\ \vdots&&\ddots&\\\
x_{rs}&0&&\lambda_{s}\end{bmatrix}\right).$

Then $Y\cap T_{2}=\emptyset$, and $Y$ intersects $X_{1}$ transversely in
$V_{r}=\operatorname{M}_{s}^{r}$.

Note that in the definition of $Y$ the matrix $a$ and the distinct nonzero
scalars $\lambda_{2},\ldots,\lambda_{s}$ remain fixed, and each of
$x_{22},x_{23},\ldots,x_{rs}$ varies over $k$. Thus $Y$ is isomorphic to the
affine space $\mathbb{A}_{k}^{(s-1)(r-1)}$ linearly embedded into $V_{r}$.
Here and subsequently

$\vec{e}_{1}=\begin{bmatrix}1\\\ 0\\\ \vdots\\\
0\end{bmatrix},\quad\ldots,\quad\vec{e}_{s}=\begin{bmatrix}0\\\ \vdots\\\ 0\\\
1\end{bmatrix}$

denote the standard basis vectors of $k^{s}$.

######  Remark 13.3.

In the case of $s=2$, the statement of Proposition 13.2 does not make sense,
since there is no $1\times 1$ matrix $a$ such that none of the standard basis
vectors of $k^{1}$ is an eigenvector of $a$. This is why we restrict to
$s\geqslant 3$ in this section.

Since Propositions 13.1 and 13.2 are geometric in nature, we will assume that
$k$ is an algebraically closed field of characteristic $0$ for the remainder
of this section.

### Proof of Proposition 13.1

Let $k\langle x_{1},\dots,x_{r}\rangle$ denote the free unital associative
$k$-algebra in $r$ non-commuting variables $x_{1},\ldots,x_{r}$. Let $C$ be
the $2$-sided ideal generated by the commutators $[x_{i},x_{j}]$.

Let $E=V_{r}\times k^{s}$ be the trivial vector bundle of rank $s$ over
$V_{r}$. For every $f\in k\langle x_{1},\ldots,x_{r}\rangle$, we define an
endomorphism $L_{f}\colon E\to E$ by $L_{f}(v)=f(a_{1},\ldots,a_{r})\cdot v$
over $(a_{1},\ldots,a_{r})\in V_{r}$. Let

$N=\bigcap_{f\in C}\,\operatorname{Ker}(L_{f})$

be the intersection of the kernels of $L_{f}$, where $f$ ranges over $C$. It
follows from the noetherian property of $E$ that $N$ is, in fact, the kernel
of the map

$L=L_{f_{1}}\times\ldots\times L_{f_{N}}\colon E\to
E\times_{V_{r}}E\times_{V_{r}}\ldots\times_{V_{r}}E$

of vector bundles over $V_{r}$ for finitely many elements
$f_{1},\ldots,f_{N}\in C$. Let $(V_{r})_{\operatorname{null}\geqslant i}$
(respectively, $(V_{r})_{\operatorname{null}=i}$) be the closed (respectively,
locally closed) subvariety of $V_{r}$ where the nullity of $L$ is $\leqslant
i$ (respectively, $=i$). Then $N$ is a vector bundle of rank $i$ over
$(V_{r})_{\operatorname{null}=i}$.

######  Lemma 13.4.

Denote the fibre of $N$ over $a=(a_{1},\ldots,a_{r})\in V_{r}$ by $N_{a}$.
Then

(a) $N_{a}$ is invariant under $a_{i}$ for each $i=1,\ldots,r$.

(b) $(V_{r})_{\operatorname{null}\geqslant 1}=X_{1}$ and
$(V_{r})_{\operatorname{null}\geqslant 2}=T_{2}$.

###### Proof.

(a) Suppose $f\in C$. Then $fx_{i}\in C$ for every $i=1,\ldots,r$.
Consequently, if $v$ lies in $N_{a}$, then $L_{f}(a_{i}v)=L_{fx_{i}}(v)=0$ for
every $f\in C$. Thus $a_{i}v$ also lies in $N_{a}$.

(b) For the first equality, note that if a $1$-dimensional subspace $V$ of
$k^{s}$ is invariant under $a_{i}$ for each $i=1,\ldots,r$, then clearly the
restrictions of $a_{1},\ldots,a_{r}$ to $V$ commute pairwise. This shows that
$X_{1}\subset(V_{r})_{\operatorname{null}\geqslant 1}$. To prove the opposite
inclusion, suppose $a\in(V_{r})_{\operatorname{null}\geqslant 1}$, Then $\dim
N_{a}\geqslant 1$ and the restrictions of $a_{1},\ldots,a_{r}$ to $N_{a}$
commute pairwise. Since we are assuming that our base field $k$ is
algebraically closed, $a_{1},\ldots,a_{r}$ have a common eigenvector in
$N_{a}$. Hence, $a\in X_{1}$.

The second equality, $(V_{r})_{\operatorname{null}\geqslant 2}=T_{2}$, follows
directly from the definition of $T_{2}$. ∎

######  Remark 13.5.

It is easy to see that $T_{2}$ is, in fact, a proper closed subvariety of
$X_{2}$; in particular, $\dim T_{2}<\dim X_{2}$.

We are now ready to finish the proof of Proposition 13.1. Let $N_{1}$ be the
restriction of $N$ to

$(V_{r})_{\operatorname{null}=1}=(V_{r})_{\operatorname{null}\geqslant
1}-(V_{r})_{\operatorname{null}\geqslant 2}=X_{1}-T_{2}.$

Then $N_{1}$ is a vector bundle of rank $1$ over $(X_{1}-T_{2})$. Its
projectivization $\mathbb{P}(N_{1})$ is the incidence variety in
$(X_{1}-T_{2})\times\mathbb{P}^{s-1}$ consisting of pairs

$((a_{1},\dots,a_{r}),L),$

where $L$ is a $1$-dimensional subspace of $k^{s}$ invariant under
$(a_{1},\ldots,a_{r})\in(X_{1}-T_{2})$. Since $N_{1}$ is a vector bundle of
rank $1$ over $(X_{1}-T_{2})$, the natural projection
$\mathbb{P}(N_{1})\to(X_{1}-T_{2})$ is an isomorphism.

It thus suffices to prove that $\mathbb{P}(N_{1})$ is smooth. To prove that
$\mathbb{P}(N_{1})$ is smooth, we will take advantage of the fact that the
natural projection map $N\to\mathbb{P}^{s-1}$ given by
$\pi\colon((a_{1},\ldots,a_{r}),L)\mapsto L$ is
$\operatorname{PGL}_{s}$-equivariant, and $\operatorname{PGL}_{s}$ acts
transitively on the target $\mathbb{P}^{s-1}$. This tells us that $\pi$ is
flat and the fibres of $\pi$ over the $k$-points of $\mathbb{P}^{s-1}$ are
pairwise isomorphic. Moreover, the fibre over any closed point
$v\in\mathbb{P}^{s-1}$ is an open subvariety of the affine space of all
$r$-tuples of matrices having $v$ as an eigenspace. Hence, it is smooth.
Therefore, $\pi\colon\mathbb{P}(N_{1})\to\mathbb{P}^{s-1}$ is flat with smooth
fibres. We conclude that $\pi$ is a smooth morphism, and consequently,
$\mathbb{P}(N_{1})$ is smooth over $\operatorname{Spec}k$, as claimed. ∎

### Proof of Proposition 13.2

######  Lemma 13.6.

The scheme-theoretic intersection of the subvarieties $Y$ and $X_{1}$ of
$V_{r}$ is a single reduced point

$p:=\left(\begin{bmatrix}0&0&\dots&0\\\ 0&&&\\\ \vdots&&a&\\\
0&&&\end{bmatrix},\begin{bmatrix}0&0&\dots&0\\\ 0&\lambda_{2}&&\\\
\vdots&&\ddots&\\\ 0&&&\lambda_{s}\end{bmatrix},\begin{bmatrix}0&0&\dots&0\\\
0&\lambda_{2}&&\\\ \vdots&&\ddots&\\\
0&&&\lambda_{s}\end{bmatrix},\dots,\begin{bmatrix}0&0&\dots&0\\\
0&\lambda_{2}&&\\\ \vdots&&\ddots&\\\ 0&&&\lambda_{s}\end{bmatrix}\right).$

###### Proof.

We claim that $p$ is the only geometric point in the intersection $X_{1}\cap
Y$. To see this, let $L$ be an algebraically closed field and let
$(a_{1},\dots,a_{r})\in(X_{1}\cap Y)(L)$, where

$a_{1}=\begin{bmatrix}0&0&\dots&0\\\ 0&&&\\\ \vdots&&a&\\\
0&&&\end{bmatrix}\quad\text{and}\quad a_{i}=\begin{bmatrix}0&0&\dots&0\\\
x_{i2}&\lambda_{2}&&0\\\ \vdots&&\ddots&\\\
x_{is}&0&&\lambda_{s}\end{bmatrix}$

for $i=2,\ldots,r$. Here $x_{ij}$ are elements of $L$. The eigenvectors of
$a_{1}$ are nonzero multiples of $\vec{e}_{1}$ and $\begin{bmatrix}0\\\
v_{2}\\\ \vdots\\\ v_{s}\end{bmatrix}$ , where $\begin{bmatrix}v_{2}\\\
\vdots\\\ v_{s}\end{bmatrix}$ is an eigenvector of the invertible matrix $a$.
The eigenvectors of $a_{i}$ for $i\geqslant 2$ are nonzero multiples of
$\vec{e}_{2},\ldots,\vec{e}_{s}$ and of $\begin{bmatrix}1\\\
-\lambda_{2}^{-1}x_{i2}\\\ \vdots\\\ -\lambda_{s}^{-1}x_{is}\end{bmatrix}$. We
conclude that the matrices $a_{1},a_{2},\dots a_{r}$ do not have a common
eigenvector unless $x_{22}=x_{23}=\dots=x_{ns}=0$. In the case where
$x_{22}=x_{23}=\ldots=x_{ns}=0$, the only eigenspace shared by
$a_{1},\ldots,a_{r}$ is $\operatorname{Span}(\vec{e}_{1})$. This proves the
claim.

It remains to show that the scheme-theoretic intersection $X_{1}\cap Y$ is
reduced, i.e., that it is isomorphic to $\operatorname{Spec}k$. Let
$(R,\mathfrak{m})$ denote a local ring containing $k$. We wish to show that
there is a unique morphism $f:\operatorname{Spec}R\to X_{1}\cap Y$ given by
composing $p\colon\operatorname{Spec}k\to X_{1}\cap Y$ with the natural
projection $\operatorname{Spec}R\to\operatorname{Spec}k$. Write $L$ for the
algebraic closure of $R/\mathfrak{m}$. As we showed above, the composite
$\operatorname{Spec}L\to\operatorname{Spec}R\to X_{1}\cap Y$ must be the
geometric point $p$. Therefore, $f:\operatorname{Spec}R\to X_{1}\cap Y$ must
correspond to an $R$-valued point of the form

$q=\left(\begin{bmatrix}0&0&\dots&0\\\ 0&&&\\\ \vdots&&a&\\\
0&&&\end{bmatrix},\begin{bmatrix}0&0&\dots&0\\\ x_{22}&\lambda_{2}&&0\\\
\vdots&&\ddots&\\\
x_{2s}&0&&\lambda_{s}\end{bmatrix},\ldots,\begin{bmatrix}0&0&\dots&0\\\
x_{r2}&\lambda_{2}&&0\\\ \vdots&&\ddots&\\\
x_{rs}&0&&\lambda_{s}\end{bmatrix}\right)$

where each $x_{ij}$ lies in $\mathfrak{m}$. The condition that $q$ lies in
$X_{1}(R)$ means that there exists a unimodular vector
$\vec{v}:=\begin{bmatrix}v_{1}\\\ \vdots\\\ v_{s}\end{bmatrix}\in R^{s}$ such
that $a_{1}\vec{v}=\mu_{1}\vec{v},\ldots,a_{r}\vec{v}=\mu_{r}\vec{v}$ for some
$\mu_{1},\ldots,\mu_{r}\in R$. If we reduce modulo $\mathfrak{m}$, then as we
noted in the previous paragraph, the only eigenspace shared by
$a_{1},\ldots,a_{r}$ is $\operatorname{Span}(\vec{e}_{1})$. In other words, if
we reduce modulo $\mathfrak{m}$, the vector $\vec{v}$ will become a scalar
multiple of $\vec{e}_{1}$. Therefore $v_{1}\in R^{\times}$, while
$v_{2},\dots,v_{s}\in\mathfrak{m}$. The condition that $\vec{v}$ is an
eigenvector of $a_{1}$ gives

$\mu_{1}\begin{bmatrix}v_{1}\\\ v_{2}\\\ \vdots\\\
v_{s}\end{bmatrix}=\begin{bmatrix}0&0&\dots&0\\\ 0&&&\\\ \vdots&&a&\\\
0&&&\end{bmatrix}\begin{bmatrix}v_{1}\\\ v_{2}\\\ \vdots\\\
v_{s}\end{bmatrix}=\begin{bmatrix}0\\\ w_{2}\\\ \vdots\\\
w_{s}\end{bmatrix},\text{ where }\begin{bmatrix}w_{2}\\\ \vdots\\\
w_{s}\end{bmatrix}=a\begin{bmatrix}v_{2}\\\ \vdots\\\ v_{s}\end{bmatrix}.$

Because $v_{1}$ is a unit, $\mu_{1}=0$, so that $\vec{w}=0$. Since $a$ is an
invertible $(s-1)\times(s-1)$-matrix, it follows that $v_{2}=\dots=v_{s}=0$.
That is, we have deduced that $\vec{e}_{1}$ is a unit multiple of $\vec{v}$.

We may now suppose $\vec{e}_{1}$ is an eigenvalue of each matrix. As in the
field case, looking at the first entry of
$a_{i}\vec{e}_{1}=\lambda_{i}\vec{e}_{1}$ for $i=1,\dots,r$ implies that the
associated eigenvalues are $0$. Therefore $x_{ij}=0$ for all $i=2,\ldots,r$
and $j=2,\ldots,s$. This proves that $q$ is uniquely determined, so that
$X_{1}\cap Y$ is reduced, as claimed. ∎

We are now ready to complete the proof of Proposition 13.2. To show that
$Y\cap T_{2}=\emptyset$, observe that $T_{2}\subset X_{1}$; see Lemma 13.4(b).
By Lemma 13.6, $Y\cap X_{1}$ is a single point $p=(a_{1},a_{2},\ldots,a_{r})$,
where

$a_{1}=\begin{bmatrix}0&0&\dots&0\\\ 0&&&\\\ \vdots&&a&\\\
0&&&\end{bmatrix}\quad\text{and}\quad
a_{2}=\ldots=a_{r}=\begin{bmatrix}0&0&\dots&0\\\ 0&\lambda_{2}&&\\\
\vdots&&\ddots&\\\ 0&&&\lambda_{s}\end{bmatrix}.$

Hence, it suffices to show that $p\not\in T_{2}$. Assume the contrary: there
exists a subspace $V\subset k^{s}$ of dimension $\geqslant 2$ which is
invariant under $a_{1}$ and $a_{2}$ and such that $a_{1}$ and $a_{2}$ commute
when restricted to $V$. Note that
$\operatorname{Span}_{k}(\vec{e}_{2},\ldots,\vec{e}_{s})$ is invariant under
both $a_{1}$ and $a_{2}$. Hence so is
$V^{\prime}=V\cap\operatorname{Span}_{k}(\vec{e}_{2},\ldots,\vec{e}_{n})$.
Since $\dim V\geqslant 2$, we conclude that $\dim V^{\prime}\geqslant 1$.
Since $a_{1}|_{V^{\prime}}$ and $a_{2}|_{V^{\prime}}$ commute, $a_{1}$ and
$a_{2}$ must therefore share an eigenvector in $V^{\prime}$. But the only
eigenvectors $a_{1}$ and $a_{2}$ share in $k^{s}$ are scalar multiples of
$\vec{e}_{1}$, a contradiction. We conclude that $p\not\in T_{2}$.

It remains to show that $X_{1}$ and $Y$ intersect transversely at $p$. Since
$p\not\in T_{2}$, the variety $X_{1}$ is smooth at $p$ by Proposition 13.1.
Since $Y$ is an affine space, $Y$ is also smooth at $p$. By definition $Y$ is
of dimension $(s-1)(r-1)$, and by Proposition 7.1(a), $X_{1}$ is of dimension
$rn^{2}-(s-1)(r-1)$. Their intersection is a single reduced point $p$, by
Lemma 13.6. Since

$\dim T_{p}(V_{r})=rn^{2}=\dim T_{p}(X_{1})+\dim T_{p}(Y),$

$X_{1}$ and $Y$ intersect transversely in $V_{r}$. ∎

##  14\. Proof of Theorem 1.5(b)

In this section, unless otherwise stated, $k$ denotes a subfield of
$\mathbb{C}$. We will show that the natural map

$\mathrm{H}^{2(s-1)(r-1)}(B\operatorname{PGL}_{s})\to\mathrm{H}^{2(s-1)(r-1)}_{\operatorname{PGL}_{s}}(U_{r})$

is not injective.

Consider the $1$-parameter subgroup
${\mathbb{G}_{m}}\hookrightarrow\operatorname{PGL}_{s}$ given by
$\lambda\mapsto\begin{bmatrix}\lambda^{-1}&0\\\ 0&I_{s-1}\end{bmatrix}$. The
group ${\mathbb{G}_{m}}$ acts on $\operatorname{M}_{s}$ by

$\lambda\cdot\begin{bmatrix}a_{11}&a_{12}&\dots&a_{1n}\\\
a_{21}&a_{22}&\dots&a_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{n1}&a_{n2}&\dots&a_{nn}\end{bmatrix}=\begin{bmatrix}a_{11}&\lambda^{-1}a_{12}&\dots&\lambda^{-1}a_{1n}\\\
\lambda a_{21}&a_{22}&\dots&a_{2n}\\\ \vdots&\vdots&\ddots&\vdots\\\ \lambda
a_{n1}&a_{n2}&\dots&a_{nn}\end{bmatrix}$

Recall that the variety $Y$ from Proposition 13.2 is isomorphic to
$\mathbb{A}^{(s-1)(r-1)}$, the affine space with linear coordinates
$(x_{22},\dots,x_{rn})$. The variety $Y$ is ${\mathbb{G}_{m}}$-invariant as a
subvariety of $V_{r}$, and the isomorphism $Y\cong\mathbb{A}^{(s-1)(r-1)}$ is
${\mathbb{G}_{m}}$-equivariant, where ${\mathbb{G}_{m}}$ acts on the linear
coordinates by

(14.1) $\lambda\colon(x_{22},\dots,x_{rn})\mapsto(\lambda x_{22},\dots,\lambda
x_{rn}).$

The normal bundle $N$ of ${\operatorname{Spec}k}\cong X_{1}\cap Y$ in
$Y\cong\mathbb{A}^{(s-1)(r-1)}$ is trivial of rank $(s-1)(r-1)$, and the total
space may be identified with $Y$ itself. From (14.1), $N$ is isomorphic to the
direct sum of $(s-1)(r-1)$ copies of the standard representation of
${\mathbb{G}_{m}}$ on $\mathbb{A}^{1}$. The top Chern class of this
representation, $c_{(s-1)(r-1)}^{{\mathbb{G}_{m}}}(N)$, is
$\vartheta^{(s-1)(r-1)}\in\mathrm{H}^{2(s-1)(r-1)}_{{\mathbb{G}_{m}}}({\operatorname{Spec}k})$,
where $\vartheta=c^{{\mathbb{G}_{m}}}_{1}(\mathbb{A}^{1})$ is a generator of
$\mathrm{H}^{2}_{{\mathbb{G}_{m}}}(\operatorname{Spec}k)=\mathrm{H}^{2}(B{\mathbb{G}_{m}})\cong\mathbb{Z}$.

######  Remark 14.1.

We describe four identifications, which we use for the rest of the section.

First, because $X_{1}\cap Y={\operatorname{Spec}k}$, we may identify
$\mathrm{H}^{*}_{\mathbb{G}_{m}}(X_{1}\cap Y)$ with $\mathbb{Z}[\vartheta]$,
where $|\vartheta|=2$.
linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,todo:
linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,There is a question
of $\vartheta$ versus $-\vartheta$. Is it necessary to specify which generator
$\vartheta$ is?—Ben W.

Second, we use the isomorphism
$i^{*}:\mathrm{H}_{\mathbb{G}_{m}}^{*}(Y)\to\mathrm{H}_{\mathbb{G}_{m}}^{*}(X_{1}\cap
Y)$ to identify $\mathrm{H}_{\mathbb{G}_{m}}^{*}(Y)$ with
$\mathbb{Z}[\vartheta]$ as well.

Third, because of the isomorphism $V_{r}\cong\mathbb{A}_{k}^{rs^{2}}$, we may
also identify $\mathrm{H}_{\mathbb{G}_{m}}^{*}(V_{r})$ with
$\mathbb{Z}[\vartheta]$.

Finally, recall from Proposition 7.1(a) that $X_{2}$ is a closed irreducible
subvariety of $V_{r}$ of codimension $2(s-2)(r-1)$. By Remark 13.5, the
codimension of $T_{2}$ in $V_{r}$ is at least $2(s-2)(r-1)+1$. We deduce
(using e.g., Lemma A.7) that for values of $j<4(s-2)(r-1)+1$, the natural
inclusion $(V_{r}-T_{2})\hookrightarrow V_{r}$ induces an isomorphism

$\mathrm{H}_{\mathbb{G}_{m}}^{j}(V_{r}-T_{2})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathrm{H}_{\mathbb{G}_{m}}^{j}(V_{r}).$

Using this, we identify $\mathrm{H}_{\mathbb{G}_{m}}^{j}(V_{r}-T_{2})$ with
$\mathrm{H}_{\mathbb{G}_{m}}^{j}(V_{r})=\mathrm{H}^{j}(B{\mathbb{G}_{m}})$ for
every $j\leqslant 4(s-2)(r-1)$.

######  Proposition 14.2.

The Gysin map $\displaystyle j_{*}:\mathrm{H}^{0}_{{\mathbb{G}_{m}}}(X_{1}\cap
Y)\to\mathrm{H}^{2(s-1)(r-1)}_{{\mathbb{G}_{m}}}(Y)$ on
${\mathbb{G}_{m}}$-equivariant cohomology, induced by the inclusion
$(X_{1}\cap Y)\hookrightarrow Y$, satisfies $j_{*}(1)=\vartheta^{(s-1)(r-1)}$.

###### Proof.

The inclusion $(X_{1}\cap Y)\hookrightarrow Y$ is isomorphic as a map of
varieties with ${\mathbb{G}_{m}}$-action to the inclusion
$\\{0\\}\to\mathbb{A}^{(s-1)(r-1)}_{k}$, where the affine space is given the
standard ${\mathbb{G}_{m}}$-action.

In order to calculate the ${\mathbb{G}_{m}}$-equivariant cohomology groups of
the spaces concerned, we compare the map with

$\\{0\\}\times(\mathbb{A}^{N+1}_{k}-\\{0\\})\to\mathbb{A}^{(s-1)(r-1)}_{k}\times(\mathbb{A}^{N+1}-\\{0\\})$

where $N>(s-1)(r-1)$ and the ${\mathbb{G}_{m}}$ action on
$\mathbb{A}_{k}^{N+1}$ is the standard one. Now we take quotients by the
${\mathbb{G}_{m}}$-action to arrive at

$z:\mathbb{P}^{N}_{k}\to\mathbb{P}^{N+(s-1)(r-1)}_{k}-\mathbb{P}^{(s-1)(r-1)-1}_{k},$

which is the inclusion of $\mathbb{P}^{N}_{k}$ as the zero-section of the
bundle $E=\mathcal{O}(1)^{(s-1)(r-1)}$.

The Gysin map $j_{*}$ above fits in a commutative diagram:

$\textstyle{\mathrm{H}^{0}_{\mathbb{G}_{m}}(X_{1}\cap
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\scriptstyle{=}$$\textstyle{\mathrm{H}^{2(s-1)(r-1)}_{\mathbb{G}_{m}}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{=}$$\textstyle{\mathrm{H}^{0}(\mathbb{P}^{N}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{z_{*}\quad\quad}$$\textstyle{\mathrm{H}^{2(s-1)(r-1)}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\scriptstyle{z^{*}}$$\textstyle{\mathrm{H}^{2(s-1)(r-1)}(\mathbb{P}^{N}_{k}).}$

The arrows marked “$=$” represent identifications.

The composite $z^{*}\circ z_{*}$ sends $1$ to the Euler class of $E$ (Remark
A.3), which is $\vartheta^{(s-1)(r-1)}$ by the multiplicativity of Euler
classes [MS74, p. 156]. ∎

######  Lemma 14.3.

The Gysin map

$\mathrm{H}^{*}_{{\mathbb{G}_{m}}}(X_{1}-T_{2})\to\mathrm{H}^{*+2(s-1)(r-1)}_{{\mathbb{G}_{m}}}(V_{r}-T_{2})$

on ${\mathbb{G}_{m}}$-equivariant cohomology, induced by the inclusion
$(X_{1}-T_{2})\hookrightarrow(V_{r}-T_{2})$, sends $1_{X_{1}-T_{2}}$ to
$\vartheta^{(s-1)(r-1)}$. In particular,
$\mathrm{H}^{0}_{{\mathbb{G}_{m}}}(X_{1}-T_{2})\to\mathrm{H}^{2(s-1)(r-1)}_{{\mathbb{G}_{m}}}(V_{r}-T_{2})$
is nonzero.

###### Proof.

Consider a square in which the vertical maps are Gysin maps associated to
closed embeddings, and in which the horizontal maps are induced by the
inclusions $Y\cap X_{1}\to X_{1}-T_{2}$ and $Y\to V_{r}-T_{2}$:

$\textstyle{\mathbb{Z}[\vartheta]\cong\mathrm{H}_{{\mathbb{G}_{m}}}^{0}(Y\cap
X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{{\mathbb{G}_{m}}}^{0}(X_{1}-T_{2})\cong\mathbb{Z}[\vartheta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}[\vartheta]\cong\mathrm{H}_{{\mathbb{G}_{m}}}^{2(s-1)(r-1)}(Y)}$$\textstyle{\mathrm{H}_{{\mathbb{G}_{m}}}^{2(s-1)(r-1)}(V_{r}-T_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

This square commutes by Lemma A.5, which applies because $Y$ and $X_{1}-T_{2}$
intersect transversely in $V_{r}-T_{2}$ according to Proposition 13.2.

Lemma 14.2 says the left vertical map sends $1_{Y\cap X_{1}}$ to
$\vartheta^{(s-1)(r-1)}$. The upper horizontal map is an isomorphism since the
varieties in the source and target are connected. Consider the maps induced by
inclusions

${\mathrm{H}^{j}_{\mathbb{G}_{m}}(V_{r})}$${\mathrm{H}^{j}_{\mathbb{G}_{m}}(V_{r}-T_{2})}$${\mathrm{H}^{j}_{\mathbb{G}_{m}}(Y).}$

Here Remark 14.1 tells us that the left arrow is an isomorphism provided
$j<4(s-2)(r-1)+1$ and the curved arrow is an isomorphism for all $j$. We
deduce that inclusion induces an isomorphism
$\mathrm{H}^{2(s-1)(r-1)}_{\mathbb{G}_{m}}(V_{r}-T_{2})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathrm{H}^{2(s-1)(r-1)}_{\mathbb{G}_{m}}(Y)$.

The assertion of the lemma is now proved by chasing the diagram. ∎

######  Lemma 14.4.

The Gysin map

$\mathrm{H}^{0}_{\operatorname{PGL}_{n}}(X_{1}-T_{2})\to\mathrm{H}^{2(s-1)(r-1)}_{\operatorname{PGL}_{n}}(V_{r}-T_{2})$

on $\operatorname{PGL}_{n}$-equivariant cohomology, induced by the inclusion
$(X_{1}-T_{2})\hookrightarrow(V_{r}-T_{2})$, is nonzero.

###### Proof.

Lemma A.6 says there is a commutative diagram

$\textstyle{\mathbb{Z}\cong\mathrm{H}_{{\mathbb{G}_{m}}}^{0}(X_{1}-T_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{\operatorname{PGL}_{s}}^{0}(X_{1}-T_{2})\cong\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{{\mathbb{G}_{m}}}^{2(s-1)(r-1)}(V_{r}-T_{2})}$$\textstyle{\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r}-T_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

in which the upper horizontal arrow takes $1$ to $1$. Lemma 14.3 and a
diagram-chase now establishes the result. ∎

######  Lemma 14.5.

The morphism
$\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r})\to\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(U_{r})$
induced by the open immersion $U_{r}\hookrightarrow V_{r}$ is not injective.

###### Proof.

There is a factorization of the inclusion $U_{r}\hookrightarrow V_{r}$ as

$U_{r}\hookrightarrow V_{r}-(T_{2}\cup X_{1})\hookrightarrow
V_{r}-T_{2}\hookrightarrow V_{r},$

and each of these inclusions is $\operatorname{PGL}_{s}$-equivariant. By
functoriality, it suffices to show

$\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r})\to\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r}-(T_{2}\cup
X_{1}))$

is not injective. Clearly, since $T_{2}\subsetneq X_{1}$, the codimension of
$T_{2}$ in $V_{r}$ is strictly greater than the codimension of $X_{1}$, i.e.,
is strictly greater than $(r-1)(s-1)$; see Proposition 7.1(a). By Remark 14.1,

$\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r})\to\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r}-T_{2})$

is an isomorphism. It is therefore sufficient to show that

$\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r}-T_{2})\to\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r}-(T_{2}\cup
X_{1}))$

is not injective. This follows from Proposition 14.4 in combination with the
Thom–Gysin sequence

$\mathrm{H}^{0}_{\operatorname{PGL}_{s}}(X_{1}-T_{2})\to\mathrm{H}^{2(s-1)(r-1)}_{\operatorname{PGL}_{s}}(V_{r}-T_{2})\to\mathrm{H}^{2(s-1)(r-1)}_{\operatorname{PGL}_{s}}(V_{r}-(T_{2}\cup
X_{1})).\qed$

######  Remark 14.6.

(i) Lemma 14.5 can fail when $s=2$. For instance, if $s=2$ and $r=2$, then one
can show that
$\mathrm{H}_{\operatorname{PGL}_{2}}^{2}(V_{2})\cong\mathrm{H}^{2}(B\mathrm{PU}_{2})=0$
by exploiting the isomorphism of Lie groups between $\operatorname{SO}_{3}$
and the projective unitary group $\mathrm{PU}_{2}$ and [Bro82, Theorem 1.5].
With some additional effort, we have checked by direct computation that the
morphism

$\mathrm{H}_{\operatorname{PGL}_{2}}^{2(r-1)}(V_{r})\to\mathrm{H}_{\operatorname{PGL}_{2}}^{2(r-1)}(U_{r})$

is injective for a few other small even values of $r$.
linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,todo:
linecolor=Green,backgroundcolor=Green!25,bordercolor=Green,It’s actually a lot
of effort—Ben W.

(ii) The argument in Lemma 14.5 actually shows the slightly stronger result
that

$\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r};\mathbb{Q})\to\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(U_{r};\mathbb{Q})$

is not injective, if $s\geqslant 3$. By contrast, when $s=2$ and $r$ is even,
there is an isomorphism
$\mathrm{H}_{\operatorname{PGL}_{2}}^{2(r-1)}(V_{r};\mathbb{Q})\cong\mathrm{H}^{2(r-1)}(B\operatorname{SO}_{3};\mathbb{Q})=0$,
using [Bro82, Theorem 1.5].

Taken together, (i) and (ii) suggest that the case where $s=2$ and $r$ is even
is substantially different from the case where $s\geqslant 3$ considered in
Lemma 14.5. On the other hand, Gant has shown that this lemma continues to
hold when $s=2$ and $r\geqslant 3$ is odd; see [Gan20, Theorem 5.2].

###### Proof of Theorem 1.5(b).

First, let us suppose the field $k$ is $\mathbb{Q}$. Set $r$ to be the largest
integer satisfying $d\geqslant 2(s-1)(r-1)$. In other words, set

(14.2) $r=\left\lfloor{\frac{d}{2(s-1)}}\right\rfloor+1.$

We claim that the natural map
$\mathrm{H}^{2(s-1)(r-1)}(B\operatorname{PGL}_{s})\to\mathrm{H}^{2(s-1)(r-1)}_{\operatorname{PGL}_{s}}(U_{r})$
is not injective. This follows from the commutative diagram

$\textstyle{\mathrm{H}^{2(s-1)(r-1)}(B\operatorname{PGL}_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathrm{H}_{\operatorname{PGL}_{s}}^{2(s-1)(r-1)}(V_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{2(s-1)(r-1)}_{\operatorname{PGL}_{s}}(U_{r})}$

along with Lemma 14.5.

We now apply Theorem 11.4 with $i=2(s-1)(r-1)$ to produce a $\mathbb{Q}$-ring
$R^{\prime}$ and an Azumaya $R^{\prime}$-algebra $B^{\prime}$ such that
$\operatorname{Kdim}R^{\prime}=i=2(s-1)(r-1)$ and