$\displaystyle\forall\sigma\in\operatorname{GW}(\kappa_{x},\omega_{\kappa_{x}/k}),\operatorname{rk}\big{(}\operatorname{Tr}^{MW}_{\kappa_{x}/k}(\sigma)\big{)}=d_{x}.\operatorname{rk}(\sigma).$

where $d_{x}=[\kappa_{x}:k]$.

######  Remark 4.2.6.

Notice in particular that every quadratic cycle which comes from the
hyperbolic map will have a degree of the form $n.h$ for $n\in\mathbb{Z}$ and
$h$ the hyperbolic form.

######  4.2.7.

Transfers in the monogenic case. Let $E/k$ be a monogenic finite extension
field. Giving a generator $\alpha\in E$ is equivalent to give a closed
embedding $x:\operatorname{Spec}(E)\rightarrow\mathbb{A}^{1}_{k}$,
corresponding to the (monogenic) minimal polynomial of $\alpha$ in $E$.
Therefore, the preceding proposition gives for any integer $q\in\mathbb{Z}$ a
well-defined transfer map:

$\operatorname{Tr}^{MW,\alpha}_{E/k}:\operatorname{K}^{MW}_{q}(E,\omega_{E/k})\rightarrow\operatorname{K}^{MW}_{q}(k),$

which a priori depends on the chosen parameter $\alpha$.

We also define an $\mathcal{L}$-twisted version, for an invertible $k$-vector
space $\mathcal{L}$, as follows:

$\begin{array}[]{rcl}\operatorname{Tr}^{MW,\alpha}_{E/k}:\operatorname{K}^{MW}_{q}(L,\omega_{E/k}\otimes\mathcal{L}_{E})&\rightarrow&\operatorname{K}^{MW}_{q}(k,\mathcal{L}),\\\
\sigma\otimes w\otimes
a&\mapsto&\operatorname{Tr}^{MW,\alpha}_{E/k}\big{(}\langle
u\rangle\sigma\otimes w\big{)}\otimes a^{\prime}\end{array}$

where $\sigma\in\operatorname{K}^{MW}_{q}(E)$, $w\in\omega_{E/k}^{\times}$,
$l\in\mathcal{L}_{E}^{\times}$ and we have written: $l=l^{\prime}\otimes u$
for $l^{\prime}\in\mathcal{L}^{\times}$, $u\in E^{\times}$ (as according to
our notation $\mathcal{L}_{E}=\mathcal{L}\otimes_{k}E$).

######  Remark 4.2.8.

  1. (1)

We will see in 4.4.13 that the above transfers are independent of the
generator $\alpha$ (and extend its definition to the non necessarily monogenic
case).

  2. (2)

Our construction is a variation on Morel’s one, as done in [Mor12, §4.2, 5.1].
The main difference is that one uses appropriate twists (by canonical sheaves)
which allows us to work over an arbitrary base field $k$, in particular
allowing inseparable extensions from the start.

######  4.2.9.

Bass and Tate method, already mentioned, was applied to Milnor K-theory
([BT73, I.5.4]). They constructed the transfer map on Milnor K-theory for
monogenic finite extensions, and later, Kato proved that these transfers
extend to arbitrary finite extensions $E/k$ ([Kat80, §1.7, Prop. 5]), giving a
transfer map393939Kato called this the norm homomorphism:

$\operatorname{Tr}^{M}_{E/k}:\operatorname{K}^{M}_{*}(E)\rightarrow\operatorname{K}^{M}_{*}(k).$

In particular, when $E/k$ is monogenic, this map coincides with Bass-Tate
morphism for any choice of generator $\alpha$ of $E/k$. As the (twisted)
hyperbolic and forgetful maps (2.4.9) are compatible with residues (2.5.8) we
easily derive from the above construction the following compatibility lemma
(extending 4.2.5).

######  Lemma 4.2.10.

Let $E/k$ be a monogenic finite extension with generator $\alpha\in E$. Then
the following diagram is commutative:

$\textstyle{\operatorname{K}^{M}_{*}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{H}}$$\scriptstyle{\operatorname{Tr}^{M}_{E/k}}$$\textstyle{\operatorname{K}^{MW}_{*}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{F}}$$\scriptstyle{\operatorname{Tr}^{MW}_{E/k,\alpha}}$$\textstyle{\operatorname{K}^{M}_{*}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{M}_{E/k}}$$\textstyle{\operatorname{K}^{M}_{*}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{H}}$$\textstyle{\operatorname{K}^{MW}_{*}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{F}}$$\textstyle{\operatorname{K}^{M}_{*}(k).}$

###  4.3. A variation on Scharlau’s quadratic reciprocity property

######  4.3.1.

Let $E/k$ be a finite extension field. Recall that Scharlau has defined in
[Sch72] (Definition p. 79) a notion of transfer maps for Witt groups,
depending on the choice of a $k$-linear map $s:E\rightarrow k$.404040These
maps are denoted by $s^{*}:W(E)\rightarrow W(k)$ in _loc. cit._ , but we will
prefer the notation $s_{*}$ (for obvious reasons). Scharlau originally
considered fields of characteristic not $2$ but the definition makes sense in
arbitrary characteristic. Moreover, one can replace non-degenerate quadratic
forms by non-degenerate symmetric bilinear forms.

Using the differential trace map
$\operatorname{Tr}^{\omega}_{E/k}:\omega_{E/k}\rightarrow k$ (see 6.2.4), it
is possible to give a uniform definition, which does not depend on such a
choice. Moreover, we will see that it coincides with the trace maps
$\operatorname{Tr}^{MW,\alpha}_{E/k}$ just defined in degree $q\leq 0$.

The definition is very similar to Scharlau’s definition, but motivated by the
form of MW-transfers (see 4.2.7), we use $\mathcal{L}$-valued inner product
spaces (see 2.1.14). Given an arbitrary $\omega_{E/k}$-valued inner product
space $\Phi:V\otimes_{E}V\rightarrow\omega_{E/k}$, one can consider the
composite map

$\operatorname{Tr}^{\omega}_{E/k}\circ\Phi:V\otimes_{k}V\rightarrow
V\otimes_{E}V\xrightarrow{\ \Phi\
}\omega_{E/k}\xrightarrow{\operatorname{Tr}^{\omega}_{E/k}}k,$

which is again a non-degenerate symmetric bilinear $k$-form. It is compatible
with isomorphisms and orthogonal sums, therefore it induces a well-defined
map:

$\begin{array}[]{rcl}\operatorname{Tr}^{\omega}_{E/k*}:\operatorname{GW}(E,\omega_{E/k})&\rightarrow&\operatorname{GW}(k)\\\
[\Phi]&\mapsto&[\operatorname{Tr}^{\omega}_{E/k}\circ\Phi].\end{array}$

As, by definition, the map
$\operatorname{Tr}^{\omega}_{E/k}:\omega_{E/k}\rightarrow k$ is $k$-linear,
one deduces that $\operatorname{Tr}^{\omega}_{E/k*}$ is
$\operatorname{GW}(k)$-linear (recall the $\operatorname{GW}(k)$-action on
both sides from 2.1.14).

######  Definition 4.3.2.

Let $E/k$ be an arbitrary finite extension fields. We call the
$\operatorname{GW}(k)$-linear morphism
$\operatorname{Tr}^{\omega}_{E/k*}:\operatorname{GW}(E,\omega_{E/k})\rightarrow\operatorname{GW}(k)$
just defined the _(differential) GW-transfer map_.

Modding out by the ideal $(h)$, one gets a (differential) W-transfer map that
we still denote:
$\operatorname{Tr}^{\omega}_{E/k*}:\operatorname{W}(E,\omega_{E/k})\rightarrow\operatorname{W}(k)$.

######  Example 4.3.3.

If $E/k$ is separable, then $\omega_{E/k}=E$ and
$\operatorname{Tr}^{\omega}_{E/k}$ is just the usual trace map:
$\operatorname{Tr}_{E/k}:E\rightarrow k$ (see 6.3.14). In particular,
$\operatorname{Tr}^{\omega}_{E/k*}=\operatorname{Tr}_{E/k*}:\operatorname{GW}(E)\rightarrow\operatorname{GW}(k),\operatorname{W}(E)\rightarrow\operatorname{W}(k)$
is the usual Scharlau transfer associated with the trace “form”
$\operatorname{Tr}_{E/k}$.

In the inseparable case on the contrary, $\operatorname{Tr}_{E/k}=0$. The link
with Scharlau traces will be explained in 4.3.5.

######  Example 4.3.4.

One can compute the GW-differential transfer maps more explicitly.

Consider a monogenic field extension $E/k$ of degree $d$, written as
$E=k[\alpha]$. Let $f$ be the minimal polynomial of $\alpha$, so that for
$I=(f)$, $E=k[t]/I$. Then, as explain in 4.1.10,
$\omega_{E/k}\simeq(I/I^{2})^{\vee}\otimes\omega_{k[t]/k}$. In particular, the
invertible $k$-vector space $\omega_{E/k}$ admits an explicit base given by
the element $\bar{f}^{*}\otimes dt$. In particular, any $\omega_{E/k}$-valued
inner product space

$\Phi:V\otimes_{E}V\rightarrow\omega_{E/k}$

can be written as $(x,y)\mapsto\phi(x,y)\otimes_{k}(\bar{f}^{*}\otimes dt)$
where $\phi:V\otimes_{E}V\rightarrow E$ is an inner product space.

With this notation, 6.3.13 gives the following computation:

$\operatorname{Tr}^{\omega}_{E/k}\circ\big{(}\phi\otimes(\bar{f}^{*}\otimes
dt)\big{)}=\tau_{E/k}^{\alpha}\circ\phi=:\tau_{E/k*}^{\alpha}(\phi),$

where we recall that the _Tate trace map_ $\tau_{E/k}^{\alpha}:E\rightarrow k$
is the $k$-linear form associated with the element $\alpha^{d-1}$ of the
$k$-base $(1,\alpha,\ldots,\alpha^{d-1})$ of $E$.

When $E/k$ is non monogenic, one writes
$E=k[\alpha_{1},\ldots,\alpha_{n}]=k[t_{1},\ldots,t_{n}]/(f)$, where
$f=(f_{1},...,f_{n})$ for monic polynomials $f_{i}\in k[t_{1},\ldots,t_{n}]$.
Then combining the notation 4.1.10 and 6.3.12, one gets the formula:

$\operatorname{Tr}^{\omega}_{E/k}\circ\Big{(}\phi\otimes\big{(}(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}\otimes(dt_{1}\wedge\ldots
dt_{n})\big{)}\Big{)}=\tau_{f}\circ\phi,$

where $\tau_{f}$ is the Sheja-Storch trace map (6.3.8) associated with the
presentation $f$ of $E/k$.

######  Remark 4.3.5.

Comparison with Scharlau transfer. A particular case of Grothendieck duality
(see 6.2.7) gives the following isomorphism:

$\omega_{E/k}\xrightarrow{\sim}\operatorname{Hom}_{k}(E,k),w\mapsto
s_{w}:=\operatorname{Tr}^{\omega}_{E/k}(-.w).$

According to 2.4.4, one has a canonical identification
$\operatorname{GW}(E,\omega_{E/k})\simeq\operatorname{GW}(E)\otimes_{\mathbb{Z}[E^{\times}]}\mathbb{Z}[\omega_{E/k}^{\times}]$.
With this notation, one can see that the above transfers incorporate all
Scharlau’s transfer maps at once: for $ \sigma\in\operatorname{GW}(E)$ and a
non-zero $w\in\omega_{E/k}$, one gets:

$\operatorname{Tr}^{\omega}_{E/k*}(\sigma\otimes w)=s_{w*}(\sigma).$

######  4.3.6.

One easily derives from the previous definition the following basic properties
of the differential GW-trace map, for a finite extension $\varphi:k\rightarrow
E$ of degree $d$:

  1. (1)

For any $\sigma\in\operatorname{GW}(E,\omega_{E/k})$, one has
$\operatorname{rk}(\operatorname{Tr}^{\omega}_{E/k}(\sigma))=d.\operatorname{rk}(\sigma)$.

  2. (2)

If $L/E$ and $E/k$ are finite extensions,
$\operatorname{Tr}^{\omega}_{L/E*}\circ\operatorname{Tr}^{\omega}_{E/k*}=\operatorname{Tr}^{\omega}_{L/k*}$
where we have hidden the canonical isomorphism
$\omega_{L/k}\simeq\omega_{L/E}\otimes_{L}\omega_{E/k}$ (4.1.7).414141This
follows from the functoriality of the differential trace map.

  3. (3)

For $\sigma\in\operatorname{GW}(E,\omega_{E/k})$,
$\sigma^{\prime}\in\operatorname{GW}(k)$, one has:
$\operatorname{Tr}^{\omega}_{E/k}(\sigma.\varphi_{*}(\sigma^{\prime}))=\operatorname{Tr}^{\omega}_{E/k}(\sigma).\sigma^{\prime}$.424242This
is generically called the projection formula, and more specifically _Frobenius
reciprocity_ in the theory of quadratic forms ([Sch72, p. 80]).

The main result for the GW-differential transfer map is the following
_quadratic reciprocity formula_ which extends to the Milnor-Witt case a
formula due to Scharlau first proved in [Sch72, Th. 4.1], with a similar
proof.

######  Theorem 4.3.7.

Let $k$ be an arbitrary field. Then the following formula holds:

(4.3.7.a)
$\sum_{x\in(\mathbb{P}^{1}_{k})_{(0)}}\operatorname{Tr}^{\omega}_{\kappa_{x}/k*}\circ\partial_{x}=0,$

as maps
$\operatorname{K}^{MW}_{1}\big{(}k(t),\omega_{k(t)/t}\big{)}\rightarrow\operatorname{GW}(k)$.
Here, the map

$\partial_{x}:\operatorname{K}^{MW}_{1}\big{(}k(t),\omega_{k(t)/k}\big{)}\rightarrow\operatorname{GW}(\kappa_{x},\omega_{\kappa_{x}/k})$

stands for the residue map associated with the discrete valuation on $k(t)$
associated to the closed point $x\in\mathbb{P}^{1}_{k}$ (see 4.2.1).

###### Proof.

The abelian group
$\operatorname{K}^{MW}_{1}\big{(}k(t),\omega_{k(t)/k}\big{)}$ is generated by
elements of the form $[f]\otimes dt$ where $f\in k(t)^{\times}$ is a rational
function on $\mathbb{P}^{1}_{k}$. So we need only to check the vanishing on
these particular elements.

Consider the prime decomposition of $f$:

(4.3.7.b) $f=u.\pi_{1}^{m_{1}}\ldots\pi_{r}^{m_{r}},$

where $u\in k^{\times}$, $m_{i}\in\mathbb{Z}$ and $\pi_{i}$ is an irreducible
monic polynomial in $k[t]$. Each polynomial $\pi_{i}$ corresponds to a closed
point $x_{i}$ in $\mathbb{A}^{1}_{k}\subset\mathbb{P}^{1}_{k}$, with residue
field $\kappa_{i}=\kappa(x_{i})=k[t]/(\pi_{i})$. With this notation, we will
write $\alpha_{i}\in\kappa_{i}$ for the obvious generator of $\alpha_{i}$
(_i.e._ corresponding to $t$).

We first remark that, computing the quadratic order of vanishing of at
$\infty$ using the uniformizer $1/t$, we find in
$\operatorname{GW}(k)$:434343Note that $\omega_{\kappa_{\infty}/k}=k$ so that
we identify $\operatorname{GW}(\kappa_{\infty},\omega_{\kappa_{\infty}/k})$
with $\operatorname{GW}(k)$. With this identification, the GW-differential
trace map $\operatorname{Tr}_{\kappa_{\infty}/k}^{\omega}$ is just the
identity.

$\partial_{\infty}([f]\otimes dt)=-d_{\epsilon}\langle
u\rangle\in\operatorname{GW}(k).$

Let us write $f_{i}=\prod_{j\neq i}f_{j}^{m_{j}}$, so that
$f=uf_{i}.\pi_{i}^{m_{i}}$. Applying formula (3.1.10.a) (with the added twist
$dt$), one gets:

$\sum_{x\in(\mathbb{P}^{1}_{k})_{(0)}}\operatorname{Tr}^{\omega}_{\kappa_{x}/k*}\circ\partial_{x}([f]\otimes
dt)=\sum_{i=1}^{r}(m_{i})_{\epsilon}\langle
u\rangle\operatorname{Tr}^{\omega}_{\kappa_{i}/k*}\big{(}\langle
f_{i}(\alpha_{i})\rangle\otimes
dt\otimes\bar{\pi}_{i}^{*}\big{)}-d_{\epsilon}\langle u\rangle.$

Let us denote by $(*)$ the right-hand side, so that we need to show that $(*)$
is $0$ in $\operatorname{GW}(k)$. The (virtual) rank of $(*)$ is

$\sum_{i=1}^{r}m_{i}.\deg(\pi_{i})-d$

which is obviously zero — according to relation (4.3.7.b).

Therefore, one needs only to show that the class of $(*)$ is zero in
$\operatorname{W}(k)$. Obviously, one can assume that $u=1$. Moreover, as
$n_{\epsilon}=0$ for $n$ even in $\operatorname{W}(k)$, one can assume that
$m_{i}=1$ for all $i$.

Let us consider the monogenic $k$-algebra $A=k[t]/(f)$, and write $\alpha$ its
generator. Recall that $A$ is a finite $k$-vector space with basis
$\mathcal{B}=(1,\alpha,\ldots,\alpha^{d-1})$. The Chinese remainder lemma
gives an isomorphism of $k$-algebras:

$\Theta:A\xrightarrow{\ \sim\
}\prod_{i=1}^{r}\kappa_{i},g\mapsto\big{(}f_{i}(\alpha_{i})g(\alpha_{i})\big{)}_{1\leq
i\leq r}.$

Applying 6.2.5, one deduces that

$\sum_{i=1}^{r}\operatorname{Tr}^{\omega}_{\kappa_{i}/k*}\big{(}\langle
f_{i}(\alpha_{i})\rangle\otimes
dt\otimes\bar{\pi}_{i}^{*}\big{)}=\operatorname{Tr}^{\omega}_{A/k*}\big{(}\langle
1\rangle\otimes dt\otimes\bar{f}^{*}\big{)}.$

We are now reduced to show the following equality in $\operatorname{W}(k)$:

(4.3.7.c) $\operatorname{Tr}^{\omega}_{A/k*}\big{(}\langle
1\rangle\otimes(dt\otimes\bar{f}^{*})\big{)}=d_{\epsilon}.$

One can apply 6.3.13 (see 4.3.4) to compute the left-hand side: if one denotes
by $\tau_{A/k}^{\alpha}:A\rightarrow k$ the Tate trace map associated with
$A/k$ and its generator $\alpha$, — that is the linear form associated with
$\alpha^{d-1}$ in the basis $\mathcal{B}$ — this inner product on the
$k$-vector space $A$ is given by the formula:

$A\otimes_{k}A\rightarrow
k,(g,g^{\prime})\mapsto\tau_{A/k}^{\alpha}(gg^{\prime}).$

One easily computes the form of the symmetric $(d\times d)$-matrix of this
symmetric bilinear in the basis $\mathcal{B}$ as:

$\left(\raisebox{0.5pt}{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern
7.5pt\raise-9.6111pt\hbox{\hbox{\kern
0.0pt\raise-2.25555pt\hbox{$\scriptstyle{0}$}}}}}\ignorespaces{}{\hbox{\kern
15.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 31.5pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
15.5pt\raise-19.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
34.0pt\raise-19.22221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-5.5pt\raise-38.44443pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1}$}}}}}}}{\hbox{\kern
15.5pt\raise-38.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
34.0pt\raise-38.44443pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\kern
26.0pt\raise-28.83333pt\hbox{\hbox{\kern
0.0pt\raise-1.62848pt\hbox{$\scriptstyle{*}$}}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces}\right).$

But the class of the corresponding inner product space in
$\operatorname{GW}(k)$ is $d_{\epsilon}=\langle 1,-1,\ldots\rangle$ as it has
a totally isotropic subspace of rank $n$ spanned by $(1,\ldots,\alpha^{n-1})$
if $d=2n$ or $d=2n+1$, and its determinant is $(-1)^{d-1}$. This proves
(4.3.7.c). ∎

######  Remark 4.3.8.

It is interesting to note that the end of the previous proof also shows the
following degree formula, for any finite degree $d$ extension $E/k$:

$\operatorname{Tr}^{\omega}_{E/k*}(\langle 1\rangle\otimes dt)=d_{\epsilon}.$

######  Remark 4.3.9.

Multiplying by $\eta$, and looking modulo the hyperbolic form h (granted the
$\operatorname{GW}(k)$-linearity of each involved operator), the equation
(4.3.7.a) gives a twisted version of Sharlau’s quadratic reciprocity formula:
for any class $\sigma\in\operatorname{W}(k(t),\omega_{k(t)/k})$ of a
$\omega_{k(t)/k}$-valued inner product space over $k(t)$, one has:

$\sum_{x\in\mathbb{P}^{1}_{k,(0)}}\operatorname{Tr}^{\omega}_{\kappa_{x}/k*}(\partial_{x}(\sigma))=0.$

In fact, using 4.3.4 and applying this equality to $\sigma=\sigma_{0}\otimes
dt$, one gets back precisely Scharlau’s formula (see also [GHKS70, §2, Satz]
in the characteristic $2$ case).

The main application of the previous theorem, taking into account the
uniqueness statement of 4.2.3 is the following comparison result between the
two transfer maps we have introduced.

######  Corollary 4.3.10.

Let $E/k$ be a monogenic finite extension field. Then for any generator
$\alpha$ of $E/k$, and any $q<0$, one has commutative diagrams

$\textstyle{\operatorname{K}^{MW}_{0}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{MW,\alpha}_{E/k}}$$\textstyle{\operatorname{K}^{MW}_{0}(k)}$$\textstyle{\operatorname{K}^{MW}_{q}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{MW,\alpha}_{E/k}}$$\textstyle{\operatorname{K}^{MW}_{q}(k)}$$\textstyle{\operatorname{GW}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{E/k*}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{GW}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{W}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{E/k*}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{W}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

where the vertical isomorphisms come from 2.4.4.

###  4.4. General trace maps

######  4.4.1.

Let $E/k$ be a finite extension with canonical module $\omega_{E/k}$.

We have already seen (4.3.2) how the Grothendieck differential trace map
induces a transfer map on twisted Grothendieck-Witt and Witt groups. We now
show how to extend these transfers to Milnor-Witt K-theory using Morel’s
fundamental square from 2.3.7. We first need a lemma.

######  Lemma 4.4.2.

Consider the above notation. For any integer $n\in\mathbb{Z}$, one has:

$\operatorname{Tr}^{\omega}_{E/k*}\big{(}\operatorname{I}^{n}(E,\omega_{E/k})\big{)}\subset\operatorname{I}^{n}(k)$

where we have used notation 2.4.10 for $\operatorname{I}^{n}$ and the transfer
map on Witt groups was defined in 4.3.2.

Using 4.3.5, the lemma follows from [Ara75, Satz 3.3]. At this point, one can
easily deduce it from our earlier computations so we give a proof for
completeness.

###### Proof.

The case $n\geq 0$ is trivial. We note the case $n=1$ is easy (use 4.3.6(1)).
For the other cases, using the functoriality of GW-transfers 4.3.6(2), one
reduces to the case where $E/k$ is monogenic, with say a fixed generator
$\alpha$. This case now follows from 4.3.10, 2.3.5 and the fact
$\operatorname{Tr}^{MW,\alpha}_{E/k}$ (defined in 4.2.7) commutes with
multiplication by $\eta$ and $h$. ∎

In particular we get well-defined transfer maps on the algebra functor
$\operatorname{I}^{*}$. As an intermediate step, we show that these transfers
are compatible with the monogenic transfers obtained so far on the Milnor-Witt
K-theory functor (4.2.7).

######  Lemma 4.4.3.

Let $E/k$ be a monogenic finite extension, with a generator $\alpha\in E$.
Then the following diagram is commutative:

$\textstyle{\operatorname{K}^{MW}_{*}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}_{E}}$$\scriptstyle{\operatorname{Tr}^{MW,\alpha}_{E/k}}$$\textstyle{\operatorname{K}^{M}_{*}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}_{k}}$$\textstyle{\operatorname{I}^{*}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{E/k*}}$$\textstyle{\operatorname{I}^{*}(k).}$

Given the previous lemma, and the construction of the morphism $\mu^{\prime}$
(see 2.3.6) this statement reduces to 4.3.10.

######  4.4.4.

According to 2.4.11, $\operatorname{K}^{MW}_{n}(E,\omega_{E/k})$ can be
identified with the abelian group made of pairs
$(\sigma,\tau)\in\operatorname{I}^{n}(E)\times\operatorname{K}_{n}(E)$ such
that $\pi(\sigma)=\mu(\tau)$. The following lemma is the last step needed to
define the transfer map associated with $E/k$ on Milnor-Witt K-theory.

######  Lemma 4.4.5.

Consider the above notation. Then one has the following equality in
$\bar{\operatorname{I}}^{n}(k)$:

$\pi\big{(}\operatorname{Tr}^{\omega}_{E/k*}(\sigma)\big{)}=\mu\big{(}\operatorname{Tr}^{M}_{E/k}(\tau)\big{)}.$

###### Proof.

By functoriality of the differential trace map (6.2.6) and of Kato’s transfer
map on Milnor K-theory, one reduces to the case of finite monogenic extensions
$E=k[\alpha]$. Then the result follows from the existence of the trace map
$\operatorname{Tr}^{MW,\alpha}_{E/k}$, and its compatibility with both Kato’s
transfer (4.2.10) and the differential transfer on $\operatorname{I}^{*}$
(4.4.3). ∎

We finally obtain the main definition of this section.

######  Definition 4.4.6.

Let $E/k$ be a finite extension with canonical module $\omega_{E/k}$. One
defines the transfer map on Milnor-Witt K-theory by the following formula:

$\textstyle{\operatorname{K}^{MW}_{*}(E,\omega_{E/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{MW}_{E/k}}$$\scriptstyle{(\mu^{\prime}_{E},\mathbf{F}_{E})}$$\scriptstyle{\sim}$$\textstyle{\operatorname{K}^{MW}_{*}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mu^{\prime}_{k},\mathbf{F}_{k})}$$\scriptstyle{\sim}$$\textstyle{\operatorname{I}^{*}(E,\omega_{E/k})\times_{\bar{\operatorname{I}}^{*}(E)}\operatorname{K}^{M}_{*}(E)}$$\textstyle{\operatorname{I}^{*}(k)\times_{\bar{\operatorname{I}}^{*}(k)}\operatorname{K}^{M}_{*}(k)}$$\textstyle{(\sigma,\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}\operatorname{Tr}^{\omega}_{E/k*}(\sigma),\operatorname{Tr}^{M}_{E/k}(\tau)\big{)}}$

well-defined according to the previous lemma. The vertical isomorphisms come
from 2.4.11.

As in the end of 4.2.7, one also defines for an invertible $k$-vector space
$\mathcal{L}$, an $\mathcal{L}$-twisted transfers:

$\operatorname{Tr}^{MW}_{E/k}:\operatorname{K}^{MW}_{q}(E,\omega_{E/k}\otimes\mathcal{L}_{E})\rightarrow\operatorname{K}^{MW}_{q}(k,\mathcal{L}).$

When we denote by $\varphi:k\rightarrow E$ the structural map of the extension
$E/k$, it is customary to use the notation
$\varphi^{*}=\operatorname{Tr}^{MW}_{E/k}$. We also call it occasionally the
trace map.444444Other terminologies that we prefer to avoid are the norm
(Kato) and corestriction (Rost).

######  Remark 4.4.7.

This trace map has all the good properties of its analog on Milnor K-theory.
It is compatible with composition (as this is the case for
$\operatorname{Tr}^{\omega}_{E/k}$ and $\operatorname{Tr}^{M}_{E/k}$). It
satisfies the so-called projection formula: for
$(\sigma,\beta)\in\operatorname{K}^{MW}_{*}(E,\omega_{E/k}\otimes\mathcal{L}_{E})\times\operatorname{K}^{MW}_{*}(k,\mathcal{M})$,
one has in $\operatorname{K}^{MW}_{*}(k,\mathcal{L}_{k}\otimes\mathcal{M})$:

$\varphi^{*}(\sigma.\varphi_{*}(\beta))=\varphi^{*}(\sigma).\beta.$

This follows from 4.3.6(3) and the corresponding formula for Milnor K-theory
(see [BT73, formula (5), p. 378]).

Finally, we note that from a geometric point of view, if one denotes by
$f:\operatorname{\mathrm{Spec}}(E)\rightarrow\operatorname{\mathrm{Spec}}(k)$
the induced morphism, one can also denote: $\varphi^{*}=f_{*}$ and
$\varphi_{*}=f^{*}$. In this way, the previous formula looks like the
”classical” projection formula (for Chow groups, cohomology,…)

As an immediate corollary of the previous definition, we obtain the following
explicit description of transfers on Milnor-Witt K-theory.

######  Corollary 4.4.8.

Let $E/k$ be a finite extension with canonical module $\omega_{E/k}$, and $n$
an integer.

  1. (1)

If $n=0$ (resp. $n<0$) then through the identification

$\operatorname{K}^{MW}_{0}(E,\omega_{E/k})=\operatorname{GW}(E,\omega_{E/k})\text{
(resp.
}\operatorname{K}^{MW}_{n}(E,\omega_{E/k})=\operatorname{W}(E,\omega_{E/k})\text{)}$

of 2.3.1, one has
$\operatorname{Tr}^{MW}_{E/k}=\operatorname{Tr}^{\omega}_{E/k*}$ where
$\operatorname{Tr}^{\omega}_{E/k}$ is the differential trace map (see 4.3.2).

  2. (2)

If $n>0$, any element $\sigma\in\operatorname{K}^{MW}_{n}(E,\omega_{E/k})$ can
be written as a sum of elements of the form $([\phi]\otimes
w,\sigma^{\prime})$ where:

     * •

$(V,\phi:V\otimes_{E}V\rightarrow E)$ is an inner product space over $E$,
$[\phi]$ is its class in $\operatorname{W}(E)$ and
$[\phi]\in\operatorname{I}^{n}(E)$,

     * •

$w\in\omega_{E/k}$ is a non-zero differential $k$-form on $E$ of maximal
degree if $E/k$ is not separable, and just a unit of $E$ if $E/k$ is
separable,

     * •

$\sigma^{\prime}=\\{u_{1},\ldots,u_{n}\\}$ is a symbol in
$\operatorname{K}^{M}_{n}(E)$, for certain units $u_{i}\in E^{\times}$.

For such an element, one has:

$\operatorname{Tr}^{MW}_{E/k}([\phi]\otimes
w,\sigma^{\prime})=\big{(}[\operatorname{Tr}^{\omega}_{E/k}\circ(\phi.w)],\operatorname{Tr}^{M}_{E/k}(\sigma^{\prime})\big{)}$

where $\operatorname{Tr}^{\omega}_{E/k}\circ(\phi.w)$ is the class in
$\operatorname{W}(k)$ (and in fact $\operatorname{I}^{n}(k)$) of the inner
product space on $V$ over $k$ with bilinear form

$(x,y)\mapsto\operatorname{Tr}^{\omega}_{E/k}\big{(}\phi(x,y).w\big{)},$

and $\operatorname{Tr}^{M}_{E/k}$ is the transfer map on Milnor K-theory.

######  Example 4.4.9.

In general, we refer the reader to 4.3.4 and 4.3.5 for the computation of the
differential trace map $\operatorname{Tr}^{\omega}_{E/k*}$ on the
Grothendieck-Witt or Witt group. One can single out the following explicit
computations.

  1. (1)

If $E/k$ is separable, then $\omega_{E/k}=E$ and
$\operatorname{Tr}^{\omega}_{E/k}=\operatorname{Tr}_{E/k}$ is the usual trace
map (6.3.14). In particular,
$\operatorname{K}^{MW}_{0}(E,\omega_{E/k})\simeq\operatorname{GW}(E)$ and for
any unit $u\in\mathbf{E}^{\times}$, $\operatorname{Tr}^{MW}(\langle
u\rangle)=[\operatorname{Tr}_{E/k}(u.-)]$ the $\operatorname{GW}$-class of the
_scaled trace form_ , $(x,y)\mapsto\operatorname{Tr}_{E/k}(uxy)$.

  2. (2)

Let $E/k$ be a finite monogeneous extension of degree $d$, with generator
$\alpha$.

According to 4.3.5, one has an isomorphism

$\omega_{E/k}\simeq\operatorname{Hom}_{k}(E,k),w\mapsto s_{w}.$

In particular, there exists a unique non-zero form $w\in\omega_{E/k}$ such
that $s_{w}$ is the $k$-linear form which maps $\alpha^{0}$ to $1$ and
$\alpha^{i}$ to $0$ for $0<i<d$.

Then for any unit $u\in E^{\times}$, and for the particular choice of $w$ made
above, one has:

$\operatorname{Tr}^{MW}_{E/k}([u]\otimes
w)=[N_{E/k}(u)]\in\operatorname{K}^{MW}_{1}(k)$

where $N_{E/k}:E^{\times}\rightarrow k^{\times}$ is the usual norm of the
finite extension $E/k$. According to the previous corollary, this follows from
the [Lam05, VII, Cor. 2.4] for the Witt part454545the computation of _loc.
cit._ extends in characteristic $2$ as well and [BT73, I.§5, Th. 5.6] for the
Milnor part.

This formula generalizes to arbitrary finite extension provided one chooses
the correct differential form $w$.

  3. (3)

Let $k$ be a field of characteristic $p>0$, $a\in k$ be an element which is
not a $p$-th root and $E=k[\sqrt[q]{a}]=k[t]/(t^{q}-a)$. Put
$\alpha=\sqrt[q]{a}\in E$. There exists a canonical non-zero element
$w=dt\otimes(\overline{t^{q}-a})^{*}$ of $\omega_{E/k}$ (see 4.1.10 with
$n=1$).

Then for any unit $u\in E^{\times}$, and again for the particular choice of
$w$ made above, one has:

$\operatorname{Tr}^{MW}(\langle u\rangle\otimes w)=[\tau^{\alpha}_{E/k}(u.-)]$

where $\tau^{\alpha}_{E/k}$ is the _Tate trace map_ associated with the $q$-th
root $\alpha$ (see 6.3.10), and $[\tau^{\alpha}_{E/k}(u.-)]$ is the
$\operatorname{GW}$-class of the scaled (Tate) trace form of the $k$-vector
space $E$:

$E\otimes_{k}E\rightarrow k,(x,y)\mapsto\tau^{\alpha}_{E/k}(uxy).$

Note in particular that one gets the following degree formula:

$\operatorname{Tr}^{MW}(\langle 1\rangle\otimes w)=[\tau^{\alpha}_{E/k}]$

In comparison with the last example, one gets the following more general
_degree formula_ in Milnor-Witt K-theory.

######  Corollary 4.4.10.

Let $E/k$ be a finite extension of degree $d$. We consider a minimal family of
generators $(\alpha_{1},...,\alpha_{n})$ and the associated presentation
$E=k[t_{1},\ldots,t_{n}]/(f_{1},\ldots,f_{n})$ as in 4.1.10. Let
$w=(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}\otimes(dt_{1}\wedge\ldots
dt_{n})$ be the canonical element of $\omega_{E/k}$ as in _loc. cit._

Then one has in $\operatorname{K}^{MW}_{0}(k)=\operatorname{GW}(k)$:

$\operatorname{Tr}^{MW}_{E/k}(\langle 1\rangle\otimes w)=d_{\epsilon}$

where we have used the notation of 2.2.18.

###### Proof.

By multiplicativity of $d_{\epsilon}$ (2.2.18), and the functoriality of the
MW-trace map, one reduces to the monogenic case. Then, it follows from 4.3.8.
∎

######  Remark 4.4.11.

In general, any element of $w^{\prime}=\omega_{E/k}$ can be written as
$w^{\prime}=u.w$. One should be careful however that if one replaces $w$ in
the above corollary by $w^{\prime}$, this completely changes the above result.
For example, in the case of 4.4.9(3), one gets

$\operatorname{Tr}^{MW}(\langle 1\rangle\otimes
w^{\prime})=[\tau^{\alpha}_{E/k}(u.-)].$

######  4.4.12.

Finally, we want to compare the previous definition of transfers on Milnor-
Witt K-theory with the original one due to Morel for finitely generated
extensions of some perfect field: [Mor12, Rem. 4.32].

Recall the construction of Morel, for a finite extension $E/k$.464646Note that
contrary to the previously known constructions, we do not to assume that $E$
and $k$ are finitely generated extensions over some base perfect field. We fix
a finite generating family $\underline{\alpha}=(\alpha_{1},\ldots,\alpha_{n})$
of $E/k$, to which we associate a tower of finite monogenic extensions
$\kappa_{i}=k[\alpha_{1},\ldots,\alpha_{i}]$:

$k\subset\kappa_{1}\subset\ldots\subset\kappa_{n}=E.$

Then we can define the following composite map, denoted by
$\operatorname{Tr}^{MW,\underline{\alpha}}_{E/k}$:

$\displaystyle\operatorname{K}^{MW}_{q}(E,\omega_{E/k})$
$\displaystyle\simeq\operatorname{K}^{MW}_{q}(E,\omega_{E/\kappa_{n-1}}\otimes\omega_{\kappa_{n-1}/k}|_{E})\xrightarrow{\operatorname{Tr}^{MW,\alpha_{n}}_{E/\kappa_{n-1}}}\operatorname{K}^{MW}_{q}(\kappa_{n-1},\omega_{\kappa_{n-1}/k})$
$\displaystyle\simeq\operatorname{K}^{MW}_{q}(\kappa_{n-1},\omega_{\kappa_{n-1}/\kappa_{n-2}}\otimes\omega_{\kappa_{n-2}/k}|_{\kappa_{n-1}})\xrightarrow{\operatorname{Tr}^{MW,\alpha_{n-1}}_{\kappa_{n-1}/\kappa_{n-2}}}\ldots$
$\displaystyle\ldots\operatorname{K}^{MW}_{q}(\kappa_{1},\omega_{\kappa_{1}/k})\xrightarrow{\operatorname{Tr}^{MW,\alpha_{1}}_{\kappa_{1}/k}}\operatorname{K}^{MW}_{q}(k)$

where the morphism $\operatorname{Tr}^{MW,x_{i}}_{\kappa_{i}/\kappa_{i-1}}$ is
the $(\omega_{\kappa_{i-1}/k})$-twisted MW-transfer associated with
$(\kappa_{i}/\kappa_{i-1},\alpha_{i})$, as defined in 4.2.7.

The main result of [Mor12, §4.2] (see Th. 4.27), is that this composite map,
at least for finitely generated extensions of some perfect base field, is
independent of the chosen family of generators. This result has also been
proved later in [Fel20b] by direct transport of the proof of Kato (again under
the same assumptions). Actually, given the method we have chosen, we get
another proof of this theorem (without any restriction on the fields
considered).

######  Proposition 4.4.13.

Consider the above notation. Then one has an equality:

$\operatorname{Tr}^{MW}_{E/k}=\operatorname{Tr}^{MW,\underline{\alpha}}_{E/k}$

where the left hand-side was defined in 4.4.6.

In particular, the computations given precedingly apply to the already known
(geometric) transfer map on Milnor-Witt K-theory.

###### Proof.

As the transfers of 4.4.6 are compatible with composition (4.4.7), one reduces
to the monogenic case. This is then a consequence (already observed) of the
definition, and lemmas 4.2.10, 4.4.3. ∎

##  5\. The Milnor-Witt premodule structure

We now summarize the properties of Milnor-Witt’s K-theory that we have
obtained in the previous sections. They correspond to the axioms of Feld’s
theory of Milnor-Witt premodules [Fel20a].474747This theory follows the model
of Rost cycle modules [Ros96], and its developments made in Schmid’s thesis
[Sch97] for application to the Witt groups.

The axioms are difficult to grasp at first sight. A general rule is that
Milnor-Witt K-theory should be viewed as a bigraded cohomology theory on
(spectrum of) fields equipped with a natural functoriality, an exceptional
functoriality (called transfers), a structure of bigraded algebra (product)
and residue maps (corresponding to boundary morphisms of localization exact
sequences).

###  5.1. Functorialities

For any triple $(E,\mathcal{L},n)$, where $E$ is a field, $\mathcal{L}$ an
invertible $E$-vector space, and $n\in\mathbb{Z}$ an integer, one has an
abelian group $\operatorname{K}^{MW}_{n}(E,\mathcal{L})$. It satisfies the
following functorial behavior (as in [Fel20a, Def. 3.1]):

  1. D1

(see 2.4.6(2)): Given any morphism $\varphi:E\rightarrow F$ of fields, one has
a morphism of abelian groups:

$\varphi_{*}:\operatorname{K}^{MW}_{n}(E,\mathcal{L})\rightarrow\operatorname{K}^{MW}_{n}(F,\mathcal{L}\otimes_{E}F).$

  2. D2

(see 4.4.6): Given a finite morphism $\varphi:E\rightarrow F$ of fields, one
has a _transfer map_ :

$\psi^{*}=\operatorname{Tr}^{MW}_{F/E}:\operatorname{K}^{MW}_{n}(F,\omega_{F/E}\otimes_{E}\mathcal{L})\rightarrow\operatorname{K}^{MW}_{n}(E,\mathcal{L})$

where $\omega_{F/E}$ is the canonical invertible $F$-vector space associated
with the finite field extension $F/E$ (see 4.1.5).

  3. D3

(see 2.4.6(1)): It has a structure of bigraded algebras. Given triples
$(E,\mathcal{L},n)$ and $(E,\mathcal{M},m)$, one has a _product_ :

$\operatorname{K}^{MW}_{n}(E,\mathcal{L})\otimes\operatorname{K}^{MW}_{m}(E,\mathcal{M})\rightarrow\operatorname{K}^{MW}_{n+m}(E,\mathcal{L}\otimes_{E}\mathcal{M}).$

In other words, $\operatorname{K}^{MW}_{*}(E,*)$ is a
$\operatorname{K}^{MW}_{*}(E,*)$-module, graded with respect to $\mathbb{Z}$
and to the set of isomorphism classes of invertible $E$-vector spaces.

  4. D4

(see 2.5.2 and 2.5.5): Let $(E,v)$ be a discretely valued field with ring of
integers $\mathcal{O}_{v}$, $\mathcal{L}$ be an invertible
$\mathcal{O}_{v}$-modules and $n\in\mathbb{Z}$ an integer. We let $\kappa_{v}$
be the residue field, $\mathcal{L}_{E}=\mathcal{L}\otimes_{\mathcal{O}_{v}}E$.
One has a morphism of abelian groups, called the _residue map_ :

$\partial_{v}:\operatorname{K}^{MW}_{n}(E,\mathcal{L}_{E})\rightarrow\operatorname{K}^{MW}_{n}(E,\omega_{v}\otimes_{\mathcal{O}_{v}}\mathcal{L})$

where $\omega_{v}=(\mathcal{M}_{v}/\mathcal{M}_{v}^{2})^{\vee}$ — the normal
sheaf of
$\operatorname{Spec}(\kappa_{v})\rightarrow\operatorname{Spec}(\mathcal{O}_{v})$.

There is a further functorial property hidden in the axioms of [Fel20a] that
we now state explicitly:

  1. D1’

(see 2.4.6(3)) Given an isomorphism $\theta:\mathcal{L}\rightarrow\mathcal{M}$
of $E$-vector spaces, one has a morphism of abelian groups:

$\theta_{*}:\operatorname{K}^{MW}_{n}(E,\mathcal{L})\rightarrow\operatorname{K}^{MW}_{n}(E,\mathcal{M}).$

######  Remark 5.1.1.

Recall from 2.4.7 that, using the category of twisted fields, one can unify
the functoriality datas of (D1) and (D1’).

######  5.1.2.

There are many similar examples that we have seen in this text.

  1. (1)

The periodized Witt ring $\operatorname{W}[t,t^{-1}]$, which therefore becomes
a $\mathbb{Z}$-graded algebra: this comes from the isomorphism $\phi$ of 2.3.3
and the fact multiplication by $\eta$ is compatible with datas D* on
$\operatorname{K}^{MW}_{*}$.

In particular, the canonical map

$\operatorname{K}^{MW}_{*}\rightarrow\operatorname{W}[t,t^{-1}]$

(essentially obtained by inverting $\eta$,) is compatible with datas D*.

  2. (2)

The graded algebra $\operatorname{I}^{*}$ associated with the fundamental
ideal $\operatorname{I}$ of the Witt group (2.2.4): a quick way of seeing that
is to use the isomorphism $\psi$ of 2.3.5 and the fact multiplication by $h$
is compatible with all datas D* on $\operatorname{K}^{MW}_{*}$.

In particular, the inclusion
$\operatorname{I}^{*}\subset\operatorname{W}[t,t^{-1}]$, as well as the
canonical map
$\mu^{\prime}:\operatorname{K}^{MW}_{*}\rightarrow\operatorname{I}^{*}$
(2.3.7) are compatible with all datas D*.

  3. (3)

The Milnor K-theory $\operatorname{K}^{M}_{*}$: a first way of seing that is
the isomorphism from 2.2.9, and again the fact that multiplication by $\eta$
is compatible with datas D* on $\operatorname{K}^{MW}_{*}$.

On the other hand, recall that in this example, one has a canonical
isomorphism
$\operatorname{K}^{M}_{*}(E,\mathcal{L})\simeq\operatorname{K}^{M}_{*}(E)$
(2.4.8). Or in other words, the data D1* is trivial: for any automorphism
$\theta:\mathcal{L}\rightarrow\mathcal{L}$ of invertible $E$-vector spaces,
the map
$\theta_{*}:\operatorname{K}^{M}_{*}(E,\mathcal{L})\rightarrow\operatorname{K}^{M}_{*}(E,\mathcal{L})$
is equal to the identity. We will say that $\operatorname{K}^{M}_{*}$ is
_orientable_.

In this case, the functoriality D* actually corresponds to the functoriality
of Rost cycle premodules [Ros96, Def. 1.1]. Moreover, the hyperbolic and
forgetful maps

$\operatorname{K}^{M}_{*}\xrightarrow{H}\operatorname{K}^{MW}_{*}\xrightarrow{F}\operatorname{K}^{MW}_{*}$

of 2.2.10 are compatible with data D*: D1 and D1’ are obvious, D2 comes from
4.4.6, D3 comes from the fact both maps are morphisms of rings, D4 was
observed in 2.5.8.

  4. (4)

The graded algebra $\bar{\operatorname{I}}^{*}$ (see again 2.2.4): a quick way
of seing that is to use the Milnor conjecture 2.2.3 and to use the preceding
point. One can also use the fact $\operatorname{I}^{*}$ is a sub-algebra of
$\operatorname{W}[t,t^{-1}]$ and therefore, all datas D* descend to the
quotient $\bar{\operatorname{I}}^{*}$, as $t$ on the right hand-side is
compatible with all datas D*.

Note that $\bar{\operatorname{I}}^{*}$ is also orientable as in the previous
point. According to the previous remarks, one sees that the canonical maps
(2.3.7)

$\operatorname{K}^{M}_{*}\xrightarrow{\mu}\bar{\operatorname{I}}^{*},\quad\operatorname{I}^{*}\xrightarrow{\pi}\bar{\operatorname{I}}^{*}$

are compatible with the datas D*.

###  5.2. Rules

The preceding functoriality satisfy a very precise list of rules, again taken
from [Fel20a, Def. 3.1]. Let $\varphi:E\rightarrow F$, $\psi:F\rightarrow L$
be morphims of fields, and respectively $\Phi:E\rightarrow F$,
$\Psi:F\rightarrow L$ be finite morphims of fields,

  1. R1a

(obvious) $(\psi\circ\varphi)_{*}=\psi_{*}\varphi_{*}$

  2. R1b

(see 4.4.7) $(\Psi\circ\Phi)^{*}=\Phi^{*}\Psi^{*}$

  3. R1c

(see 5.3.9 more generally) Assume that $\Phi$ or $\psi$ are separable. Then:

$\psi_{*}\Phi^{*}=\sum_{x\subset F\otimes_{E}L}\Phi_{x}^{*}\psi_{x*}$

where $x$ is a prime ideal, $\kappa(x)=(F\otimes_{E}L)/x$ its residue field,
$\Phi_{x}:E\rightarrow\kappa(x)$ and $\psi_{x}:E\rightarrow\kappa(x)$ the
induced map, and we have used the fact $\omega_{F/E}=F$,
$\omega_{\kappa(x)/E}=\kappa(x)$.

######  Remark 5.2.1.

The statement (R1c) will be reinforced in _loc. cit._ , following an idea of
[Fel20b]. Note however that this formula is enough to develop the theory of
Chow-Witt groups (especially pullbacks). Hovewer, one can give a firect proof
of (R1c) from our definition of Milnor-Witt transfers (4.4.6): on reduces to
the corresponding formulas for Milnor K-theory (see [BT73, (5.8)]) and for the
differential trace map (see [Con00], last statement of 3.4.1, or use the
explicit computation of 6.3.12).

Next, one assumes $\sigma,\sigma^{\prime},\beta$ are elements (”Milnor-Witt
symbols”) of the relevant Milnor-Witt K-group:

  1. R2a

(obvious)
$\varphi_{*}(\sigma.\sigma^{\prime})=\varphi_{*}(\sigma).\varphi_{*}(\sigma^{\prime})$

  2. R2b

(see 4.4.7) $\Phi^{*}\big{(}\Phi_{*}(\sigma)\ .\ \beta\big{)}=\sigma\ .\
\Phi^{*}(\beta)$

  3. R2c

(see 4.4.7) $\Phi^{*}\big{(}\sigma\ .\
\Phi_{*}(\beta)\big{)}=\Phi^{*}(\sigma)\ .\ \beta$

Finally, one considers discretely valued fields $(E,v)$, $(F,w)$,
$\mathcal{O}_{v}$, $\mathcal{O}_{w}$, (resp.
$\mathcal{M}_{v}$$\mathcal{M}_{w}$) the corresponding valuation ring (resp.
maximal ideal). In (R3a,c,d) (resp. R3b), we consider in addition a morphism
$\varphi:E\rightarrow F$ (resp. $\Phi:E\rightarrow F$).

  1. R3a

(formula (Res2) of 2.5.2) Assume that $w\circ\varphi=v$. Thus, one has an
induced morphism $\varphi:\kappa(v)\rightarrow\kappa(w)$ and an induced
isomorphism of invertible $\kappa(v)$-vector spaces:

$\displaystyle\theta:\omega_{v}=(\mathcal{M}_{v}/\mathcal{M}_{v})^{\vee}\otimes_{\kappa(v)}\kappa(w)$
$\displaystyle\rightarrow(\mathcal{M}_{w}/\mathcal{M}_{w})^{\vee}=\omega_{w}$
$\displaystyle\bar{\pi}^{*}$
$\displaystyle\mapsto\overline{\varphi(\pi)}^{*}.$

Then:
$\partial_{w}\circ\varphi_{*}=\theta_{*}\circ\bar{\varphi}\circ\partial_{v}$.

  2. R3b

(see [Dég23]) Let $B$ be the integral closure of $\mathcal{O}_{v}$ in $F$.
Then a prime ideal of $B$ is the same thing as a discrete valuation $w$ of $F$
extending $v$ (with eventually a ramification index). For such a valuation
$w$, one considers the commutative diagram

$\textstyle{\kappa_{w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta}$$\textstyle{\mathcal{O}_{w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{w}}$$\textstyle{\mathcal{O}_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$

and the canonical isomorphism of invertible $\kappa_{w}$-vector spaces (apply
4.1.8 with $\Theta$):

$\theta:\omega_{w}\otimes_{\mathcal{O}_{w}}\omega_{\mathcal{O}_{w}/\mathcal{O}_{v}}\xrightarrow{\
\sim\ }\omega_{\kappa_{w}/\kappa_{v}}\otimes_{\kappa_{v}}\omega_{v}$

Then one has:

$\partial_{v}\circ\Phi^{*}=\sum_{w/v}\Phi_{w}^{*}\circ\theta_{*}\circ\partial_{w}.$

  3. R3c

(formula (Res2) of 2.5.2) Assume $w\circ\varphi=0$. Then
$\partial_{w}\circ\varphi_{*}=0$.

  4. R3d

(formula (Res2) of 2.5.2) Assume $w\circ\varphi=0$ and let
$\varphi:E\rightarrow\kappa(w)$ be the morphism induced by
$\varphi:F\rightarrow E$. Let $\pi$ be a prime of $w$, and consider the
resulting trivialization (sending $\bar{\pi}^{*}$ to $1$):

$\theta^{\pi}:\omega_{v}\rightarrow E$

so that $\partial_{w}^{\pi}=\theta^{\pi}_{*}\circ\partial_{w}$, and:
$s_{w}^{\pi}(\sigma)=\theta^{\pi}_{*}\circ\partial_{w}([\pi].\sigma)$.

Then $s_{w}^{\pi}\circ\varphi_{*}=\bar{\varphi}_{*}$.

  5. R3e

(see 2.5.4) The following formulas hold:

$\displaystyle\partial_{v}([u].\sigma)$
$\displaystyle=\epsilon[\bar{u}].\partial_{v}(\sigma),$
$\displaystyle\partial_{v}(\eta.\sigma)$
$\displaystyle=\eta.\partial_{v}(\sigma)$

where $u\in\mathcal{O}_{v}^{\times}$ is a unit of $v$.

######  Remark 5.2.2.

Axiom (R3b) is certainly the most difficult. It was first proved in [Fas08,
Cor. 10.4.5] with different conventions than ours, using the Gersten-Witt
complex and in characteristic not $2$. It is also proved when
$\mathcal{O}_{v}$ is an essentially smooth $k$-algebra by Morel in [Mor12, Th.
5.26]. We will give a proof in [Dég23], using the formalism of Rost-Schmid
complexes in dimension greater than $1$.

The last axiom is tautological for Milnor-Witt K-theory, but useful for
computations. Let us first remark that, given an invertible $E$-vector space
$\mathcal{L}$, one has an isomorphism:

$E^{\times}\rightarrow\mathrm{Aut}_{E}(\mathcal{L}),u\mapsto(l\mapsto u.l).$

Given an $E$-automorphism $\Theta$ of $\mathcal{L}$, one denotes by
$\delta_{\Theta}\in E^{\times}$ the corresponding unit.

  1. R4a

For any automorphism $\Theta$ of an invertible $E$-vector space $\mathcal{L}$,
and any $\sigma\in\operatorname{K}^{MW}_{*}(E,\mathcal{L})$, one has
$\Theta_{*}(\sigma)=\langle\delta_{\Theta}\rangle.\sigma$.

As already mentioned, datas D* and relation R* on $\operatorname{K}^{MW}_{*}$,
once restricted to finitely generated extensions of a perfect field $k$ and
valuations trivial on $k$ in D4, defines of MW-premodule in the sense of Feld
([Fel20a]). In fact, we can forget about the existence of a base field $k$, so
we introduce the following obvious extension of Feld’s axioms, inspired by
[DFJ].

######  Definition 5.2.3.

An application $M$ from fields to $\mathbb{Z}$-graded abelian groups equipped
with datas D* and satisfying the relations R* stated above will be called an
_absolute (cohomological) MW-premodule_.

Morphisms of absolute MW-premodules are given by transformations natural with
respect to datas D1, D1’, D2 and commuting with datas D3 and D4.

Thus $\operatorname{K}^{MW}_{*}$ is an absolute MW-premodule.

######  5.2.4.

We give a more theoretical view on the structure of a MW-premodule $M$. First,
$M$ is a $\mathbb{Z}$-graded ”enriched” functor from (twisted) fields to
$\mathbb{Z}$-graded abelian groups $\mathscr{A}b^{\mathbb{Z}}$:

  1. (1)

covariant with respect to morphisms of fields (D1+R1a), and more precisely
with the category of _twisted fields_ (+D1’+R4, see 2.4.7),

  2. (2)

contravariant with respect to finite field extensions up to a twists (D2+R1b).

These functorialities correspond to the fact $\operatorname{K}^{MW}_{*}$ is a
_twisted homology theory_ with respect to fields (_i.e._ a twisted cohomology
with respect integral $0$-dimensional schemes). In particular, the second
functoriality correspond to the _exceptional functoriality_ aka Gysin
morphisms.

One can define a monoidal structure on such bi-functors in such a way that
$\operatorname{K}^{MW}_{*}$ is a monoid. Because of the exceptional
functoriality, this is non-trivial and one must use the equivalence of [Fel21,
Th 4.0.1] to get the right formulas. Then data D3 as well as relations R2a,
R2, R2c, can be stated by saying that $M$ is a left
$\operatorname{K}^{MW}_{*}$-module.484848Beware also that that if one wants to
a symmetric monoidal structure on the both the category and on
$\operatorname{K}^{MW}_{*}$, one has to work with graded invertible vector
spaces over fields instead of just invertible vector spaces (see [Fel20a,
§3.2] and [Fas20, Def. 1.18] and the discussion thereafter). This issue will
be discussed when considering products on Chow-Witt groups ([Dég23]).

Finally, data D4 is the trace of a localization exact sequence. In fact, it
should be interpreted as a boundary map in such an exact sequence (see 2.5.9
for $\operatorname{K}^{MW}_{*}$). Indeed, this is exactly the case when one
considers the localization long exact sequence of Section 3.3 applied to the
spectrum of the valuation ring $\mathcal{O}_{v}$, and the immersion of the
closed point.

######  Example 5.2.5.

Further examples of absolute MW-premodules comes from the setting of 5.1.2.
Indeed, one deduces from the construction of datas D* on $W[t,t^{-1}]$,
$\operatorname{I}^{*}$, $\operatorname{K}^{M}_{*}$,
$\bar{\operatorname{I}}^{*}$ that they all satisfy the above relations. In
particular, they all define absolute MW-premodules.

Moreover, one has the following commutative diagram of MW-premodules:

$\textstyle{\operatorname{K}^{MW}_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{K}^{MW}_{*}[\eta^{-1}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\sim}$$\textstyle{\operatorname{W}[t,t^{-1}]}$$\textstyle{\operatorname{K}^{MW}_{*}/(h)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{K}^{W}_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{\sim}$$\textstyle{\operatorname{I}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

from 2.3.5 and

$\textstyle{\operatorname{K}^{MW}_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}}$$\scriptstyle{\operatorname{F}}$$\textstyle{\operatorname{K}^{MW}_{*}/(\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{K}^{M}_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\operatorname{I}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{\operatorname{I}}^{*}}$$\textstyle{\operatorname{K}^{M}_{*}/(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$

from 2.3.7 and 2.4.11. The last diagram is a cartesian square, and $\mu$ is
the _Milnor map_ (2.2.3).

One has a last morphism, the hyperbolic map:

$\operatorname{K}^{M}_{*}\xrightarrow{\operatorname{H}}\operatorname{K}^{MW}_{*}$

and we recall that $\operatorname{F}\circ\operatorname{H}=2.\operatorname{Id}$
(see 2.2.11).

###  5.3. Stronger axioms and quadratic multiplicites

In this section, we formulate following Feld [Fel20a, Fel20b], stronger forms
of properties (R1c) and (R3a) involving multiplicities, as in the theory of
cycle modules [Ros96]. Note that the main difficulty compared to Rost’s theory
is the necessity to describe what happens on twists. Compare to the formula
given by Feld, we make explicit the isomorphisms needed to get a coherent
formula.

######  5.3.1.

We start with the stronger form of (R3a). We consider a ramified extension
$\varphi:E\rightarrow F$ of valued field $(E,v)$, $(F,w)$ with ramification
index $e>0$: $w\circ\varphi=e.v$. We still denote by
$\varphi:\mathcal{O}_{v}\rightarrow\mathcal{O}_{w}$ the induced morphism on
ring of integers, and by $\bar{\varphi}:\kappa_{v}\rightarrow\kappa_{w}$ the
induced map on residue field.

Let us choose prime $\pi_{v}\in\mathcal{O}_{E}$, $\pi_{w}\in\mathcal{O}_{F}$
respectively for $v$ and $w$. This defines a unique isomorphism of
$\kappa_{w}$-vector spaces:

$\theta:\omega_{v}\otimes_{\kappa_{v}}\kappa_{w}\rightarrow\omega_{w},\bar{\pi}_{v}^{*}\otimes
1\mapsto\bar{\pi}_{w}^{*}$

where $\omega_{v}=(\mathcal{M}_{v}/\mathcal{M}_{v}^{2})^{\vee}$ (resp.
$\omega_{w}=(\mathcal{M}_{w}/\mathcal{M}_{w}^{2})^{\vee}$) are the respective
normal sheaves.

Note that there exists a uniquely defined unit $u\in\mathcal{O}_{F}^{\times}$
such that $\varphi(\pi_{v})=u.\pi_{w}^{e}$.

######  Proposition 5.3.2 (Property R3a+).

Consider the above hypothesis and notation. Then the following diagram
commutes:

$\textstyle{\operatorname{K}^{MW}_{*}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{v}}$$\scriptstyle{\varphi_{*}}$$\textstyle{\operatorname{K}^{MW}_{*}(\kappa(v),\omega_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\bar{u}\rangle.e_{\epsilon}.\bar{\varphi}_{*}\otimes\theta_{*}}$$\textstyle{\operatorname{K}^{MW}_{*}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{w}}$$\textstyle{\operatorname{K}^{MW}_{*}(\kappa(w),\omega_{w}).}$

Moreover, the right vertical map does not depend on the choice of primes
$\pi_{v}$ and $\pi_{w}$.

###### Proof.

Consider an element $\sigma\in\operatorname{K}^{MW}_{*}(E)$. As all maps
commute with multiplication by $\eta$, one reduces to consider a symbol of the
form $\sigma=[u_{1},...,u_{n}]$. By using relation (MW2) of Milnor-Witt
K-theory, the fact $w(\varphi(\pi_{E}))>0$, and the properties of the residue
map, one reduces to the case where $\sigma=[\pi_{F},u_{2},...,u_{n}]$, with
$u_{i}\in\mathcal{O}_{E}^{\times}$. We compute the composite of the maps
through the left-down right corner:

$\partial_{w}\big{(}\varphi_{*}(\sigma)\big{)}=\partial_{w}\big{(}[u.\pi_{F}^{e},\varphi(u_{2}),...,\varphi(u_{n})]\big{)}=\langle\bar{u}\rangle.e_{\epsilon}.[\bar{\varphi}(\bar{u}_{2}),...,\bar{\varphi}(\bar{u}_{n})]\otimes\bar{\pi}_{F}^{*}.$

where the last equality follows from 2.5.2(Res2). Another application of _loc.
cit._ gives
$\partial_{v}(\sigma)=[\bar{u}_{2},...,\bar{u}_{n}]\otimes\bar{\pi}_{E}^{*}$,
and so the first assertion follows.

For the second assertion, we write $\pi^{\prime}_{E}=u_{E}\pi_{E}$,
$\pi^{\prime}_{F}=u_{F}\pi_{F}$, with $u_{E}$, $u_{F}$ units. Then a
straightforward computation reduces to show the equality in
$\operatorname{GW}(\kappa_{w})$:

(5.3.2.a)
$\langle\bar{u}_{F}\rangle.e_{\epsilon}=\langle\bar{u}_{F}^{e}\rangle.e_{\epsilon}.$

If $e$ is odd, one gets
$\langle\bar{u}_{F}^{e}\rangle=\langle\bar{u}_{F}\rangle$ and therefore
(5.3.2.a) is true. If $e=2n$ is even, $e_{\epsilon}=n.h$. But for any unit
$a\in\kappa_{w}^{\times}$, one has: $\langle a\rangle.h=h$ (2.1.11(GW3)). Thus
(5.3.2.a) holds true in that latter case. ∎

######  Remark 5.3.3.

In the preceding proposition, one cannot avoid in general the presence of the
correcting unit $\bar{u}$ in the formula of the right vertical map. Using
property R4a, it is possible to give a more compact definition of this map.
Indeed, working in the abelian group

$\mathbb{Z}[Hom_{\kappa_{w}}(\omega_{v}\otimes_{\kappa_{v}}\kappa_{w},\omega_{w})]=\mathbb{Z}[Hom_{\kappa_{v}}(\omega_{v},\omega_{w})],$

one can define the element:

$\theta_{u}^{e}=\sum_{i=0}^{e-1}\delta_{\bar{u}^{(-1)^{i}}}\circ\theta$

with the notation of (R4a). With that definition, the formula of the preceding
proposition reads:

$\partial_{w}\circ\varphi_{*}=(\bar{\varphi}_{*}\otimes\theta_{u}^{e})\circ\partial_{v}.$

This last formula agrees with the computation of the
$\mathbb{A}^{1}$-homotopical defect of the purity isomorphism done in [Fel21,
Th. 2.2.2].

######  5.3.4.

The preceding formula has interesting corollaries. Let us set up the notation
before stating the first one.

We let $\varphi:E\rightarrow F$ be an arbitrary field extension,
$\varphi^{\prime}:E(t)\rightarrow F(t)$ the induced extension. A closed point
$x\in\mathbb{A}^{1}_{E,(0)}$ corresponds to a monic irreducible polynomial
$\pi_{x}\in E[t]$ and we denote by $v_{x}$ the corresponding $\pi_{x}$-adic
valuation on $E(t)$. One can consider the prime decomposition in $F[t]$:

$\varphi^{\prime}(\pi_{x})=\prod_{y/x}\pi_{y}^{e_{y/x}}.$

The product runs over a finite family of closed points
$y\in\mathbb{A}^{1}_{F,{(0)}}$, corresponding to the irreducible polynomial
$\pi_{y}\in F[t]$, and the integers $e_{y/x}$ are some multiplicities.
Equivalently, the $\pi_{y}$-adic valuations $v_{y}$ on $F(t)$ runs over the
extensions of the valuation $v_{x}$, such that
$v_{y}\circ\varphi^{\prime}_{*}=e_{y/x}.v_{x}$. As for the preceding
proposition, we consider $\omega_{x}$ and $\omega_{y}$ the respective normal
sheaves associated with $v_{x}$ and $v_{y}$ respectively. Then one considers
the isomorphism
$\theta_{y}:\omega_{x}\otimes_{\kappa_{x}}\kappa_{y}\rightarrow\omega_{y}$,
sending $\bar{\pi}_{x}^{*}\otimes 1$ to $\bar{\pi}_{y}^{*}$. We let
$\varphi_{y}:\kappa(x)\rightarrow\kappa(y)$ be the induced morphism.

######  Corollary 5.3.5.

Consider the above notation. Then the following diagram commutes:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{K}^{MW}_{*}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{*}}$$\textstyle{\operatorname{K}^{MW}_{*}(E(t))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{E}}$$\scriptstyle{\varphi^{\prime}_{*}}$$\scriptstyle{(2)}$$\textstyle{\bigoplus_{x\in\mathbb{A}^{1}_{E,(0)}}\operatorname{K}^{MW}_{*}(\kappa(x),\omega_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{y/x}e_{y/x,\epsilon}.\varphi_{y*}\otimes\theta_{y*}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{K}^{MW}_{*}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{K}^{MW}_{*}(F(t))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{F}}$$\textstyle{\bigoplus_{y\in\mathbb{A}^{1}_{F,(0)}}\operatorname{K}^{MW}_{*}(\kappa(y),\omega_{y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the two horizontal sequences are the split short exact sequences deduced
from 3.2.1.

###### Proof.

The commutativity of the left-hand square is trivial. For square (2), we
consider an element $\sigma\in\operatorname{K}^{MW}_{*}(E(t))$. As all maps
involved commute with $\eta$, one can assume $\sigma=[f_{1},...,f_{n}]$,
$f_{i}\in E(t)$. Let $S\subset\mathbb{A}^{1}_{E,(0)}$ be the finite set of
points such that the family $(\pi_{x})_{x\in S}$ is exactly made of the
irreducible polynomials appearing in the prime decomposition of the $f_{i}$.
Thus, $d_{E}(\sigma)=\sum_{x\in S}\partial_{v_{x}}(\sigma)$.

Similarly, let $T\subset\mathbb{A}^{1}_{F,(0)}$ be the finite set such that
the family $(\pi_{x})_{x\in T}$ is made of the irreducible polynomials
appearing in the prime decomposition of the $\varphi^{\prime}(f_{i})$. Then
$d_{F}(\varphi^{\prime}(\sigma))=\sum_{y\in T}\partial_{v_{x}}(\sigma)$.

With this notation, the conclusion comes from applying 5.3.2 to each point
$x\in S$ and then take the sum of the resulting formulas. ∎

######  Remark 5.3.6.

Considering the graded Chow-Witt groups as defined in 3.1.5, the right
vertical map of the diagram can be seen as the definition of a pullback map

$f^{*}:C^{1}(\mathbb{A}^{1}_{E})_{*}\rightarrow C^{1}(\mathbb{A}^{1}_{F})_{*}$

associated to the flat morphism
$f:\mathbb{A}^{1}_{F}\rightarrow\mathbb{A}^{1}_{E}$ (note that in this
particular case, though $f$ is not of finite type, it is quasi-finite). In
fact the commutativity of square (2) gives (after adding twists with a line
bundle $\mathcal{L}$ over $\mathbb{A}^{1}_{E}$) a well-defined morphism of
complexes, called the _flat pullback_ :

$f^{*}:C^{*}(\mathbb{A}^{1}_{E},\mathcal{L})\rightarrow
C^{1}(\mathbb{A}^{1}_{F},f^{-1}\mathcal{L})_{*}$

The definition of pullbacks on Chow-Witt groups associated with any flat
morphism is still missing, and we intend to come back to that point after the
work [DFJ].

Using the Bass-Tate approach to transfers in the monogenic case, and
especially the characterization obtained in 4.2.3, one deduces from the
commutativity of square (2) in the previous theorem the following result,
which we state as a lemma for the next statement.

######  Lemma 5.3.7.

Let $\varphi:E\rightarrow F$ be an arbitrary field extension, and consider the
notation of the previous corollary. Then the following square is commutative:

$\textstyle{\bigoplus_{x\in\mathbb{P}^{1}_{E,(0)}}\operatorname{K}^{MW}_{*}(\kappa(x),\omega_{\kappa(x)/E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{x}\operatorname{Tr}^{MW}_{\kappa(x)/E}}$$\scriptstyle{\sum_{y/x}e_{y/x,\epsilon}.\varphi_{y*}\otimes\theta_{y*}}$$\textstyle{\operatorname{K}^{MW}_{*}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{*}}$$\textstyle{\bigoplus_{y\in\mathbb{P}^{1}_{F,(0)}}\operatorname{K}^{MW}_{*}(\kappa(y),\omega_{\kappa(x)/E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{y}\operatorname{Tr}^{MW}_{\kappa(y)/F}}$$\textstyle{\operatorname{K}^{MW}_{*}(F)}$

where the sum on the vertical left hand-side runs over the point
$y\in\mathbb{P}^{1}_{F}$ which lies above $x\in\mathbb{P}^{1}_{E}$, $e_{y/x}$
is defined as in the previous corollary and $e_{\infty/\infty}=1$. We have
abused notation by denoted $\theta_{y}$ the isomorphism induced by the one of
the previous corollary; explicitly:

$\displaystyle\theta_{y}:\omega_{\kappa(x)/E}\simeq\omega_{x}\otimes\omega_{\mathbb{A}^{1}_{\kappa(x)}/\kappa(x)}$
$\displaystyle\rightarrow\omega_{y}\otimes\omega_{\mathbb{A}^{1}_{\kappa(y)}/\kappa(y)}\simeq\omega_{\kappa(y)/F}$
$\displaystyle\bar{\pi}_{x}^{*}\otimes dt$
$\displaystyle\mapsto\bar{\pi}_{y}^{*}\otimes dt.$

Indeed, it suffices to apply the preceding corollary, 4.2.3 together with
(R3a) for the case of the valuation at infinity on $E(t)$.

######  5.3.8.

We are now ready to state the stronger form of axiom (R1c), without any
assumption of separability. Namely, we consider a commutative square of rings:

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\varphi}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R}$

where $E$, $F$, $L$ are fields,494949only the positive characteristics case is
relevant $\Phi$ is finite, and we have put: $R=F\otimes_{E}L$. Then $R$ is a
non necessarily reduced ring.

We choose a presentation $L=F[t_{1},...,t_{n}/(\pi_{1},\ldots,\pi_{n})]$ as in
4.1.10. In other words, letting $\alpha_{i}$ be the image of $t_{i}$ in the
the above quotient, one has $L=F[\alpha_{1},...,\alpha_{n}]$. Moreover,
$\pi_{i}$ is a polynomial with coefficients in $F$ involving only the
variables $t_{1},...,t_{i}$, and $\pi_{i}(\alpha_{1},...,\alpha_{i-1},t_{i})$
is the minimal polynomial of $\alpha_{i}$ over
$F[\alpha_{1},...,\alpha_{i-1}]$.

Then, $\pi_{1}$ seen as a polynomial in $t_{1}$ can be uniquely factored in
$F$ as:

$\pi_{1}=\prod_{j\in J_{1}}\pi_{1,j}^{e_{1,j}}$

where $\pi_{1,j}$ is an irreducible polynomial in $F[t_{1}]$, and $e_{1,j}$ a
positive integer. Arguing inductively, one obtains the following presentation
of $R$:

$R=\prod_{x\in
X}F[t_{1},...,t_{n}]/\big{(}\pi_{1,x}^{e_{1,x}},...,\pi_{n,x}^{e_{n,x}}\big{)}$

such that for any $(i,x)\in[1,n]\times I$, $\pi_{i,x}$ is a polynomial in
$(t_{1},...,t_{i})$ such that $\pi_{i,x}(\alpha_{1},...,\alpha_{i-1},t_{i})$
is a prime divisor of $\pi_{i}(\alpha_{1},...,\alpha_{i-1},t_{i})$ in
$F[\alpha_{1},...,\alpha_{i-1},t_{i}]$. Moreover, the indexing set is
$X=\operatorname{Spec}(R)$, the set of prime ideals of $R$.

Let us fix a point $x\in X$. As a prime ideal of $R$, one can write
$x=(\pi_{1,x},...,\pi_{n,x})$. The local Artinian ring $R_{x}$ is of length:

$e_{x}:=\lg(R_{x})=e_{x,1}...e_{x,n}.$

Moreover, the residue field $\kappa(x):=R/x$ is finite over $F$ and one can
define an isomorphism of $F$-vector spaces:

$\displaystyle\theta_{x}:\omega_{L/E}\otimes_{L}F$
$\displaystyle\xrightarrow{\ \sim\ }\omega_{\kappa(x)/F},$
$\displaystyle(\bar{\pi}_{1}\wedge...\wedge\bar{\pi}_{n})^{*}\otimes(dt_{1}\wedge...\wedge
dt_{n})$
$\displaystyle\longmapsto(\bar{\pi}_{1,x}\wedge...\wedge\bar{\pi}_{n,x})^{*}\otimes(dt_{1}\wedge...\wedge
dt_{n})$

The following result gives a more precise form of [Fel20b, Th. 3.8].

######  Theorem 5.3.9 (Property R1c+).

Consider the above notation. Then the following diagram is commutative:

$\textstyle{\operatorname{K}^{MW}_{*}(L,\omega_{L/E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{*}}$$\scriptstyle{\sum_{\mathfrak{p}}e_{\mathfrak{p},\epsilon}.\varphi_{\mathfrak{p}*}\otimes\theta_{\mathfrak{p}*}}$$\textstyle{\operatorname{K}^{MW}_{*}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{*}}$$\textstyle{\bigoplus_{\mathfrak{p}\subset
R}\operatorname{K}^{MW}_{*}(\kappa(\mathfrak{p}),\omega_{\kappa(\mathfrak{p})/F})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{\mathfrak{p}}\Phi_{\mathfrak{p}}^{*}}$$\textstyle{\operatorname{K}^{MW}_{*}(F)}$

where $\mathfrak{p}$ runs over the prime ideals of $R=L\otimes_{E}F$, and the
map $\Phi_{\mathfrak{p}}:F\rightarrow\kappa(\mathfrak{p})$,
$\varphi_{\mathfrak{p}}:L\rightarrow\kappa(\mathfrak{p})$ are induced
respectively by $\Phi$, $\varphi$.

Moreover, the left-hand vertical map is independent on the chosen
parametrization of $L/E$.

###### Proof.

The proof follows that of _loc. cit._ By multiplicativity of the symbol
$?_{\epsilon}$ (see the end of 2.2.18), and compatibility of the isomorphism
$\theta_{x}$ with the number of variables $n$, one reduces to the case where
$L/E$ is monogenic, _i.e._ $n=1$ with our previous notation. To simplify the
notation, we write $t$, $\alpha$, $\pi_{x}$, … for $t_{1}$, $\alpha_{1}$,
$\pi_{1}$, etc…

Then the first statement to be proved is a particular case of 5.3.7, obtained
by considering only the point $x\in\mathbb{A}^{1}_{E,(0)}$ such that
$L=\kappa(x)$.

Finally, the second statement follows from (the last statement of) 5.3.2. ∎

######  Remark 5.3.10.

This theorem is in fact the projection formula $f^{*}p_{*}=q_{*}g^{*}$ in
(graded) Chow-Witt groups for the cartesian square:

$\textstyle{\mathbb{P}^{1}_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{g}$$\textstyle{\operatorname{Spec}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathbb{P}^{1}_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\operatorname{Spec}(E).}$

This is in fact one of the main properties needed for a flat pullback. Compare
to [Fel20b], we have avoided the need of a (perfect) base field, and we have
described the isomorphisms needed to change the twists (_i.e._ the map
$\theta_{\mathfrak{p}*}$).

Surprisingly, $f$ is induced by an arbitrary field extension, not necessarily
finitely generated.

##  6\. Appendix: coherent duality, traces and residues

###  6.1. Categorical duality and traces

######  6.1.1.

We recall the classical categorical framework for expressing duality. We refer
the reader to [Mac71], Section XI.1 (resp. VII.7), for references on symmetric
monoidal categories (resp. closed symmetric monoidal categories). To simplify
the exposition, we will apply MacLane’s coherence theorem (_loc. cit._ , XI.1,
Th. 1) and consider that any composite of coherence isomorphisms (_i.e._
expressing unity, associativity, commutativity of the symmetric monoidal
structure) is an identity.

The historical references for the next two definitions are [DP80] and [SR72].

######  Definition 6.1.2.

Let $(\mathcal{C},\otimes,\mathbbm{1})$ be a symmetric monoidal category and
$M$ an object of $\mathcal{C}$. We say that $M$ is _dualizable_ 505050Other
possible terminologies are _strongly dualizable_ or _rigid_. if there exists
an object $M^{\vee}$ and morphisms:

$\operatorname{\pi}:M\otimes
M^{\vee}\rightarrow\mathbbm{1},\operatorname{\delta}:\mathbbm{1}\rightarrow
M^{\vee}\otimes M,$

respectively called the duality _pairing_ and _co-pairing_ , such that the
following composite maps

$\displaystyle
M\xrightarrow{\operatorname{Id}\otimes\operatorname{\delta}}M\otimes
M^{\vee}\otimes M\xrightarrow{\operatorname{\pi}\otimes\operatorname{Id}}M$
$\displaystyle
M^{\vee}\xrightarrow{\operatorname{\delta}\otimes\operatorname{Id}}M^{\vee}\otimes
M\otimes
M^{\vee}\xrightarrow{\operatorname{Id}\otimes\operatorname{\pi}}M^{\vee}$

are the identity. One says that $M^{\vee}$ is the _dual_ of $M$.

The triple $(M^{\vee},\operatorname{\pi},\operatorname{\delta})$ uniquely
determines $M^{\vee}$ as the dual of $M$. It follows from the definition that
the functor $\tau_{M}=(M\otimes-)$ is both left and right adjoint to the
functor $\tau_{M^{\vee}}(M^{\vee}\otimes-)$. In particular, $\tau_{M^{\vee}}$
is the internal Hom functor with source $M$, and one gets a canonical
isomorphism, bifunctorial in $M$ and $N$:

$M^{\vee}\otimes N\simeq\operatorname{\underline{Hom}}(M,N).$

When the monoidal category $\mathcal{C}$ is closed, there exists a canonical
isomorphism $M^{\vee}\simeq\operatorname{\underline{Hom}}(M,\mathbbm{1})$,
characterized as being the evaluation at $\mathbbm{1}$ of the unique
isomorphism $\tau_{M^{\vee}}\simeq\operatorname{\underline{Hom}}(M,-)$.

######  Example 6.1.3 (exercice).

Let $A$ be a (commutative) ring, and $A\mkern-2.0mu\operatorname{-mod}$ the
closed symmetric monoidal category of $A$-modules. Then the following
conditions are equivalent:

  1. (i)

$M$ is dualizable in $A\mkern-2.0mu\operatorname{-mod}$;

  2. (ii)

$M$ is a finitely generated and projective $A$-module.

######  Example 6.1.4.

The preceding example generalizes to a quasi-compact quasi-separated scheme
$X$. Let $\operatorname{\mathrm{D}}(\mathcal{O}_{X})$ be the derived category
of $\mathcal{O}_{X}$-modules, endowed with its closed symmetric monoidal
structure via the derived tensor product. Let $K$ be an object of
$\operatorname{\mathrm{D}}(\mathcal{O}_{X})$. Then the following conditions
are equivalent (see e.g. [Stack, Ex. 0FPC, Lem. 0FPD, Prop. 09M1]):

  1. (i)

$K$ is dualizable in $\operatorname{\mathrm{D}}(\mathcal{O}_{X})$;

  2. (ii)

$K$ is a perfect complex of $\mathcal{O}_{X}$-modules;

  3. (iii)

$K$ is compact.

######  Definition 6.1.5.

Consider the above setting and let $M$ be a dualizable object with dual
$(M^{\vee},\operatorname{\pi},\operatorname{\delta})$. We define the _trace_
of an endomorphism $f:M\rightarrow M$ as the following element of the ring
$\operatorname{End}_{\mathcal{C}}(\mathbbm{1})$:

$\operatorname{tr}_{M}(f):\mathbbm{1}\xrightarrow{f^{\prime}}M\otimes
M^{\vee}\xrightarrow{\operatorname{\pi}}\mathbbm{1}$

where $f^{\prime}$ is obtained by adjunction from $f$. This defines a map:

$\operatorname{tr}_{M}:\operatorname{End}_{\mathcal{C}}(M)\rightarrow\operatorname{End}_{\mathcal{C}}(\mathbbm{1}).$

######  Remark 6.1.6 (exercice).

One can derive the following formulas for traces:

  * •

$\operatorname{tr}_{M}(f\circ g)=\operatorname{tr}_{M}(g\circ f)$.

  * •

$\operatorname{tr}_{M\otimes N}(f\otimes
g)=\operatorname{tr}_{M}(f)\otimes\operatorname{tr}_{N}(g)$.

  * •

$\operatorname{tr}_{M}(\lambda.f)=\lambda.\operatorname{tr}_{M}(f)$,
$\lambda\in\operatorname{End}_{\mathcal{C}}(\mathbbm{1})$.

######  Example 6.1.7.

Consider the setting of 6.1.3. Obviously, $\mathbbm{1}=A$ and
$\operatorname{End}_{A}(A)=A$, as a ring.

Given a dualizable $A$-module $M$, the trace map
$\operatorname{Tr}_{M}:\operatorname{End}_{A}(M)\rightarrow A$ defined above
coincide with the classical notion for number theory. In particular, when $M$
admits a (global) $A$-basis $(f_{1},\ldots,f_{n})$, through the induced
isomorphism $\operatorname{End}_{A}(M)\simeq\mathcal{M}_{n}(A)$, the map
$\operatorname{Tr}_{M}$ is the usual trace map of matrices.

######  6.1.8.

Consider again the abstract situation of a symmetric monoidal category
$(\mathcal{C},\otimes,\mathbbm{1})$, and a dualizable object $M$ of
$\mathcal{C}$ with dual $(M^{\vee},\operatorname{\pi},\operatorname{\delta})$.

We remark that the trace map $\operatorname{tr}_{M}$ is induced by an internal
trace map:

$\underline{\operatorname{tr}}_{M}:\operatorname{\underline{Hom}}(M,M)\simeq
M^{\vee}\otimes M\simeq M\otimes M^{\vee}\xrightarrow{\pi}\mathbbm{1}.$

This means that
$\operatorname{tr}_{M}=\operatorname{\underline{Hom}}(\underline{\operatorname{tr}}_{M},\mathbbm{1})$.

Assume now that $M$ admits a product map $\mu:M\otimes M\rightarrow M$ (for
example, $M$ is a monoid, [Mac71, VII.3]). Then one gets a $\mu$-trace
morphism:

$\operatorname{Tr}^{\mu}_{M}:M\xrightarrow{\mu^{\prime}}\operatorname{\underline{Hom}}(M,M)\xrightarrow{\underline{\operatorname{tr}}_{M}}\mathbbm{1}$

As a particular case, one gets back the following classical definition from
algebra:

######  Definition 6.1.9.

Let $A$ be a ring and $B$ be a commutative $A$-algebra which is projective and
finitely generated as an $A$-module.

Then $B$ is a dualizable $A$-module and we define the _trace morphism_

$\operatorname{Tr}_{B/A}:B\rightarrow A$

as the $A$-linear map associated above with respect to the multiplication map
$B\otimes_{A}B\rightarrow B$.

Concretely, the trace of an element $b\in B$ is the trace of the endomorphism
$\gamma_{b}$ such that $\gamma_{b}(x)=b.x$. It can be computed locally by
choosing bases of the $A$-module $B$ and using the trace of matrices. The
local definitions then glue using faithfully flat descent.

Let us recall the following classical result.

######  Proposition 6.1.10.

Let $B/A$ be a finitely generated projective ring extension. Then the
following conditions are equivalent:

  1. (i)

$B/A$ is étale.

  2. (ii)

For every prime ideal $\mathfrak{q}$ in $B$, $\mathfrak{p}$ being its inverse
image in $A$, $L=B/\mathfrak{q}$, $K=A/\mathfrak{p}$, one has:
$\operatorname{Tr}_{L/K}\neq 0$.

  3. (iii)

The bilinear form $B\otimes_{A}B\rightarrow A,x\otimes
y\mapsto\operatorname{Tr}_{B/A}(xy)$ is non degenerate - _i.e._ induces by
adjunction an isomorphism $B\rightarrow B^{\vee}$ of $A$-modules.

In particular, a finite field extension $L/K$ is separable if and only if
$\operatorname{Tr}_{L/K}\neq 0$.

In particular, the trace map of an inseparable extension is not interesting.
This justifies the use of a finer duality theory, which was introduced by
Grothendieck. We recall the abstract setting to end-up this section.

######  Definition 6.1.11.

Let $(\mathcal{C},\otimes,\mathbbm{1})$ be a closed symmetric monoidal
category. Let $K$ be an object of $\mathcal{C}$, and write
$\mathrm{D}_{K}(M)=\operatorname{\underline{Hom}}(M,K)$. The evaluation map
$M\otimes\operatorname{\underline{Hom}}(M,K)$ induces by adjunction a
canonical map $\omega_{M}:M\rightarrow\mathrm{D}_{K}\circ\mathrm{D}_{K}(M)$.

One says that $K$ is _dualizing_ if the natural transformation
$\omega:\operatorname{Id}_{\mathcal{C}}\rightarrow\mathrm{D}_{K}\circ\mathrm{D}_{K}$
is an isomorphism.

In words, $\mathrm{D}_{K}(M)$ is called the weak $K$-dual of $M$, and the
definition asks that any object $M$ is isomorphic to its double weak $K$-dual,
by the canonical map $\omega_{M}$.

######  Remark 6.1.12.

In the original definition, that of a dualizing complex ([Har66, Definition,
p. 258]), one had additional assumptions (finite injective dimension and lower
boundedness). One has progressively dismissed this kind of assumptions, in
order to extend Grothendieck’s theory to other context (torsion étale sheaves
[ILO14, VII, 6.1.1], constructible pro-étale sheaves [BS15, 6.7.20],
$D$-modules, motivic homotopy [Ayo07, 2.3.73] and motivic complexes [CD19,
4.4.24]).

######  6.1.13.

Consider a dualizing object $K$ of $\mathcal{C}$ as in the above definition.
Then:

  1. (1)

The map $\mathbbm{1}\rightarrow\operatorname{\underline{Hom}}(K,K)$, deduced
from $\operatorname{Id}_{K}$ by adjunction, is an isomorphism.

  2. (2)

For any object $M$, $N$ in $\mathcal{C}$, one has an isomorphism:

$\mathrm{D}_{K}\big{(}M\otimes\mathrm{D}_{K}(N)\big{)}\simeq\operatorname{\underline{Hom}}(M,N).$

  3. (3)

An object $K^{\prime}$ of $\mathcal{C}$ is dualizing if and only if there
exists a $\otimes$-invertible object $L$ such that $K^{\prime}=K\otimes L$.
Moreover, in this case, one has
$L\simeq\mathrm{D}_{K^{\prime}}(K)=\operatorname{\underline{Hom}}(K,K^{\prime})$.

  4. (4)

If $M$ is a dualizable object in $\mathcal{C}$ with dual $M^{\vee}$, then
$\mathrm{D}_{K}(M)\simeq M^{\vee}\otimes K$.

  5. (5)

The dualizing object $K$ is rigid if and only if it is invertible.

We give the arguments for completeness:

  1. (1)

use the isomorphism $\mathbbm{1}\simeq D_{K}(D_{K}(\mathbbm{1}))$ and
$\operatorname{\underline{Hom}}(\mathbbm{1},-)\simeq\operatorname{Id}_{C}$ (as
a right adjoint to the functor $(\mathbbm{1}\otimes-)$).

  2. (2)

Use the sequence of isomorphisms:

$\mathrm{D}_{K}\big{(}M\otimes\mathrm{D}_{K}(N)\big{)}=\operatorname{\underline{Hom}}\big{(}M\otimes\mathrm{D}_{K}(N),K\big{)}\simeq\operatorname{\underline{Hom}}\big{(}M,\mathrm{D}_{K}\mathrm{D}_{K}(N)\big{)}\simeq\operatorname{\underline{Hom}}(M,N).$

  3. (3)

$\Leftarrow$: use $\operatorname{\underline{Hom}}(M,N\otimes
L)\simeq\operatorname{\underline{Hom}}(M\otimes
L^{-1},N)\simeq\operatorname{\underline{Hom}}(M,N)\otimes L$.
$\Rightarrow$: one proves $M\mapsto M\otimes\mathrm{D}_{K}(K^{\prime})$ is an
equivalence. It suffices to apply the equivalence $\mathrm{D}_{K}$, point (2)
to reduce to the fact $D_{K^{\prime}}$ is an equivalence.

  4. (4)

Follows from the definition.

  5. (5)

Follows from point (4).

######  Example 6.1.14.

In the category of locally compact abelian groups, the unit circle
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$ is a dualizing object. We will se more
examples in the next section.

###  6.2. Grothendieck differential trace map and duality

######  6.2.1.

Let $f:X\rightarrow S$ be a morphism of quasi-compact and quasi-separated
schemes. We have an adjoint pair:

$\mathbf{L}f^{*}:\operatorname{\mathrm{D}}(\mathcal{O}_{S})\leftrightarrows\operatorname{\mathrm{D}}(\mathcal{O}_{X}):\mathbf{R}f_{*}.$

We say that a complex $K$ of $\operatorname{\mathrm{D}}(\mathcal{O}_{X})$ is
quasi-coherent is its cohomology sheaves are quasi-coherent. We let
$\operatorname{D_{qc}}(X)$ be the full sub-category of
$\operatorname{\mathrm{D}}(\mathcal{O}_{X})$ made of quasi-coherent
complexes.515151For Noetherian schemes, this category is equivalent to the
derived category of the abelian category of quasi-coherent sheaves: see
[Stack, Prop. 36.8.3]. Both functors $\mathbf{L}f^{*}$ and $\mathbf{R}f_{*}$
preserves quasi-coherent complexes (see [Stack, 36.3.8, 36.4.1]). The
following theorem is one of the essential part of Grothendieck’s duality
theory.

######  Theorem 6.2.2.

Consider the above assumptions.

  1. (1)

The functor
$\mathbf{R}f_{*}:\operatorname{D_{qc}}(X)\rightarrow\operatorname{D_{qc}}(S)$
admits a left adjoint. When $f$ is _in addition proper_ , we denote this left
adjoint by
$f^{!}:\operatorname{D_{qc}}(S)\rightarrow\operatorname{D_{qc}}(X)$.

  2. (2)

If $f$ is proper smoothable lci with canonical sheaf $\omega_{X/S}$ and
relative dimension $f$ (see 4.1.5), there exists a canonical isomorphism:

$\mathfrak{p}_{f}:\omega_{X/S}[d]\rightarrow f^{!}(\mathcal{O}_{S})$

with the notation of the preceding point. It is called the purity isomorphism
associated to $f$.

###### Proof.

The first statement is Neeman’s theorem (see [Nee96], [Stack, 48.3.1]).

We could not find an appropriate reference for point (2). However, it follows
from the results of [Har66], with some complements brought by many years of
improvement. Let us summarize the arguments from the litterature. We now erase
the symbols $\mathbf{L}$ and $\mathbf{R}$ for readability.

First, we fix a factorisation of $f$ as $X\xrightarrow{i}P\xrightarrow{p}S$
such that $i$ is a regular closed immersion and $p$ is a smooth morphism. We
will consider the functors $i^{!}$ and $p^{!}$ restricted to
$\operatorname{D^{+}_{qc}}(X)$, which is legitimate thanks to [Stack, 48.3.5].

According to [Stack, 48.9.2], there exists a canonical isomorphism (uniquely
characterized by the adjoint property) of functors $i^{!}\simeq i^{\flat}$
where
$i^{\flat}:\operatorname{D^{+}_{qc}}(P)\rightarrow\operatorname{D^{+}_{qc}}(X)$
is the functor defined in [Har66, III. §6]. As $i$ is a regular closed
immersion, there exists a canonical isomorphism of functors by [Har66, III.
Cor. 7.3]:

$\mathfrak{p}_{i}:\omega_{X/P}[-m]=\omega_{X/P}[-m]\otimes
i^{*}(\mathcal{O}_{P})\simeq i^{\flat}(\mathcal{O}_{P})\simeq
i^{!}(\mathcal{O}_{P})$

where $m$ is the codimension of $i$.

As $p$ is proper and smooth, there exists a canonical isomorphism of functors,
as defined in [Nee20, 4.1.6]:

$\mathfrak{p}_{p}:\omega_{P/S}[n]=\big{(}\omega_{P/S}[n]\otimes
p^{*}(\mathcal{O}_{S})\big{)}\simeq p^{!}(\mathcal{O}_{S})$

where $n$ is the dimension of $p$. In particular, the complex
$p^{!}(\mathcal{O}_{S})$ is perfect.

We now build the desired map as the following composition:

$\displaystyle\omega_{X/S}[d]\simeq\omega_{X/P}[-m]\otimes
i^{*}(\omega_{P/S}[n])\xrightarrow{\mathfrak{p}_{i}\otimes
i^{*}(\mathfrak{p}_{p})}$ $\displaystyle i^{!}(\mathcal{O}_{P})\otimes
i^{*}(p^{!}(\mathcal{O}_{S}))$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{\simeq}}i^{!}(\mathcal{O}_{P}\otimes
p^{!}(\mathcal{O}_{S}))\simeq f^{!}(\mathcal{O}_{S})$

where the isomorphism $(*)$ exists as $i$ is lci ([Stack, 48.8.1]).

To justify the word “canonical”, one needs to prove that the above isomorphism
does not depend on the choice of the factorisation. The steps for this fact
are well-known. The main points may be found in [Har66, §III]: 2.2, 6.2, 6.4,
and most of all 8.1 (see also the proof of Th. 3.3.2 [DJK21]). ∎

######  Remark 6.2.3.

In fact, the purity isomorphism can be generalized in the coherent context as
follows. For any bounded quasi-coherent complex $K$, one get an isomorphism by
the following composite maps:

$f^{!}(K)\xrightarrow{(*)}f^{*}(K)\otimes
f^{!}(\mathcal{O}_{S})\xrightarrow{\mathfrak{p}_{f}}f^{*}(K)\otimes\omega_{X/S}[d].$

The isomorphism $(*)$ follows from [Har66, III, 8.8]. This is specific to the
coherent case. The analogue isomorphism does not hold in other six functors
formalism such as the étale $\ell$-adic or motivic one, unless further
restrictions to $f$ are assumed (e.g. $f$ is smooth, or a nil-immersion).

######  Definition 6.2.4.

Assuming $f$ is proper smoothable lci of relative dimension $d$, we will
denote by

$\operatorname{Tr}^{\omega}_{f}:\mathbf{R}f_{*}(\omega_{X/S})[d]\rightarrow\mathcal{O}_{S}$

the map obtained by adjunction from $\mathfrak{p}_{f}$ and call it the
_differential trace map_ associated with $f$.

When $f$ is finite lci, taking global sections, we will also consider the
induced trace map:

$\operatorname{Tr}^{\omega}_{X/S}:\Gamma(X,\omega_{X/S})\rightarrow\Gamma(S,\mathcal{O}_{S}).$

Finally, if $X/S$ is the spectrum of a finite lci ring extension $B/A$ the
above map will be denoted by:

$\operatorname{Tr}^{\omega}_{B/A}:\omega_{B/A}\rightarrow A.$

######  Remark 6.2.5.

The above definition makes clear several properties of the differential trace
map. It is functorial with respect to flat base change in $X$, and compatible
with disjoint sum in $X$.

This mean in particular that if we have an isomorphism of finite lci
$A$-algebras:

$\Theta:B\xrightarrow{\sim}\prod_{i\in I}B_{i},$

the following diagram is commutative:

$\textstyle{\omega_{B/A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{*}}$$\scriptstyle{\sim}$$\scriptstyle{\operatorname{Tr}^{\omega}_{B/A}}$$\textstyle{\bigotimes_{i\in
I}\omega_{B_{i}/A}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\otimes_{i}\operatorname{Tr}^{\omega}_{B_{i}/A}}$$\textstyle{A}$

######  Remark 6.2.6.

The compatibility with composition of the trace map is more involved. Consider
a facotrisation $X\xrightarrow{g}Y\xrightarrow{h}S$ of $f$ by proper
smoothable and lci morphisms, of respective dimension $n$ and $m$. First
recall that there exists a canonical isomorphism (see 4.1.7):

$\psi:\omega_{X/S}\simeq\omega_{X/Y}\otimes(f^{*}\omega_{Y/S}).$

The compatibility with composition of the differential trace map is expressed
by the following commutative diagram (again we cancel the symbols $\mathbf{R}$
and $\mathbf{L}$ for readability):

$\textstyle{f_{*}(\omega_{X/S})[d]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{X/S}}$$\scriptstyle{\sim}$$\scriptstyle{\psi}$$\textstyle{\mathcal{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{*}\big{(}\omega_{X/Y}\otimes(g^{*}\omega_{Y/S})\big{)}[d]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{h_{*}\big{(}g_{*}(\omega_{X/Y})[n]\otimes\omega_{Y/S}\big{)}[m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{*}\big{(}\operatorname{Tr}^{\omega}_{X/Y}\otimes\operatorname{Id}\big{)}}$$\textstyle{h_{*}(\omega_{Y/S})[m]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{Y/S}}$$\textstyle{\mathcal{O}_{S}}$

The second vertical map is obtained by the so-called projection formula, which
hols here either because $g$ is proper or even simply as $\omega_{Y/S}$ is an
invertible sheaf. This statement follows from [Har66, III, 10.5] (see also
[Con00, Th. 3.4.1]).

In the affine case, $X=\operatorname{\mathrm{Spec}}(C)$,
$Y=\operatorname{\mathrm{Spec}}(B)$, $S=\operatorname{\mathrm{Spec}}(A)$, $f$,
$g$ and $h$ being finite, the diagram takes the following simpler form:

$\textstyle{\omega_{C/A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{C/A}}$$\scriptstyle{\sim}$$\scriptstyle{\psi}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{C/B}\otimes_{B}\omega_{B/A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{C/B}\otimes\operatorname{Id}}$$\textstyle{\omega_{B/A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{B/A}}$$\textstyle{A.}$

######  6.2.7.

A particular case of duality. For any quasi-coherent complex $K$ over $X$, and
any proper smoothable lci morphism $f:X\rightarrow S$, the adjunction property
of the pair $(\mathbf{R}f_{*},f^{!})$ gives an isomorphism:

$\displaystyle\operatorname{Hom}_{\operatorname{\mathrm{D}}(\mathcal{O}_{X})}(K,\omega_{X/S}[d])$
$\displaystyle\xrightarrow{\sim}\operatorname{Hom}_{\operatorname{\mathrm{D}}(\mathcal{O}_{S})}(\mathbf{R}f_{*}(K),\mathcal{O}_{S}),$
$\displaystyle\big{(}u:K\rightarrow\omega_{X/S}[d]\big{)}$
$\displaystyle\mapsto\big{(}\operatorname{Tr}_{X/S}^{\omega}\circ\mathbf{R}f_{*}(u)\big{)}.$

In the case of a finite lci ring extension $B/A$, and for a $B$-module $M$,
this boils down to an isomorphism:

$\displaystyle\operatorname{Hom}_{B}(M,\omega_{B/A})$
$\displaystyle\xrightarrow{\sim}\operatorname{Hom}_{A}(M,A),$
$\displaystyle\big{(}u:M\rightarrow\omega_{B/A}\big{)}$
$\displaystyle\mapsto\big{(}\operatorname{Tr}_{B/A}^{\omega}\circ u\big{)}.$

Finally for $M=B$, we get an isomorphism between $A$-linear forms on $B$ and
elements of $\omega_{B/A}$:

$\displaystyle\omega_{B/A}$
$\displaystyle\xrightarrow{\sim}\operatorname{Hom}_{A}(B,A),$ $\displaystyle
w$
$\displaystyle\mapsto\big{(}\psi_{w}:\lambda\mapsto\operatorname{Tr}_{B/A}^{\omega}(\lambda.w)\big{)}.$

######  Example 6.2.8.

We end-up this section with a few classical examples of duality in the case of
coherent sheaves.

  1. (1)

A concrete case of duality is obtained when $S$ is the spectrum of any field
$k$, $X$ a proper smoothable lci $k$-scheme. In that case, the first
isomorphis of 6.2.7 applied to $K[n]$ where $K$ is a bounded complex $K$ with
coherent cohomology, gives an isomorphism of $k$-vector spaces:

$\operatorname{Ext}^{d-n}_{\mathcal{O}_{X}}(K,\omega_{X/k})\xrightarrow{\sim}H^{n}(X,K)^{*}$

The trace map $\operatorname{Tr}_{X/k}^{\omega}$ induces what I will call the
_Gysin map_ associated with $f$:

$f_{!}:H^{d}(X,\omega_{X/S})\rightarrow k$

and the above duality isomorphism corresponds to the _Poincaré duality_
(perfect) pairing:

$\displaystyle\operatorname{Ext}^{d-n}_{\mathcal{O}_{X}}(K,\omega_{X/k})\otimes
H^{n}(X,K)$ $\displaystyle\rightarrow k$ $\displaystyle(x,y)\mapsto
f_{!}(a.b).$

  2. (2)

In the case of a proper smoothable lci morphism $f:X\rightarrow S$, one can
interpret Grothendieck duality, for $K=\mathcal{O}_{X}$, by saying that
$\mathbf{R}f_{*}(\mathcal{O}_{X})$ is dualizable (6.1.2, 6.1.4) with dual
given by $\mathbf{R}f_{*}(\omega_{X/S})[d]$. One of the pairing coming from
this duality is a relative version of the Poincaré duality pairing:

$\mathbf{R}f_{*}(\mathcal{O}_{X})\otimes\mathbf{R}f_{*}(\omega_{X/S})[d]\rightarrow\mathbf{R}f_{*}(\omega_{X/S})[d]\xrightarrow{\operatorname{Tr}_{X/S}^{\omega}}\mathcal{O}_{S}$

where the first map comes from the fact $\mathbf{R}f_{*}$ is weakly monoidal
(as the right adjoint of a monoidal functor).

  3. (3)

Of course, the theory can be considerably generalized \- but we will only need
the case of finite field extensions! Indeed, Grothendieck’s main objective was
to obtain duality for any proper morphism
$f:X\rightarrow\operatorname{Spec}(k)$. He achieved this by constructing a
dualizing complex $K_{X}=\omega_{X/k}$, which is no longer an invertible sheaf
in general (except if $X$ is _Gorenstein_ , see [Har66, V, 9.3]). We refer the
reader to [Har66, Con00] or [LH09, Chap. 1].

###  6.3. Grothendieck and Sheja-Storch Residues

######  6.3.1.

We consider a commutative diagram of schemes:

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{f}$$\textstyle{S}$

such that $f$ is finite lci, $p$ is smooth of relative dimension $n$, and $i$
is a closed immersion with ideal sheaf $\mathcal{I}\subset\mathcal{O}_{P}$.
The hypothesis implies that $i$ is regular of codimension $n$.

Recall that we can associate to the above commutative diagram a canonical
isomorphism (4.1.7):

$\Theta:\omega_{X/S}\simeq\omega_{X/P}\otimes_{\mathcal{O}_{X}}i^{*}\omega_{P/S}\simeq(\Lambda^{n}\mathcal{I}/\mathcal{I}^{2})^{\vee}\otimes_{\mathcal{O}_{P}}\Omega^{n}_{P/S}.$

######  Definition 6.3.2.

Consider a global differential $n$-form $w\in\Gamma(P,\Omega^{n}_{P/S})$ and a
global regular parametrization $(f_{1},\ldots,f_{n})$ of $\mathcal{I}$. We
define the Grothendieck residue (symbol) of $w$ at $(f_{1},...,f_{n})$ as the
element of $\Gamma(S,\mathcal{O}_{S})$:

$\operatorname{Res}_{P/S}\left[\begin{matrix}w\\\ f_{1}\ldots
f_{n}\end{matrix}\right]=\operatorname{Tr}_{X/S}^{\omega}\big{(}(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}\otimes
i^{*}(w)\big{)}$

where we have used the differential trace map of 6.2.4 and we have considered
the element $(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}\otimes i^{*}(w)$
as an element of $\omega_{B/A}$ via the isomorphism $\Theta$.

This definition agrees with that of [Har66, III, §9] and that of [Con00,
Appendix A, (A.1.4)].525252The sign in the latter can be explained as:
$(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}=\bar{f}_{n}^{*}\wedge\ldots\wedge\bar{f}_{1}^{*}=(-1)^{n(n-1)/2}\bar{f}_{1}^{*}\wedge\ldots\wedge\bar{f}_{n}^{*}.$

######  6.3.3.

We now explain a method of Sheja and Storch to compute the above residue, and
therefore the differential trace map. Our reference is [Kun08, §8].

We will restrict to the affine case. Let $B$ be a finite projective
$A$-algebra: in other words, $B$ is dualizable as an $A$-module, see 6.1.3.

In what follows, we will use a set of indeterminates
$\underline{t}=(t_{1},...,t_{n})$, and denote by
$A[\underline{t}]=A[t_{1},...,t_{n}]$ to simplify notations. We assume that
$B$ is a complete intersection $A$-algebra: there exists elements
$\alpha=(\alpha_{1},...,\alpha_{n})\in B^{n}$ which generate $B$ as an
$A$-algebra and the kernel of the surjective map

$A[\underline{t}]\rightarrow B,t_{i}\mapsto\alpha_{i}$

admits a regular parametrization $I=(f_{1},...,f_{n})$ for polynomials
$f_{i}\in A[\underline{t}]$. We say that $f=(f_{1},...,f_{n})$ is a
_presentation_ of the lci $A$-algebra $B$.

Putting $S=\operatorname{Spec}(A)$, $X=\operatorname{Spec}(B)$,
$P=\mathbb{A}^{n}_{S}=\operatorname{Spec}(A[\underline{t}])$, we are therefore
in the conditions of 6.3.1. Consider the composite map535353Geometrically,
this map corresponds to the graph $\gamma_{i}:X\rightarrow X\times_{S}P$ of
the closed immersion $i:X\rightarrow P$. As $i$ is regular and $P/S$ is
smooth, $\gamma_{i}$ is regular. Algebraically, it is just the map evaluating
$t_{i}$ at $\alpha_{i}$.:

$\sigma:B[\underline{t}]\rightarrow B\otimes_{A}B\xrightarrow{\mu}B$

where $\mu$ is the multiplication map, and the first arrow is the natural
surjection coming from the identification
$B\otimes_{A}B=B\otimes_{A}A[t]/I=B[t]/I$. We consider the ideals:

$\displaystyle J$ $\displaystyle=\operatorname{Ker}(\mu)\subset
B\otimes_{A}B,$ $\displaystyle K$
$\displaystyle=\operatorname{Ker}(\sigma)\subset B[t].$

Then $K$ admits a regular parametrization,
$K=(t_{1}-\alpha_{1},...,t_{n}-\alpha_{n})$ and one gets $J=K/I$ as obviously
$I\subset K$ as ideals of $B[\underline{t}]$. Therefore, there exists
polynomials $c_{ij}\in B[t]$ such that

$\forall i\in[1,n],f_{i}=\sum_{j=1}^{n}c_{ij}.(t_{j}-\alpha_{j}).$

Then the element

$\Delta_{f}=\det(c_{ij}\big{(}\alpha)_{1\leq i,j\leq n}\big{)}\in
B\otimes_{A}B$

is independent of the chosen polynomials $c_{ij}$ (see [Kun08, Lemma 4.10]).

######  Definition 6.3.4.

Consider the above notation: $B/A$ is a finite projective complete
intersection and $f=(f_{1},...,f_{n})\in A[t]^{n}$ is a fixed presentation of
$B/A$. Then the element $\Delta_{f}\in B\otimes_{A}B$ is called the
_Bezoutian_ associated with the presentation $f$ of $B/A$.

######  Remark 6.3.5.

This definition is of course an extension of the classical Bezoutian (or
rather the determinant of the Bezout matrix) arising from Euler and Bezout
elimination theory, which corresponds to the case where $A=k$ is a field and
$n=2$.

######  6.3.6.

Consider again the setting of 6.3.3. We put
$B^{*}=\operatorname{Hom}_{A}(B,A)$, which is the (canonical) dual of the
dualizable $A$-module $B$ (see 6.1.2 and what follow).545454Following the
usage, we identify the set of morphisms $\operatorname{Hom}_{A}$ with the
internal Hom-functor in the category of $A$-modules. As $B$ is dualizable, the
canonical map:

$\displaystyle\Phi:B\otimes_{A}B$
$\displaystyle\rightarrow\operatorname{Hom}_{A}(B^{*},B),$ $\displaystyle
b\otimes b^{\prime}$
$\displaystyle\mapsto(\varphi\mapsto\varphi(b).b^{\prime})$

is an isomorphism. The following lemma is now a formality (see [Kun08, 8.13];
beware to translate the notation: $I$ (resp. $\omega_{B/A}$) in _loc. cit._ is
what we denote by $J$ (resp. $B^{*}$) here.)

######  Lemma 6.3.7.

Consider the above notations. Recall that
$J=\operatorname{Ker}(B\otimes_{A}B\xrightarrow{\mu}B)$, seen as an ideal of
$B\otimes_{A}B$. Then
$\Phi(\mathrm{Ann}(J))\subset\operatorname{Hom}_{A}(B^{*},B)$ and $\Phi$
induces an isomorphism:

$\mathrm{Ann}(J)\xrightarrow{\sim}\operatorname{Hom}_{A}(B^{*},B).$

With this lemma in hands, we see that there exists a unique $A$-linear map
$\tau_{f}:B\rightarrow A$, equivalently $\tau_{f}\in B^{*}$ such that:

(6.3.7.a) $\Phi(\Delta_{f})(\tau_{f})=1$

######  Definition 6.3.8.

Consider the above notation, as in 6.3.4. We call the $A$-linear map
$\tau_{f}:B\rightarrow A$ the _Sheja-Storch trace map_ associated with the
presentation $f$ of $B/A$.

######  Example 6.3.9.

We consider the _monogenic_ case:

$B=A[\alpha]=A[t]/(f)$

where $f$ is a monic polynomial in one variable $t$:

$f(t)=a_{0}+\ldots.a_{n-1}.t^{n-1}+t^{n}.$

Thus $B$ is a free $A$-module with basis $1,\alpha,...,\alpha^{n-1}$. Then one
can compute $\Delta_{f}$ explicitly and one finds that

$\tau_{f}=(\alpha^{n-1})^{*},\alpha^{i}\mapsto\delta_{n-1}^{i}.$

######  Remark 6.3.10.

The map $\tau_{f}$ does depend on the chosen generator $\alpha$ of $B/A$, or
more explicitly on the chosen presentation of $B/A$. Therefore, it is sometime
customary to put:

$\tau_{B/A}^{\alpha}=\tau_{f}.$

In view of [Tat52, §1, (2)], corresponding to the case where $B/A$ is an
inseparable extension fields, the map $\tau_{f}$ is sometimes called the _Tate
trace map_ (cf. [Kun08]).

######  Theorem 6.3.11.

Consider the assumption of the above definition.

Then the $A$-linear map $\tau_{f}:B\rightarrow A$ is not $B$-torsion, and in
fact is a $B$-basis of $\operatorname{Hom}_{A}(B,A)$.

In other words, the symmetric bilinear form

$\varphi_{f}:B\otimes_{A}B\rightarrow A,b\otimes
b^{\prime}\mapsto\tau_{f}(bb^{\prime})$

is non-degenerate: the associated map

$B\rightarrow
B^{*}=\operatorname{Hom}_{A}(B,A),\lambda\mapsto\lambda.\tau_{f}=\varphi_{f}(b,-)$

is an isomorphism.

###### Proof.

Using the definitions of 6.3.3, one obtains that $\mathrm{Ann}(J)$ is a
principal ideal domain generated by the Bezoutian $\Delta_{f}$ (a result
attributed to Wiebe, see [Kun08, Cor. 4.12]). According to the previous lemma,
$\mathrm{Ann}(J)\simeq\operatorname{Hom}_{A}(B^{*},B)$ is also an invertible
$B$-module. So $\Delta_{f}$ is a $B$-basis of the $\mathrm{Ann}(J)$. Relation
(6.3.7.a) then implies that $\tau_{f}$ is a $B$-basis of $B^{*}$ as expected.
The other assertions are formal consequences of this fact. ∎

We are now ready to state the link between the concrete construction of Sheja
and Storch and the theory of Grothendieck residue symbols (6.3.2).

######  Proposition 6.3.12.

Recall the situation of the previous theorem and definition:

  * •

$B$ is a complete intersection, finite and projective $A$-algebra

  * •

$f$ is a presentation of $B/A$: $f=(f_{1},...,f_{n})$ is regular sequence of
elements of $R=A[t_{1},...,t_{n}]$, $I=(f_{1},...,f_{n})$ and $B=R/I$.

Recall that we have a canonical isomorphism (see 4.1.9)

$\Theta:\omega_{B/A}\simeq\Lambda^{n}(I/I^{2})^{\vee}\otimes_{R}\Omega^{n}_{R/A}.$

Then, for any $\lambda\in R$, with image $\bar{\lambda}$ in $B=R/I$, we get:

$\operatorname{Res}_{R/A}\left[\begin{matrix}\lambda.dt_{1}\wedge\ldots\wedge
dt_{n}\\\ f_{1}\ldots f_{n}\end{matrix}\right]=\tau_{f}(\bar{\lambda}).$

In other words, if we let
$w=(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}\otimes
i^{*}(dt_{1}\wedge\ldots dt_{n})$ seen as an element of $\omega_{B/A}$ via the
isomorphism $\Theta$, for any $b\in B$, one gets:

(6.3.12.a) $\operatorname{Tr}^{\omega}_{B/A}(b.w)=\tau_{f}(b).$

Or equivalently, with the notation of 6.2.7: $\psi_{w}=\tau_{f}$.

###### Proof.

In the case where $A=k$ is a field (the only case we will need!), this is
[Kun08, Prop. 8.32]. In general, one can reduce to this case by base change:
we need to compare two trace maps associated with
$f:X=\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)=S$ which is
finite and syntomic. Both traces $\operatorname{Tr}_{B/A}^{\omega}$ and
$\tau_{f}$ are compatible under arbitrary base change (as syntomic finite
morphisms are stable under base change) so that we can reduce to residue
fields of $S$.

Alternatively, both definition of residues, via respectively Grothendieck and
Sheja-Storch methods, satisfy the properties (R1)-(R10) of [Har66, III, §9]
(see respectively [Con00, Appendix A] and [HL79, Hop83]). This uniquely
characterizes the residue symbol. ∎

In view of 6.3.9, we deduce the following comparison of the Grothendieck
differential trace map and the Tate trace map (6.3.10).

######  Corollary 6.3.13.

Suppose $B/A$ is a monogenic extension ring, of the form $B=A[t]/I$ where
$I=(f)$ for a monic polynomial $f\in A[t]$. We identify $\omega_{B/A}$ with
the $B$-module $(I/I^{2})^{*}\otimes_{A[t]}\Omega_{A[t]/A}$ (via the
isomorphism $\Theta$ of 4.1.9).

Then for any $b\in B$, one gets:

(6.3.13.a) $\operatorname{Tr}^{\omega}_{B/A}(b.\bar{f}^{*}\otimes
dt)=\tau_{B/A}^{\alpha}(b)$

with the notation of 6.3.10.

######  Corollary 6.3.14.

Let $B$ be a finite étale $A$-algebra. Then $\omega_{B/A}=B$ and the following
diagram commutes:

$\textstyle{\omega_{B/A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}^{\omega}_{B/A}}$$\textstyle{A}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Tr}_{B/A}}$

where $\operatorname{Tr}_{B/A}$ is the ”usual” trace map (6.1.9).

###### Proof.

This is asserted without proof in [Har66, Remark p. 187]. As both trace maps
are compatible with composition, the proof reduces to the case where $B/A$ is
monogenic, $B=A[\alpha]=A[t]/(f)$, $f$ being a monic polynomial in one
variable $t$ such that $f^{\prime}(\alpha)\in B^{\times}$. Note that under the
identification

$\Theta:B=\omega_{B/A}\simeq(I/I^{2})^{*}\otimes\Omega^{1}_{A[t]/A}$

one has $\Theta^{-1}(\bar{f}\otimes dt)=f^{\prime}(\alpha)^{-1}$ (as explained
in 4.1.10, separable case). 6.3.9 shows that $\tau_{f}=(\alpha^{n-1})^{*}$
where $n$ is the degree of $f$ in $t$.

Therefore, the relation of the corollary follows from the previous corollary
and the “Euler formula” (see e.g. [NS03, Prop. 1]):

$\operatorname{Tr}_{B/A}\big{(}f^{\prime}(\alpha)^{-1}\lambda\big{)}=(\alpha^{n-1})^{*}(\lambda).$

∎

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