# Notes on Milnor-Witt K-theory

F. Déglise

(Date: May 2023)

###### Abstract.

These notes are devoted to the foundations of Milnor-Witt K-theory of fields
of arbitrary characteristic and without any perfectness assumptions. Extending
the fundamental work of Morel, we establish all its functorial properties as
stated in Feld’s theory of Milnor-Witt modules, with a special attention about
twists. The main new result is a computation of transfers in the general (in
particular inseparable) case in terms of Grothendieck (differential) trace
maps. These notes are used as the foundation for an expository work on Chow-
Witt groups. They are built upon a series of talks given at the Spring School
“Invariants in Algebraic Geometry”, organized by Daniele Faenzi, Adrien
Dubouloz and Ronan Terpereau.

###### Contents

  1. 1 Introduction
    1. 1.1 Milnor’s K-theory and conjectures
    2. 1.2 Barge and Morel obstruction theory
    3. 1.3 Basic aim and scope of these notes
    4. 1.4 Transfers
    5. 1.5 Reading guide
    6. 1.6 Acknowledgement
  2. 2 Milnor-Witt K-theory and Grothendieck-Witt groups
    1. 2.1 Grothendieck-Witt groups and symmetric bilinear forms
    2. 2.2 Definition by generators and relations
    3. 2.3 Relations with quadratic forms
    4. 2.4 Twists
    5. 2.5 Residues
  3. 3 A detour on Chow-Witt groups of Dedekind schemes
    1. 3.1 Chow-Witt groups, quadratic divisors and rational equivalence
    2. 3.2 Homotopy invariance over a field
    3. 3.3 Localization exact sequences
    4. 3.4 Twisted Chow-Witt groups of the projective line
  4. 4 Transfers
    1. 4.1 Cotangent complexes and canonical sheaves
    2. 4.2 The quadratic degree map
    3. 4.3 A variation on Scharlau’s quadratic reciprocity property
    4. 4.4 General trace maps
  5. 5 The Milnor-Witt premodule structure
    1. 5.1 Functorialities
    2. 5.2 Rules
    3. 5.3 Stronger axioms and quadratic multiplicites
  6. 6 Appendix: coherent duality, traces and residues
    1. 6.1 Categorical duality and traces
    2. 6.2 Grothendieck differential trace map and duality
    3. 6.3 Grothendieck and Sheja-Storch Residues

##  1\. Introduction

###  1.1. Milnor’s K-theory and conjectures

One is retrospectively amazed by John Milnor’s cornerstone work [Mil70] on
K-theory and quadratic forms. At first sight, Milnor failed to define higher
K-groups of a field $K$ as one now knows that the K-theory he defines only
agrees with higher K-theory up to degree $2$, by Matsumoto’s theorem.111Among
his inspiration, Milnor cites Moore and Matsumoto’s works. See 2.2.1 and 2.2.2
for recall. On the other hand, the graded ring defined by Milnor was to be
recognized more than 20 years later222By a famous theorem of Totaro [Tot92].
as an invariant as fundamental as algebraic K-theory: it is the $(n,n)$-part
of the motivic cohomology ring of $k$. So that Milnor’s definition was truly
the first appearance of motivic cohomology, in its symbolic guise.

Moreover, Milnor had the brilliant insight of relating his new K-theory ring
to two apparently unconnected classical invariants: Galois cohomology of $K$
with coefficients in the $2$-torsion ring $\mathbb{Z}/2$, and the graded
algebra associated with the fundamental ideal of the Witt ring of $K$. One
motivation came from the theory of characteristic classes [MS74]333Recall that
this book is based on Milnor’s 1957 lectures!, and in particular from the so-
called Stiefel-Whitney classes $w_{n}(\xi)\in H^{n}(X,\mathbb{Z}/2)$ of a real
vector bundle $\xi$ over a smooth manifold $X$. Among the inspirations of
Milnor was a very short note [Del62] of Delzant defining an algebraic analogue
of these classes, for a field of characteristic not $2$:

$w^{D}_{n}:\operatorname{GW}(K)\rightarrow H^{n}(G_{K},\mathbb{Z}/2)$

where $\operatorname{GW}(K)$ is the Grothendieck group of quadratic $K$-vector
spaces444Though Delzant was obviously inspired by the Witt ring introduced in
[Wit37], this is the first occurrence of $\operatorname{GW}(K)$ in the
literature. See 2.1.2 more generally. and the right hand-side is the
$\mathbb{Z}/2$-cohomology of the absolute Galois group of $K$, or equivalently
the étale $\mathbb{Z}/2$-cohomology of $\operatorname{Spec}(K)$.

Milnor remarks that Delzant’s Stiefel-Whitney classes factorize through his
K-groups modulo $2$, thus obtaining a factorization of $w_{n}^{D}$ as:

$\operatorname{GW}(K)\xrightarrow{w_{n}}\operatorname{K}^{M}_{n}(K)/2\xrightarrow{h_{K}}H^{n}(G_{K},\mathbb{Z}/2)$

where $h_{K}$ is sometimes called the _norm residue homorphism_ or the _Galois
symbol_. The first of the Milnor conjectures states that $h_{K}$ is an
isomorphism for all $n>0$.555This was later generalized by Bloch and Kato by
replacing $2$ with an arbitrary prime. A complete proof of the Bloch-Kato
conjecture is available in [HW19], and a detailed account of the history of
the Milnor conjecture can be found in Section 1.7 of _loc. cit._

Even more remarkably, Milnor recognizes a way to go backward the map $w_{n}$,
and formulates another conjecture666see question 4.3 of [Mil70] or 2.2.3 of
the present paper relating the Witt ring with his K-theory ring. This question
was solved more than 30 years later by Orlov, Vishik, Voevodsky in [OVV07],
after Voevodsky’s proof of the first Milnor conjecture and his revolutionary
idea of introducing _motivic homotopy theory_.

###  1.2. Barge and Morel obstruction theory

Extending Milnor’s ideas on characteristic classes, and building on the ideas
of motivic homotopy, Jean Barge and Fabien Morel introduced in [BM00] an
algebraic-geometrical analogue of the Euler class of real oriented vector
bundles (see [MS74, §9, Def. p. 98]). The technical innovation of their
definition is the construction of an appropriate algebraic analogue of the
integral singular cohomology of a real manifold, a cohomology ring that they
call the _Chow groups of oriented cycles_ , which one now simply calls the
_Chow-Witt groups_.

While Milnor K-theory modulo $2$ is a suitable receptacle for Stiefel-Whitney
classes (over fields), as seen above, Barge and Morel had the idea of gluing
above this $2$-torsion ring the integral information coming from the
fundamental ideal $\operatorname{I}(K)$ and the Milnor K-ring
$\operatorname{K}^{M}_{*}(K)$: see [BM00, section 1] or 2.3.7 here. The
resulting graded ring, denoted by $J^{*}(E)$ in _loc. cit._ is now called the
_Milnor-Witt K-theory_ of $K$ and denoted by $\operatorname{K}^{MW}_{*}(K)$.
The study of this functor on fields is the main subject of the present
expository notes.

Before diving into the motivation for this work, let us come back briefly to
the work of Barge and Morel: using works of Rost [Ros96] and Schmid [Sch97],
they define the Chow-Witt groups of a smooth $k$-scheme as the cohomology of a
Gersten-like complex with coefficients in $\operatorname{K}^{MW}_{*}$, which
is now called the _Rost-Schmid complex_. Then they define the Euler class of
an _oriented_ algebraic vector bundle and make various conjectures about it
(Conjecture p. 289, Rem. 2.4 in _loc. cit._).777The conjecture was partially
solved by Morel in [Mor12, Th. 3.12] and was pushed much further by works of
Asok and Fasel; see [AF21] for a survey, and in particular §4.2 for a review
on (Barge-Morel) Euler classes.

###  1.3. Basic aim and scope of these notes

The aim of these notes is to lay the ground for the theory of quadratic cycles
and Chow-Witt groups, based on Feld’s axiomatic approach [Fel20a]. We will
therefore present the theory of the Milnor-Witt ring of fields, with an
emphasis on its functoriality. As a motivation, the hasty reader can take a
look at Section 5, and in particular the first paragraph, for the
functoriality properties we are aiming. We improve the theory known so far by
removing any assumption on the fields considered.888Usually, one considers
fields of characteristic not $2$, and sometimes one assumes they are finitely
generated over some perfect base field.

Our presentation of Milnor-Witt K-theory mixes two historical approaches. The
first one, due to Barge and Morel as already mentioned above, gives the
construction as a gluing of Witt’s theory of quadratic forms999Or rather
Grothendieck-Witt theory of inner product spaces in order to allow fields of
characteristic two; and Milnor’s K-ring. See 2.3.7 for the statement. The
second one, due to Hopkins and Morel, is much closer to Milnor’s viewpoint,
and gives a beautiful presentation of the Milnor-Witt K-theory ring in terms
of explicit generators and relations (2.2.7). In fact, the richness of the
theory comes from the comparison of these two approaches. This is based on
Milnor’s conjecture: here we refer to [GSZ16] in characteristic not $2$, and
to [Car] in characteristic $2$. Except for references to these two papers, and
to basics on Witt rings ([MH73]) and Milnor K-theory ([BT73], [Kat82]), these
notes are self-contained.

As explained in the previous section, the justification for defining Milnor-
Witt K-theory in addition to Milnor K-theory is to be able to develop an
algebraic orientation theory. The concrete trace of this orientation theory is
the existence of twists on the former K-theory which we consider as part of
the structure almost from the start. The first phenomenon that demonstrates
the need to consider twists is the construction of residues associated to a
discretely valued field $(K,v)$ with residue field $\kappa_{v}$:

$\partial_{v}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(\kappa_{v},\omega_{v}).$

Here the twists is given by the _normal space_ $\omega_{v}$ associated with
the valuation $v$; see 2.5.3 and therein for more. In our opinion, the
consideration of twists shades lights also to Witt’s theory as explained in
2.5.7.

###  1.4. Transfers

In fact, most of the paper is devoted to the study of transfer maps on Milnor-
Witt K-theory. This is no wonder as it was a famous problem left open by Bass
and Tate for Milnor K-theory ([BT73, I.§5]), only resolved by Kato in [Kat80,
§1.7, Prop. 5]. Given a finite field extension $E/k$, or $\varphi:k\rightarrow
E$, the transfer map has the form (see 4.4.6):

(1.4.0.a)
$\operatorname{Tr}_{E/k}^{MW}=\varphi^{*}:\operatorname{K}^{MW}_{n}(E,\omega_{E/k})\rightarrow\operatorname{K}^{MW}_{n}(k),$

where $\omega_{E/k}$ is the determinant of the cotangent complex of $E/k$ (we
call this the canonical module see 4.1.5). Here again, the twist is essential,
though it is trivial for (and only for) separable extensions.

For finitely generated field extension over a perfect base field of
characteristic not $2$, these transfer maps were introduced by Morel in
[Mor12, Chap. 4], in the more general context of strongly
$\mathbb{A}^{1}$-invariant sheaves but mostly avoiding twists. The theory was
recast for Milnor-Witt K-theory, still with the same restriction on fields, by
Feld in [Fel20b].

There are two methods to define transfers on functors defined on fields. The
first one is to follow the approach of Bass and Tate via residue maps and what
is called after Rost the _Weil reciprocity formula_ (see 4.2.4(2) for the case
of the projective line). The second one is by gluing known transfer maps. This
is closer to the approach of Fasel for defining pushforward on Chow-Witt
groups (see [Fas08, Cor. 10.4.5]).

Bass-Tate method. In these notes, we develop the two approaches. For the Bass-
Tate method, we have chosen to introduce Chow-Witt groups of Dedekind schemes.
This serves both as an illustration of the theory and as convenient language
to express the Weil reciprocity formula: this amounts to a computation of
Chow-Witt groups of quadratic $0$-cycles of the projective line. We state it
here for the convenience of the reader:

######  Theorem.

(see 3.4.4) Let $k$ be a field, and $\mathcal{L}$ be an invertible sheaf on
the projective line $\mathbb{P}^{1}_{k}$. We let $\mathcal{L}_{\infty}$ be the
restriction of $\mathcal{L}$ over the point at $\infty$ and $\omega_{\infty}$
be the conormal sheaf of the immersion
$i_{\infty}:\\{\infty\\}\rightarrow\mathbb{P}^{1}_{k}$.

Then the Chow-Witt group (in $\mathbb{G}_{m}$-degree $0$) of quadratic
divisors of $\mathbb{P}^{1}_{k}$ with coefficients in $\mathcal{L}$ are given
(induced by pushforward along $i_{\infty}$) by:

$\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\mathcal{L})\simeq\begin{cases}\operatorname{GW}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})&\deg(\mathcal{L})\text{
even,}\\\ \mathbb{Z}&\deg(\mathcal{L})\text{ odd.}\end{cases}$

This theorem perfectly illustrates the role of twists in Milnor-Witt K-theory
and Chow-Witt groups, which is the major difference with Milnor K-theory and
usual Chow groups. It was first proved by Fasel in [Fas13], though we lift any
restriction on the base field. Applying the above result when $\mathcal{L}$ is
the canonical sheaf $\omega=\mathcal{O}(-2)$ on $\mathbb{P}^{1}_{k}$ gives the
degree map on Chow-Witt groups:

$\deg:\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\omega)\rightarrow\operatorname{GW}(k).$

This degree map actually encompasses all the transfer maps for Milnor-Witt
K-theory for any monogenic extension field of $k$.101010We have formulated
here the results for $\mathbb{G}_{m}$-degree $1$ for the sake of clarity. It
is important to note that to get transfer maps in other degrees, one needs to
consider the whole graduation on the Rost-Schmid complex, corresponding to the
fact that Chow-Witt groups are the $0$-th $\mathbb{G}_{m}$-graded part of a
bigraded group (see 3.1.5 for more details). This is the geometric
interpretation of the method of Bass and Tate.111111see [BT73, (5.4)], §(5.4),
and especially diagram (3) The problem with this approach is to show that
these transfers are independent of the chosen generators, and that one can
extend the definition to any finite field extension.

Glueing and differential traces. Instead of proving this directly, as is done
by Morel and Feld, we use the second mentioned approach. On the one hand, one
has well-defined transfer maps on Milnor K-theory after Kato. For the
Grothendieck-Witt part (based on inner product forms to deal with
characteristic $2$), we show that one can define twisted transfer maps by
directly using the trace morphism that follow from Grothendieck duality
formalism, and which we call the differential trace map (see 6.2.4):

$\operatorname{Tr}_{E/k}^{\omega}:\omega_{E/k}\rightarrow k.$

Using this map and a classical method of Scharlau, one deduces transfers for
twisted Grothendieck-Witt groups. The advantage of these transfers is that
they do not depend on any choice, and yet can be compared precisely to
Scharlau’s ones (see 4.3.5).121212Note that this kind of construction of
twisted transfers for Witt groups (of $\mathbb{Z}[\frac{1}{2}]$-schemes) has
been previously considered by several authors including [Gil02, Nen07, CH11].
Our treatment is simple and direct, and is well-suited for explicit
computations, as explained below. Moreover, they can be glued appropriately to
Kato’s transfers on Milnor K-theory and induces the desired transfers
(1.4.0.a) on twisted Milnor-Witt K-theory.

The important result is to compare these glued transfers with the one obtained
by the Bass-Tate method. According to the uniqueness property of the latter,
this involves to check a _twisted form of the quadratic reciprocity law_ for
the transfers based on the differential trace maps. This is an enhancement of
the classical result of Scharlau, now valid in arbitrary characteristics (see
4.3.7 for the exact statement). As a consequence, our definition agrees with
that of Morel, and we also reprove the independence theorem of Morel and Feld
(see 4.4.13).

The nice feature of the second way of defining transfers is that it is well-
suited for computing trace maps. In particular, the differential trace map can
be computed explicitly using a method of Sheja and Storch. This links the
trace maps of Milnor-Witt K-theory with the method of Kass and Wickelgren for
computing (local) $\mathbb{A}^{1}$-brouwer degree ([KW19]). One can express
these computations in term of _Bezoutian_ (see 6.3.4 and 6.3.8) and this is
deeply linked with methods developed in [BMP21].131313This will be made more
precise in the second part of these notes [Dég23]. As an example let us state
the following formula which complete the étale case which was already known
(from [Hoy14, Lemma 5.8]):

######  Theorem (see 4.4.9(3)).

Let $k$ be a field of positive characteristic $p>0$, and $a\in k^{\times}$ be
an element which is not a power of $p$. Consider the inseparable extension
$E=k[\sqrt[q]{a}]$ of $k$, $q=p^{n}$.

Let $\tau^{\alpha}_{E/k}$ be _Tate trace map_ associated with $E/k$ and the
choice of $\alpha=\sqrt[q]{a}$ (see [Tat52, §1, (2)] and 6.3.10). Let
$w=dt\otimes(\overline{t^{q}-a})^{*}$ be the non zero element of the canonical
module $\omega_{E/k}$. Then for any unit $u\in E^{\times}$, the following
formula holds in $\operatorname{GW}(k)$:

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

where the right hand-side denotes the class of the inner product:
$(x,y)\mapsto\tau^{\alpha}_{E/k}(uxy)$.

This formula was actually our main motivation for writing these notes. It
illustrates again the importance of the twists in computation with Milnor-Witt
K-theory. In particular, in the above formula, changing $w$ usually completely
modify the result of the computation. We give further examples of this
phenomenon in 4.4.9, as well as an analogue of the degree formula for Milnor
K-theory: 4.4.10.

###  1.5. Reading guide

For readers who like axiomatic presentations, it is a good idea to start with
the list of structural maps of Milnor-Witt K-theory (Section 5.1) and the
basic rules they satisfy (Section 5.2).

This work is roughly divided intro three parts, with some preliminaries.

The preliminaries consist first of recall on the theory of quadratic forms
over fields, but more precisely on inner products spaces to deal with
arbitrary characteristics. The main results and computations concerning the
Grothendieck-Witt and Witt rings are recalled in Section 2.1.

Secondly, after a quick recollection on cotangent complexes and canonical
sheaves (or modules) in Section 4.1, we have given in an appendix (Section 6)
some reminders on Grothendieck coherent duality theory, and explain the link
with the work of Sheja and Storch which allows explicit computations. In
particular, we define explicitly what we call the _differential trace map_ ,
and give various interesting properties and formulas: the expression of
Grothendieck residue symbols (6.3.2), the computations in term of explicit
presentations and Bezoutians: see 6.3.12 and (6.3.12.a).

The first part concerns the definition and basic properties of the Milnor-Witt
K-ring of a field141414See 2.2.8 for the case of rings.: this is essentially
Section 2. As mentioned, we start with the presentation in term of generators
and relation (2.2.7), which is the more elementary and then relate it to the
presentation in terms of Milnor K-theory and the fundamental ideal (2.3.7).
Recall this relation is a (non-trivial) consequence of the second Milnor
conjecture (stated in 2.2.3).
Explicit computations are given in 2.3.1, 2.3.8, 2.3.9, 2.3.10 and 2.3.11. The
main specificity of Milnor-Witt K-theory are twists. We introduce them in a
second step, and use them to define residue maps in the end of Section 2.

The remaining of the paper is mainly devoted to transfers. However, we start
by introducing in Section 3 the Chow-Witt groups of arbitrary Dedekind
schemes, both as an illustration (for example of the use of twists) and as an
essential tool for the Bass-Tate method. Recall that compare to classical
intersection theory, Chow-Witt groups are twisted by a line bundle, and comes
with a bigraduation: the first one is codimension151515only 0 and 1 for
Dedekind schemes! and the second one is a $\mathbb{G}_{m}$-grading, which can
be explained by the existence of Tate twists for motives.161616Beware that
$\mathbb{G}_{m}$-twists refers in practice to twists by $\mathbb{Z}(1)[1]$ in
motivic notation; moreover, the bigraded Chow-Witt groups do not correspond to
the bigraded Milnor-Witt motivic cohomology of [BCD+22]: it is only related to
these groups, relation that one can see by looking at the $E_{2}$-page of the
coniveau filtration. The main result of the section is the computation of the
twisted Chow-Witt groups of the projective line (partially stated above):
3.4.4. The two essential tools used here are $\mathbb{A}^{1}$-invariance of
Chow-Witt groups and the localization long exact sequence. The link with usual
Chow groups is explained in 3.1.13 and 3.1.14.

The core study of transfers is done in Section 4. As a quick glance at the
table of contents shows, it follows the plan explained above, based on the own
hand on the quadratic degree map of 4.2.2, and on the other hand on the
differential trace map, used for defining transfers of Grothendieck-Witt rings
in 4.3.2.

The fourth part (Section 5) is a synthetic account of the previous parts,
making use of Feld’s axiomatic of Milnor-Witt modules. The only formula left
open is rule R3b whose proof is more sound using the full axiomatic of Rost-
Schmid complexes, and therefore will be treated in [Dég23] (but see 5.2.2 for
references to the literature). Note finally that we formulate and prove
stronger rules in Section 5.3, following and improving formulas due to Feld,
which are very relevant for quadratic intersection theory, and in particular
for the notion of _quadratic multiplicities_.

###  1.6. Acknowledgement

These notes benefited from the exchanges, discussions and interest of Jean
Fasel, Adrien Dubouloz, Niels Feld, Baptiste Calmès, Stephen McKean, Robin
Carlier. There is a special thanks to Fabien Morel, who made all this
possible, and with whom I had the pleasure of discussing subjects deeply
involved in these notes during and after my PhD, at the very moment when some
of the mathematics presented here was being invented. The author is supported
by the ANR project ”HQDIAG”, ANR-21-CE40-0015.

##  2\. Milnor-Witt K-theory and Grothendieck-Witt groups

###  2.1. Grothendieck-Witt groups and symmetric bilinear forms

######  2.1.1.

(cf. [MH73]) Let $K$ be a field. An inner product space or simply _inner
space_ $(E,\phi)$ over $K$ is a finite $K$-vector space $E$ with a bilinear
form

$\phi:E\otimes_{K}E\rightarrow K$

which is symmetric and non degenerate: $E\rightarrow
E^{\vee},x\mapsto\phi(x,-)$ is an isomorphism. The dimension of $E/K$ is
called the _rank_ of the inner space $(E,\phi)$. A morphism
$(E,\phi)\rightarrow(F,\psi)$ of inner spaces is a $K$-linear morphism
$f:E\rightarrow F$ such that $\phi(f(u),f(v))=\psi(u,v)$.

The category of inner spaces admits direct sums and tensor products:

$\displaystyle(E,\phi)\perp(F,\psi)\rightarrow(E\oplus F,\phi+\psi)$
$\displaystyle(E,\phi)\otimes(F,\psi)\rightarrow(E\otimes_{K}F,\phi.\psi).$

Therefore the set $I_{K}$ of isomorphism classes 171717This is indeed a set,
in bijection with  $\sqcup_{n\geq 0}\mathrm{Sym}_{n}(K)/\sim$ where
$\mathrm{Sym}_{n}(K)$ is the set of invertible symmetric $(n\times
n)$-matrices with coefficients in $K$, and $\sim$ is the congruence relation
on such matrices: $M\sim N$ if $M=PNP^{t}$; of inner spaces over $K$ is
commutative monoid for $\oplus$, and a commutative semi-ring for
$\oplus,\otimes$. The following definition comes from Milnor and Husemoller
[MH73]. It is a variant of the fundamental definition of Witt [Wit37]181818now
called the Witt group, see below, using the Grothendieck construction, that
apparently first appeared in the short work of Delzant [Del62].

######  Definition 2.1.2.

The _Grothendieck-Witt ring_ $\operatorname{GW}(K)$ of $K$ is the group
completion of the monoid $(I_{K},\oplus)$, with products induced by the tensor
product $\otimes$.

The rank of inner spaces induces a ring map:

(2.1.2.a) $\operatorname{GW}(K)\xrightarrow{\ \operatorname{rk}\ }\mathbb{Z}.$

######  Remark 2.1.3.

If the characteristic of $K$ is different from $2$, for any $K$-vector space
$V$, there is a one-to-one correspondence between the symmetric bilinear form
$\phi$ on $V$ and the quadratic forms $q$.191919$q(x)=\phi(x,x)$,
$\phi(x,y)=\frac{1}{2}(q(x+y)-q(x)-q(y))$ ! Then the Grothendieck-Witt ring
can be defined in terms of isomorphism classes of quadratic forms.

This is no longer true in characteristic $2$, but the definition based on
inner spaces is the correct one for $\mathbb{A}^{1}$-homotopy. Nevertheless,
one abusively use terms such as _quadratic_ intersection theory, in any
characteristic.

######  Example 2.1.4.

  1. (1)

Let $u$ be a unit in $K$. Then $K\otimes K\rightarrow K,(x,y)\mapsto u.xy$ is
an inner space of rank $1$. Its class in the Grothendieck-Witt ring is denoted
by $\langle u\rangle$. Obviously, $\langle uv^{2}\rangle=\langle u\rangle$.
Therefore, one has a canonical map:

$Q(K):=K^{\times}/(K^{\times})^{2}\rightarrow\operatorname{GW}(K).$

The group $Q(K)$ will be called the group of _quadratic classes_ of $K$.

  2. (2)

Given units $u_{i}\in K^{\times}$, we put $\langle
u_{1},\ldots,u_{n}\rangle=\langle u_{1}\rangle+\ldots+\langle u_{n}\rangle$.

A bilinear form on a framed $K$-vector space is defined by a symmetric
invertible matrix. The above element of $\operatorname{GW}(K)$ is represented
by the $K$-vector space $K^{n}$ and the diagonal matrix with coefficients
$u_{i}$.

######  Example 2.1.5.

  1. (1)

If $K$ is an algebraically closed field the rank map
$\operatorname{rk}:\operatorname{GW}(K)\rightarrow\mathbb{Z}$ is an
isomorphism.202020This is obvious in characteristic not $2$, as any inner
space admits an orthogonal base, and every element in $K$ is a square. More
generally, $\operatorname{rk}$ is an isomorphism whenever $(-1)$ is a square
in $K$ (see 2.1.13).

  2. (2)

It is well-known that a quadratic form over a real vector space is determined
by its signature. In other words, any $\sigma\in\operatorname{GW}(\mathbb{R})$
can be uniquely written as $\sigma=p.\langle 1\rangle+q.\langle-1\rangle$,
$\operatorname{rk}(\sigma)=p+q$ and the signature of $\sigma$ is defined as
the pair $(p,q)$. The map
$\operatorname{GW}(\mathbb{R})\rightarrow\mathbb{Z}\oplus\mathbb{Z},\sigma\mapsto(p,q)$
is an isomorphism.

  3. (3)

Let $K=\mathbb{F}_{q}$ be a finite field, $q=p^{n}$. Then the following
sequence of abelian groups is exact:

$\displaystyle 0\rightarrow Q(\mathbb{F}_{q})$
$\displaystyle\rightarrow\operatorname{GW}(\mathbb{F}_{q})\xrightarrow{\operatorname{rk}}\mathbb{Z}\rightarrow
0$ $\displaystyle\bar{u}$ $\displaystyle\mapsto 1-\langle u\rangle$

where $Q(\mathbb{F}_{q})$ is the group quadratic classes of $\mathbb{F}_{q}$
(2.1.4(1)). Note that this fits with item (1) above !

The preceding sequence is obviously split. Moreover, the abelian group
$\mathbb{F}_{q}^{\times}$ is cyclic or order $(q-1)$. In particular,
Lagrange’s theorem implies that $Q(\mathbb{F}_{q})$ is zero if $q$ is even,
and $\mathbb{Z}/2$ if $q$ is odd. Consequently:

$\operatorname{GW}(\mathbb{F}_{q})=\begin{cases}\mathbb{Z}&q\text{ even,}\\\
\mathbb{Z}/2\oplus\mathbb{Z}&q\text{ odd.}\end{cases}$

Consider the notations of 2.1.4. The element $h=\langle 1,-1\rangle$ is called
the (class of the) _hyperbolic form_. One can recover the following famous
definition (and extension in arbitrary characteristic) of Witt.

######  Definition 2.1.6.

One defines the Witt ring of a field $K$ as the quotient ring:

$\operatorname{W}(K)=\operatorname{GW}(K)/(h).$

The hyperbolic form being of rank $2$, the map (2.1.2.a) induces a morphism of
rings:

$W(K)\rightarrow\mathbb{Z}/2$

which is again called the _rank map_.

######  Remark 2.1.7.

2.1.2, as well as the previous one, can be extended to an arbitrary
(commutative) ring $A$ intead of a field $K$ (see [Kne77, I.§4, Prop. 1] for
the Grothendieck-Witt ring, and [MH73, I. 7.1] for the Witt ring): instead of
finite $K$-vector spaces, on considers finitely generated projective
$R$-modules $M$ equiped with a non-degerate symmetric bilinear form

$\phi:M\otimes_{R}M\rightarrow
R\mid\phi^{\prime}:M\xrightarrow{\sim}\operatorname{Hom}_{R}(M,R)=M^{\vee}$

and considers the Grothendieck group $\operatorname{GW}(R)$ associated with
the monoid of isomorphisms classes of $(M,\phi)$.

It follows from Minkovski convex body theorem (see [MH73, 4.4]) that one can
define an isomorphism of rings, called the _signature_ ,

$\sigma:\operatorname{W}(\mathbb{Z})\xrightarrow{\sim}\mathbb{Z}.$

The map $\sigma$ associates to the class of $(M,\phi)$ the signature of
$(M\otimes_{\mathbb{Z}}\mathbb{R},\phi\otimes_{\mathbb{Z}}\mathbb{R})$.

As $h$ is non-$\mathbb{Z}$-torsion in $\operatorname{GW}(\mathbb{Z})$ (because
it is not so in $\operatorname{GW}(\mathbb{R})$), one obtains that
$\operatorname{GW}(\mathbb{Z})$ is a free rank $2$ abelian group, and:

$\operatorname{GW}(\mathbb{Z})=\mathbb{Z}.\langle 1\rangle\oplus\mathbb{Z}.h.$

To get a presentation as a ring, we consider the element:
$\epsilon=-\langle-1\rangle$. Then one deduces from the above isomorphism an
isomorphism of rings:

$\operatorname{GW}(\mathbb{Z})=\mathbb{Z}[\epsilon]/(\epsilon^{2}-1).$

We will retain that this ring always acts (by functoriality) on rings of the
form $\operatorname{GW}(K)$, and more generally on the invariants of
$\mathbb{A}^{1}$-homotopy theory such as the Milnor-Witt K-theory.

######  Definition 2.1.8.

We define the _fundamental ideal_ of $\operatorname{W}(K)$ as:

$\operatorname{I}(K):=\operatorname{Ker}\big{(}\operatorname{rk}:\operatorname{GW}(K)\rightarrow\mathbb{Z}\big{)}\simeq\operatorname{Ker}\big{(}\operatorname{rk}:\operatorname{W}(K)\rightarrow\mathbb{Z}/2\big{)}.$

Typical elements of $\operatorname{I}(K)$ are given by the following _Pfister
forms_ associated with $u\in K^{\times}$:

$\langle\langle u\rangle\rangle:=1-\langle u\rangle.$

######  Remark 2.1.9.

  1. (1)

According to [MH73, 3.3], $\operatorname{I}(F)$ is the only (prime) ideal of
$\operatorname{W}(F)$ with residue field $\mathbb{F}_{2}$.

  2. (2)

This ideal is of fundamental (historical) importance as it is central to the
Milnor conjecture on quadratic forms: see 2.2.3.

######  Example 2.1.10.

Consider the case of a finite field $K=\mathbb{F}_{q}$, $q=p^{n}$. According
to 2.1.5, one gets that

$\operatorname{I}(\mathbb{F}_{q})=Q(\mathbb{F}_{q})=\begin{cases}0&q\text{
even,}\\\ \mathbb{Z}/2&q\text{ odd.}\end{cases}$

So if $q$ is even, $\operatorname{W}(\mathbb{F}_{q})=\mathbb{Z}/2$, via the
rank morphism. If $q$ is odd, one has (applying again the preceding example) a
short exact sequence:

$0\rightarrow\mathbb{Z}/2\rightarrow\operatorname{W}(\mathbb{F}_{q})\xrightarrow{\operatorname{rk}}\mathbb{Z}/2\rightarrow
0$

which is split if $q=1\mod 4$, and non split if $q=3\mod 4$. In fact, as a
ring, one gets more precisely:

$\operatorname{GW}(\mathbb{F}_{q})=\begin{cases}\mathbb{Z}/2&q\text{ even,}\\\
\mathbb{Z}/2[t]/(t-1)^{2}&q=1\mod 4,\\\ \mathbb{Z}/4&q=3\mod 4.\end{cases}$

In any case, one deduces that $\operatorname{I}^{n}(\mathbb{F}_{q})=0$ if
$n>1$.

The following result is an elaboration of Witt’s theorems on quadratic forms
([Wit37]).

######  Theorem 2.1.11.

The abelian group $\operatorname{GW}(K)$ admits a presentation whose
generators are given by symbols $\langle u\rangle$ for $u\in K^{\times}$
(mapping to the elements of 2.1.4) with relations:

  * (GW1)

$\langle uv^{2}\rangle=\langle u\rangle$.

  * (GW2)

$\langle u,v\rangle=\langle u+v,(u+v)uv\rangle$, $u+v\neq 0$

where we have used the notation $\langle u,v\rangle:=\langle u\rangle+\langle
v\rangle$.

Moreover, the relation (GW2) implies the following one:

  * (GW3)

$\langle u,-u\rangle=\langle 1,-1\rangle$.

The analogous presentation for the Witt group is well-known (see [MH73, Lem.
1.1]). We refer the reader to [Car, Th. 1.6] for a direct proof.212121Here is
the trick to get relation (GW3) from (GW2). One can assume $u\neq-1$, and one
writes using (GW2):  $\langle-u,u+1\rangle=\langle
1,-u(u+1)\rangle,\langle-1,1+u\rangle=\langle u,-u(1+u)\rangle.$ Taking the
difference of these two equalities gives (GW3).

######  Remark 2.1.12.

  1. (1)

If one wants a presentation of $\operatorname{GW}(K)$ as a ring, one needs
only adding the relation $\langle uv\rangle=\langle u\rangle\langle v\rangle$.

  2. (2)

One can take as generators of the abelian group $GW(K)$ the symbols
$\langle\bar{u}\rangle$ where $\bar{u}\in Q(K)$ is a quadratic class of a unit
$u\in K^{\times}$. Then one has only a single relation, given by (GW2) (with
$u$, $v$ replaced by $\bar{u}$, $\bar{v}$).

  3. (3)

Modulo the hyperbolic plane, one recovers the classical presentation of the
Witt group $\operatorname{W}(K)$: it is generated by symbols $\langle
u\rangle$ for a unit $u\in K^{\times}$ subject to the relations:

     * (W0)

$\langle 1,-1\rangle=0$

     * (W1)

$\langle uv^{2}\rangle=\langle u\rangle$

     * (W2)

$\langle u,v\rangle=\langle u+v,(u+v)uv\rangle$, $u+v\neq 0$.

Again, one can start with symbols $\langle\bar{u}\rangle$ of a quadratic class
$\bar{u}\in Q(K)$, in which case relation (W1) is unnecessary.

######  Example 2.1.13.

One deduces from the above presentation the following remarkable property of
the Witt group of a field $K$. The following conditions are equivalent:

  1. (1)

the rank map $\operatorname{rk}:\operatorname{W}(K)\rightarrow\mathbb{Z}/2$ is
an isomorphism;

  2. (2)

$Q(K)=1$, _i.e._ every unit in $K$ is a square.

######  2.1.14.

Twists. In what follows, it will be crucial to consider $\mathcal{L}$-valued
inner space for an arbitrary invertible $K$-vector space $\mathcal{L}$. These
are $K$ vector spaces $V$ with a symmetric bilinear forms
$V\otimes_{K}V\rightarrow\mathcal{L}$ such that the adjoint map
$V\rightarrow\operatorname{Hom}_{K}(V,\mathcal{L})$ is an isomorphism.

Then one can define as above, using again the orthogonal sum, a
$\operatorname{GW}(K)$-module $\operatorname{GW}(K,\mathcal{L})$, called the
_$\mathcal{L}$ -twisted Grothendieck-Witt group of $K$_. After modding out by
$h$, one obtains a $\operatorname{W}(K)$-module
$\operatorname{W}(K,\mathcal{L})$, the _$\mathcal{L}$ -twisted Witt group of
$K$_.

The tensor product of $K$-vector spaces induces an action of the ring
$\operatorname{GW}(K)$ on $\operatorname{GW}(K,\mathcal{L})$, and more
generally an exterior product:

$\operatorname{GW}(K,\mathcal{L})\otimes\operatorname{GW}(K,\mathcal{L}^{\prime})\rightarrow\operatorname{GW}(K,\mathcal{L}\otimes\mathcal{L}^{\prime}),$

and similarly for the twisted Witt ring.

######  Remark 2.1.15.

Both abelian groups $\operatorname{GW}(K,\mathcal{L})$ and
$\operatorname{W}(K,\mathcal{L})$ are _non canonically_ isomorphic to their
untwisted counter-parts. However, these twists can be interpreted as local
orientations in the theory of Chow-Witt groups.

###  2.2. Definition by generators and relations

######  2.2.1.

Milnor K-theory. Let us first recall that the Milnor K-theory
$\operatorname{K}^{M}_{*}(K)$ of a field $K$ is defined as the
$\mathbb{Z}$-graded algebra generated by symbols $\\{a\\}$ in degree $+1$ for
$a\in K^{\times}$ modulo the relations:

  1. (M1)

$\\{a,1-a\\}=0$

  2. (M2)

$\\{ab\\}=\\{a\\}+\\{b\\}$

where we have put $\\{a_{1},\ldots,a_{n}\\}=\\{a_{1}\\}\ldots\\{a_{n}\\}$.

Note in particular that: $\operatorname{K}^{M}_{0}(K)=\mathbb{Z}$,
$\operatorname{K}^{M}_{1}(K)=K^{\times}$.

######  Remark 2.2.2.

In general, there is a canonical _symbol map_ with values in (Quillen)
algebraic K-theory:

$\operatorname{K}^{M}_{n}(K)\rightarrow\operatorname{K}_{n}(K)$

which is an isomorphism if $n\leq 2$. The case $n\leq 1$ is easy, but the case
$n=2$ is a difficult theorem due to Matsumoto (see [Mat69]). The cokernel of
the symbol map is called the _indecomposable part_ of algebraic K-theory.

We now have all the tools to formulate the Milnor conjecture, now a theorem
due to Orlov, Vishik and Voevodsky:

######  Theorem 2.2.3 (Kato, Orlov-Vishik-Voevodsky).

Let $K$ be an arbitrary field and consider the notation of 2.1.8.

Then for any $n\geq 0$, the map
$K^{\times}\rightarrow\operatorname{I}(K),u\mapsto\langle\langle
u\rangle\rangle$ induces a ring morphism:

$\mu:\operatorname{K}^{M}_{*}(K)/2\operatorname{K}^{M}_{*}(K)\rightarrow\oplus_{n\geq
0}\operatorname{I}^{n}(K)/\operatorname{I}^{n+1}(K)$

which is an isomorphism.

See [Mil70, Question 4.3] for the statement of the conjecture, [Kat82] for the
proof when $K$ is of characteristic $2$ and [OVV07] (or [Mor05]) for the proof
in the other cases.

######  2.2.4.

Notation.– It is customary to denote by $\operatorname{I}^{*}(K)$ the
$\mathbb{Z}$-graded $\operatorname{W}(K)$-algebra where we conventionally put
$\operatorname{I}^{n}(K)=\operatorname{W}(K)$ for $n\leq 0$, and the product
is induced by that of $\operatorname{W}(K)$.

Then $\operatorname{I}(K)$ induces an ideal in $\operatorname{I}^{*}(K)$ and
we denote by $\bar{\operatorname{I}}^{*}(K)$ the quotient $\mathbb{Z}$-graded
$\operatorname{W}(K)$-algebra,222222One also finds the notation $i^{n}(K)$ for
$\bar{\operatorname{I}}^{n}(K)$; so that
$\bar{\operatorname{I}}^{n}(K)=\operatorname{I}^{n}(K)/\operatorname{I}^{n+1}(K)$
if $n\geq 0$, and $0$ otherwise. Note that it is clear that the action of
$\operatorname{W}(K)$ factors through the rank map so that
$\bar{\operatorname{I}}^{*}(K)$ is actually a $\mathbb{Z}/2$-algebra.

With this notation and the previous theorem, the morphism $\mu$ defined by
Milnor takes the form of an isomorphism of $\mathbb{Z}$-graded algebras over
$\mathbb{Z}/2$:

$\mu:\operatorname{K}^{M}_{*}(K)/2\rightarrow\bar{\operatorname{I}}^{*}(K).$

######  Example 2.2.5.

The case $n=0$ is trivial. In the case $n=1$, the map takes the form
$\mu_{1}:Q(K)\rightarrow\bar{\operatorname{I}}^{1}(K)=\operatorname{I}(K)/\operatorname{I}^{2}(K)$,
where $Q(K)$ is the group of quadratic classes. Then an explicit inverse is
given by the discriminant map

$d:\bar{\operatorname{I}}^{1}(K)\rightarrow
Q(K),[(E,\phi)]\mapsto(-1)^{r(r-1)/2}.\det(M_{\phi})$

where $(E,\phi)$ is an inner space of even rank $r$, and $M_{\phi}$ is any
matrix that represents it. See [Mil70, Th. 4.1] and [MH73, def. p.77]. For the
case $n=2$, and the interpretation of $\bar{\operatorname{I}}^{2}(K)$ in terms
of Clifford invariants of quadratic forms, we refer the reader to [MH73],
Theorem III.5.8 and its proof.

######  2.2.6.

The following definition, due to Hopkins and Morel (see [Mor04, Section 5]),
gives an extension of Milnor’s theory which mixes generators and relations of
Milnor K-theory and of the Grothendieck-Witt ring:

######  Definition 2.2.7.

Let $K$ be any field. We define the _Milnor-Witt ring_ , or _Milnor-Witt
K-theory_ , $\operatorname{K}^{MW}_{*}(K)$ of $K$ as the $\mathbb{Z}$-graded
ring with the following presentations:

Generators are given by symbols $[a]$ of degree $+1$ for $a\in K^{\times}$,
and a symbol $\eta$ of degree $-1$ called the _Hopf element_. Let us introduce
the following notations to formulate the relations:

$\displaystyle[a_{1},\ldots,a_{n}]$ $\displaystyle=[a_{1}]\ldots[a_{n}]$
$\displaystyle h$ $\displaystyle=2+\eta[-1]/$

Relations are given as follows, whenever they make sense:

  1. (MW1)

$[a,1-a]=0$

  2. (MW2)

$[ab]=[a]+[b]+\eta.[a,b]$

  3. (MW3)

$\eta[a]=[a]\eta$

  4. (MW4)

$\eta h=0$

Obviously, Milnor-Witt K-theory is a covariant functor with respect to
inclusion morphisms of fields. Given such a map $\varphi:K\rightarrow L$, the
natural morphism of $\mathbb{Z}$-graded ring (homogeneous of degree $0$):

$\varphi_{*}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(L).$

This map is sometimes called the _corestriction_.

######  Remark 2.2.8.

Given any ring $A$, the preceding definition makes sense so that we can define
the ring $\operatorname{K}^{MW}_{*}(A)$.232323See [Sch17, Def. 4.10] for more
developments. The resulting $\mathbb{Z}$-graded ring is covariantly functorial
in the ring $A$. This extended definition is useful for example when $A$ is a
local ring as we will see in 2.5.9.

Note that one can directly compute this ring when $A=\mathbb{Z}$:

$\operatorname{K}^{MW}_{*}(\mathbb{Z})=\mathbb{Z}\big{[}\epsilon,\eta,[-1]\big{]}/(\epsilon^{2}-1,\eta[-1]-\epsilon+1)$

where $\epsilon$, $\eta$, $[-1]$ are respectively in degree $0$, $-1$ and $1$.
In particular,

$\operatorname{K}^{MW}_{0}(\mathbb{Z})=\mathbb{Z}[\epsilon]/(\epsilon^{2}-1)=\operatorname{GW}(\mathbb{Z})$

according to 2.1.7. The ring $\operatorname{K}^{MW}_{*}(\mathbb{Z})$ always
acts on rings $\operatorname{K}^{MW}_{*}(K)$ (and more generally on invariants
of $\mathbb{A}^{1}$-homotopy theory).

######  2.2.9.

Relation with Milnor K-theory. One immediately remark that if one adds
$\eta=0$ to the above relations (MW*), one recovers the relations (M*) of
Milnor K-theory. In other words, sending the generators $\\{a\\}$ to the class
of $[a]$ in $\operatorname{K}^{MW}_{*}(K)/(\eta)$ induces an isomorphism of
$\mathbb{Z}$-graded algebras:

$\operatorname{F}:\operatorname{K}^{M}_{*}(K)\xrightarrow{\sim}\operatorname{K}^{MW}_{*}(K)/(\eta).$

In particular, for any integer $q\in\mathbb{Z}$, one deduces an exact sequence
of abelian groups:

(2.2.9.a)
$\operatorname{K}^{MW}_{q+1}(K)\xrightarrow{\gamma_{\eta}}\operatorname{K}^{MW}_{q}(K)\xrightarrow{\operatorname{F}}\operatorname{K}^{M}_{q}(K)\rightarrow
0.$

In the other direction, one can look at the morphism of $\mathbb{N}$-graded
algebras

$(K^{\times})^{\otimes,*}\mapsto\operatorname{K}^{MW}_{*}(K),u_{1}\otimes\ldots\otimes
u_{q}\mapsto h.[u_{1},...,u_{q}](q\geq 0).$

Because of relation (MW4), this maps factors through relation (M1) and (M2),
and therefore induces a well-defined morphism of $\mathbb{Z}$-graded algebras:

$\operatorname{H}:\operatorname{K}^{M}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(K).$

Following [BCD+22, Chap. 2, §1]), we introduce the following terminology for
these two maps:

######  Definition 2.2.10.

The morphisms of $\mathbb{Z}$-graded algebras242424both homogeneous of degree
$0$
$\operatorname{F}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{M}_{*}(K)$
and
$\operatorname{H}:\operatorname{K}^{M}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(K)$
are respectively called the _forgetful_ and _hyperbolic maps_.

######  2.2.11.

By definition, each of the above maps are uniquely characterized by the
properties:

$\displaystyle\operatorname{F}(\eta)=0,\quad\operatorname{F}([a])=\\{a\\}.$
$\displaystyle H(\\{a\\})=h.[a].$

Moreover, one deduces the following relations252525Use that $h$ modulo $\eta$
is equal to $2$ in $\operatorname{K}^{MW}_{*}(K)$ for the first one:

$\displaystyle F\circ H$ $\displaystyle=2.\operatorname{Id}$ $\displaystyle
H\circ F$ $\displaystyle=\gamma_{h}.$

######  Remark 2.2.12.

. In particular, one can remark that the forgetful map induces a split
epimorphism:

$\operatorname{K}^{MW}_{*}(K)[1/2]\rightarrow\operatorname{K}^{M}_{*}(K)[1/2].$

This fact will be made more precise in 2.3.9.

Let us come back to the study of the general groups
$\operatorname{K}^{MW}_{*}(K)$. One obtains the following presentation of each
individual graded parts, as abelian groups:

######  Proposition 2.2.13.

Consider an arbitrary field $K$ and an integer $n\in\mathbb{Z}$. Then the
abelian group $\operatorname{K}^{MW}_{n}(K)$ is generated by symbols of the
form:

$[\eta^{r},a_{1},\ldots,a_{n+r}],r\geq 0,a_{i}\in K^{\times}$

modulo the following three relations:

  1. _(MW1ab)_

$[\eta^{r},a_{1},\ldots,a_{n+r}]=0$ if $a_{i}+a_{i+1}=1$ for some $i$

  2. _(MW2ab)_

$[\eta^{r},a_{1},\ldots,a_{i}b_{i},\ldots,a_{n+r}]=[\eta^{r},a_{1},\ldots,a_{i},\ldots,a_{n+r}]+[\eta^{r},a_{1},\ldots,b_{i},\ldots,a_{n+r}]+[\eta^{r+1},a_{1},\ldots,a_{i},b_{i},\ldots,a_{n+r}]$

  3. _(MW4ab)_

$[\eta^{r},a_{1},\ldots,-1,\ldots,a_{n+r-1}]=-2[\eta^{r},a_{1},\ldots,\cancel{-1},\ldots,a_{n+r-1}]$
for $r\geq 2$

See [Car] for the proof.

######  Corollary 2.2.14.

Assume that $n\geq 1$, then the abelian group $\operatorname{K}^{MW}_{n}(K)$
is generated by the elements $[u_{1},\ldots,u_{n}]$ for an $n$-uplet of units
$u_{i}\in K^{\times}$.

This simply follows from the previous proposition by using relation (MW2ab).

######  Remark 2.2.15.

In particular, the abelian group $\operatorname{K}^{MW}_{1}(K)$ is generated
by symbols $[u]$ for $u\in K^{\times}$. However, beware that the map
$\iota:K^{\times}\rightarrow\operatorname{K}^{MW}_{1}(K),u\mapsto[u]$ is not a
morphism of group, except when $K=\mathbb{F}_{2}$. Indeed, one can express the
addition law in $\operatorname{K}^{MW}_{1}(K)$ by the formula:

$[u]+[v]=[uv]-\eta.[u,v],$

and $\eta.[u,v]$ is not zero in general. Note also that the forgetful map

$\operatorname{K}^{MW}_{1}(K)\xrightarrow{\operatorname{F}}\operatorname{K}^{M}_{1}(K)=K^{\times}$

is a surjective morphism of abelian group, but $\iota$ is a splitting of
$\operatorname{F}$ only after forgetting the group structure. In fact, we will
give an explicit description of this group in 2.3.8.

######  2.2.16.

Following Morel, one considers the following important element in Milnor-Witt
K-theory:

$\epsilon=-\langle-1\rangle\in\operatorname{K}^{MW}_{0}(K).$

Then relation (MW4) can be rewritten as $\epsilon.\eta=\eta$. Moreover, the
defect of commutavity of the multiplicative structure of Milnor-Witt K-theory
can be precisely expressed in terms of $\epsilon$ as follows.

######  Proposition 2.2.17.

For any field $K$, one has the relation:

$\forall(\alpha,\beta)\in\operatorname{K}^{MW}_{n}(K)\times\operatorname{K}^{MW}_{m}(K),\alpha\beta=\epsilon^{nm}.\beta\alpha.$

One says that the $\mathbb{Z}$-graded algebra $\operatorname{K}^{MW}_{*}(K)$
is _$\epsilon$ -commutative_. To prove this formula, one reduces to the case
$\alpha=[a]$, $\beta=[b]$ for units $a$, $b$ (see [Car, Cor. 1.5]).

######  2.2.18.

Quadratic multiplicities. One associates to any integer $n\in\mathbb{Z}$ the
following element of $K_{0}^{MW}(K)$:

$n_{\epsilon}=\begin{cases}\sum_{i=0}^{n-1}(-\epsilon)^{i}&n\geq 0\\\
\epsilon.(-n)_{\epsilon}&n<0.\end{cases}$

An equivalent computation:

$n_{\epsilon}=\begin{cases}m.h&n=2m\\\ m.h+1&n=2m+1\end{cases}$

Beware that the induced arrow
$\mathbb{Z}\rightarrow\operatorname{K}^{MW}_{0}(K),n\mapsto n_{\epsilon}$ is a
monoid morphism for multiplication

$(nm)_{\epsilon}=n_{\epsilon}m_{\epsilon}$

but not for the addition (compute $3_{\epsilon}$ and $4_{\epsilon}$).

######  Remark 2.2.19.

  1. (1)

A principle of quadratic enumerative geometry (see [Lev20]) is that, under a
careful choice of orientations, degrees of classical enumerative geometry
should be replaced by $\epsilon$-degrees as defined above.

  2. (2)

With the previous notation, relation (MW4) translates to:

$2_{\epsilon}.\eta=0$

This should remind the reader of the fact the classical/topological Hopf map
$\eta:S^{3}\rightarrow S^{2}$ induces a $2$-torsion element in the stable
homotopy groups of spheres, which account for the isomorphism:

$\pi^{st}_{3}(S^{2})=\mathbb{Z}/2.\eta$

where the left hand-side group is the third stable homotopy group of $S^{2}$.

  3. (3)

In negative degree, the quadratic multiplicities $n_{\epsilon}$ became
drastically simpler! Indeed, modulo $h$ or equivalently in
$\operatorname{W}(K)$,

$n_{\epsilon}=\begin{cases}1&n\text{ odd}\\\ 0&n\text{ even.}\end{cases}$

###  2.3. Relations with quadratic forms

Using the presentation obtained in the lemma just above, together with the
presentation of Grothendieck-Witt groups 2.1.11, Morel deduces the following
computation (for full details, see [Car, Prop. 1.9, Lem. 1.3]):

######  Proposition 2.3.1.

The following map is well-defined

$\operatorname{GW}(K)\rightarrow\operatorname{K}^{MW}_{0}(K),\langle
a\rangle\mapsto 1+\eta.[a]$

and induces an isomorphism of rings.

For any $n>0$, the multiplication map:
$\operatorname{K}^{MW}_{0}(K)\xrightarrow{\eta^{n}}\operatorname{K}^{MW}_{-n}(K)$
induces an isomorphism:

$\operatorname{W}(K)=\operatorname{GW}(K)/(h)\rightarrow\operatorname{K}^{MW}_{-n}(K).$

Finally, for any $n\geq 0$, the abelian group $\operatorname{K}^{MW}_{n}(K)$
is generated by symbols of the form $[a_{1},\ldots,a_{n}]$ for units $a_{i}\in
K^{\times}$.

As a consequence, we will view the elements of $\operatorname{GW}(K)$ as
elements in degree $0$ of Milnor-Witt K-theory. Note moreover, that
$\operatorname{GW}(K)$ lands in the center of the ring
$\operatorname{K}^{MW}_{*}(K)$.

######  Example 2.3.2.

The notation $h\in\operatorname{K}^{MW}_{0}(K)$ in relation (MW4) was
therefore justified, as it corresponds to the hyperbolic form in
$\operatorname{GW}(K)$. Note that relation (GW3) in 2.1.11 can be written as:

(2.3.2.a) $\forall u\in K^{\times},\langle u\rangle.h=h.$

Remark also that $h^{2}=2.h$ (direct computation).

Recall from 2.2.3 that given a unit $u\in K^{\times}$, one defines the
_Pfister form_ associated with $u$ as the element $\langle\langle
u\rangle\rangle=1-<u>$ of $\operatorname{W}(K)$.

######  Corollary 2.3.3.

Let $W(K)[t,t^{-1}]$ be the $t$-periodic $\mathbb{Z}$-graded algebra with $t$
a formal variable in degree $1$.

Then there exists a unique morphism of $\mathbb{Z}$-graded algebra

$\phi:\operatorname{K}^{MW}_{*}(K)[\eta^{-1}]\rightarrow
W(K)[t,t^{-1}],[u]\mapsto-\langle\langle u\rangle\rangle.t,\eta\mapsto t^{-1}$

and it is an isomorphism.

###### Proof.

The uniqueness of $\phi$ is obvious. We need to show that it is is well-
defined. First note that relation (MW4) implies that $h=0$ in
$\operatorname{K}^{MW}_{*}(K)[\eta^{-1}]$. Thus, it suffices to show that the
elements $-\langle\langle u\rangle\rangle.t$ and $t^{-1}$ of $W(K)[t,t^{-1}]$
satisfy the relation (MW1), (MW2) and (MW3). Relation (MW1) follows from
relation (W2) in the Witt ring (see 2.1.12(2)). Relation (MW2) follows from
the rule $\langle u\rangle\langle v\rangle=\langle uv\rangle$ in the Witt
ring, while relation (MW3) is obvious.

Finally, the preceding proposition shows that multiplication by $\eta$ induces
an isomorphism on the negative part of the $\mathbb{Z}$-graded algebra
$\operatorname{K}^{MW}_{*}(K)$. In particular, the canonical map
$\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(K)[\eta^{-1}]$
is an isomorphism in negative degree. On the other hand,
$\phi(1+\eta[u])=1-t^{-1}\langle\langle u\rangle\rangle.t=1-(1-\langle
u\rangle)=\langle u\rangle$. Therefore, applying again the preceding
proposition, one deduces that $\phi$ is an isomorphism in negative degree. As
both the source and target of $\phi$ are $\mathbb{Z}$-periodic, one deduces
that $\phi$ is an isomorphism in all degrees. ∎

######  2.3.4.

As in [Mor04], one can define the Witt K-theory of $K$ as the quotient
$\mathbb{Z}$-graded algebra:

$\operatorname{K}^{W}_{*}(K)=\operatorname{K}^{MW}_{*}(K)/(h).$

Indeed, the relations (MW*) correspond to the relations of _loc. cit._ ,
Definition 3.1.

On the other hand, one can consider the sub-algebra $\operatorname{I}^{*}(K)$
of $\operatorname{W}(K)[t,t^{-1}]$ generated by $\operatorname{I}(K).t$
(recall 2.1.8 and the notation of 2.2.4). The main result of _loc. cit._ is
that 2.2.3 implies the following finer comparison result.

######  Theorem 2.3.5.

Consider the preceding notation. Then there exists a unique morphism $\psi$ of
$\mathbb{Z}$-graded algebras that fits into the commutative diagram

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

where $\nu$ is the canonical map (use relation (MW4)), and the right-hand
vertical one is the obvious inclusion.

Moreover, $\psi$ is an isomorphism.

This theorem was first proved in [Mor04] when the characteristic of $K$ is
different from $2$. We refer the reader to [GSZ16, Th. 3.8] for a proof in the
latter case, and to [Car] for the proof in the characteristic $2$ case.

######  2.3.6.

As an application of the previous theorem, one deduces a canonical map:

$\mu^{\prime}_{n}:\operatorname{K}^{MW}_{n}(K)\longrightarrow\operatorname{K}^{W}_{n}(K)\xrightarrow{(-1)^{n}.\psi_{n}}\operatorname{I}^{n}(K)$

which can be uniquely characterized, as a morphism
$\mu^{\prime}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{I}^{*}(K)$
of $\mathbb{Z}$-graded algebras, as the map which sends $[u]$ to the Pfister
form $\langle\langle u\rangle\rangle\in\operatorname{I}^{1}(K)$ and the
element $\eta$ to the class $\langle 1\rangle$ in
$\operatorname{I}^{-1}(K)=\operatorname{W}(K)$.

######  Corollary 2.3.7.

The following commutative square of $\mathbb{Z}$-graded algebras is cartesian:

$\textstyle{\operatorname{K}^{MW}_{*}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\mu^{\prime}}$$\textstyle{\operatorname{K}^{M}_{*}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\operatorname{I}^{*}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\bar{\operatorname{I}}^{*}(K)}$

Here $F$ is the forgetful map (2.2.10) and $\mu$ is the map defined by Milnor
(2.2.3 and 2.2.4).262626recall it sends a generator $\\{u\\}$ to the class of
the Pfister form $\langle\langle u\rangle\rangle\in\operatorname{I}^{1}(K)$
modulo $\operatorname{I}^{2}(K)$

###### Proof.

Indeed, the above square in degree $n$ fits into the commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{K}^{W}_{n+1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\gamma}_{\eta,n}}$$\scriptstyle{(-1)^{n}.\psi_{n}}$$\textstyle{\operatorname{K}^{MW}_{n}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{n}}$$\scriptstyle{\mu^{\prime}_{n}}$$\textstyle{\operatorname{K}^{M}_{n}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{n}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{I}^{n+1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{I}^{n}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{\operatorname{I}}^{n}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

and the result follows as $\mu^{\prime}$ and $F$ are surjective and $\psi$ is
an isomorphism. ∎

######  Example 2.3.8.

Looking at degree $1$, we deduce the following explicit description of
$\operatorname{K}^{MW}_{1}(K)$, for any field $K$. The group
$\operatorname{K}^{MW}_{1}(K)$ is made of pairs $([\varphi],u)$ where
$\varphi$ is the Witt-class of an inner space $\phi:V\otimes_{L}V\rightarrow
K$ of even rank, $u\in K^{\times}$ is a unit such that $d(\varphi)=\bar{u}\in
Q(K)$ where $d$ is the discriminant of $\varphi$ (see 2.2.5). In other words,
an element of $\operatorname{K}^{MW}_{1}(K)$ is given by the Witt class of an
inner space over $K$ of even rank and a lift of its discriminant in
$K^{\times}$.

In this description, for any unit $u\in K^{\times}$, the symbol
$[u]\in\operatorname{K}^{MW}_{1}(K)$ is sent to the pair $(\langle\langle
u\rangle\rangle,\\{u\\})$.

######  Example 2.3.9.

. As $\bar{\operatorname{I}}^{*}(K)$ is $2$-torsion, one deducing from the
previous corollary the following interesting fact which extends 2.2.12. After
inverting $2$, the canonical maps $F$ and $\mu^{\prime}$ of the previous
corollary induces an isomorphism of $\mathbb{Z}$-graded rings:

$\operatorname{K}^{MW}_{*}(K)[1/2]\xrightarrow{F\times\mu^{\prime}}\operatorname{K}^{M}_{*}(K)[1/2]\times\operatorname{I}^{*}(K)[1/2].$

######  Corollary 2.3.10.

  1. (1)

One has an equality of ideals of $\operatorname{K}^{MW}_{*}(K)$:

$\operatorname{Ker}(\gamma_{\eta})=(h)=\mathrm{Im}(\operatorname{H}).$

In particular the sequence (2.2.9.a) can be extended into a long exact
sequence:

(2.3.10.a)
$\operatorname{K}^{M}_{*}(K)\xrightarrow{\operatorname{H}}\operatorname{K}^{MW}_{*}(K)\xrightarrow{\gamma_{\eta}}\operatorname{K}^{MW}_{*}(K)\xrightarrow{F}\operatorname{K}^{M}_{*}(K)\rightarrow
0$

which can be truncated and gives the short exact sequence:

$0\rightarrow\operatorname{I}^{*}(K)\xrightarrow{\bar{\gamma}_{\eta}}\operatorname{K}^{MW}_{*}(K)\xrightarrow{F}\operatorname{K}^{M}_{*}(K)\rightarrow
0$

such that $\bar{\gamma}_{\eta}$ is homogeneous of degree $-1$.

  2. (2)

Moreover, the forgetful map
$\operatorname{F}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{M}_{*}(K)$
identifies the principal ideal $(h)$ with the principal ideal
$2.\operatorname{K}^{M}_{*}(K)$ generated by $2$ in the Milnor ring of $K$.
One deduces a short exact sequence:

$0\rightarrow
2.\operatorname{K}^{M}_{*}(K)\xrightarrow{\operatorname{\tilde{H}}}\operatorname{K}^{MW}_{*}(K)\xrightarrow{\mu^{\prime}}\operatorname{I}^{*}(K)\rightarrow
0$

where $\operatorname{\tilde{H}}$ sends $2\in\operatorname{K}^{M}_{0}(K)$ to
$h$, and for $n>0$, sends a $2$-divisible symbol
$\\{a_{1},...,a_{n}\\}\in\operatorname{K}^{M}_{n}(K)$ to the element
$[u_{1},...,u_{n}]\in\operatorname{K}^{MW}_{n}(K)$.

Finally, $\mu^{\prime}\circ\bar{\gamma}_{\eta}$ is equal in degree $n$ to
$(-1)^{n}.i_{n}$ where $i_{n}:I^{n+1}(K)\rightarrow I^{n}(K)$ is the canonical
inclusion.

###### Proof.

Indeed, the preceding theorem implies that $\nu$ is injective, which implies
that $\operatorname{Ker}(\gamma_{\eta})=(h)$ as ideals of
$\operatorname{K}^{MW}_{*}(K)$. This concludes the first assertion as, by
construction, the image of $H$ is the ideal $(h)$. The first two exact
sequences follows directly, taking into account the isomorphism
$\psi:\operatorname{K}^{W}_{*}(K)\rightarrow\operatorname{I}^{*}(K)$. The last
exact sequence follows from the preceding corollary. ∎

######  Example 2.3.11.

We finish this subsection with a computation that easily follows from 2.3.7
and 2.1.13. If every unit in $K$ admits a square root, one has:

$\operatorname{K}^{MW}_{n}(K)=\begin{cases}\mathbb{Z}&n=0\\\
\mathbb{Z}/2&n<0\\\ K_{n}^{M}(K)&n>0.\end{cases}$

Recall also that if $K$ is algebraically closed, for all $n>1$,
$\operatorname{K}^{MW}_{n}(K)=\operatorname{K}^{M}_{n}(K)$ is divisible. These
are therefore huge groups !

###  2.4. Twists

We now introduce twists on Milnor-Witt K-theory, along the lines of 2.1.14. As
already mentioned (see Section 1.2), they will account for the local
orientations that appears on quadratic cycles (see Section 3.1 and in
particular 3.1.8). Moreover, they are necessary to obtain canonical residue
maps (see 2.5.3).

######  Definition 2.4.1.

Let $K$ be a field, and $\mathcal{L}$ be an invertible (_i.e._ of dimension
one over $K$) $K$-vector space. Consider the set:
$\mathcal{L}^{\times}:=\mathcal{L}-\\{0\\}$. The actions of $K^{\times}$ on
$\operatorname{K}^{MW}_{*}(K)$, via the map
$K^{\times}\rightarrow\operatorname{K}^{MW}_{0}(K),a\mapsto\langle a\rangle$,
and on $L^{\times}$ by scalar multiplication, give two morphisms of rings:

$\displaystyle\mathbb{Z}[K^{\times}]$
$\displaystyle\rightarrow\operatorname{K}^{MW}_{*}(K),$
$\displaystyle\mathbb{Z}[K^{\times}]$
$\displaystyle\rightarrow\mathbb{Z}[\mathcal{L}^{\times}].$

We define the _$\mathcal{L}$ -twisted Milnor-Witt K-theory_ of $K$ (or simply
the Milnor-Witt K-theory of the pair $(K,\mathcal{L})$) in degree
$n\in\mathbb{Z}$ as the following abelian group:

$\operatorname{K}^{MW}_{n}(K,\mathcal{L}):\operatorname{K}^{MW}_{n}(K)\otimes_{\mathbb{Z}[K^{\times}]}\mathbb{Z}[\mathcal{L}^{\times}].$

Elements of $\operatorname{K}^{MW}_{n}(K,\mathcal{L})$ are therefore formals
sums of elements of the form $\sigma\otimes l$ where
$\sigma\in\operatorname{K}^{MW}_{n}(K)$ and $l\in\mathcal{L}^{\times}$.

######  2.4.2.

We will identify the untwisted group $\operatorname{K}^{MW}_{*}(K)$ with
$\operatorname{K}^{MW}_{*}(K,K)$ via the obvious isomorphism:

$\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(K,K),\sigma\mapsto\sigma\otimes
1.$

Further, given any choice of $l\in\mathcal{L}^{\times}$, we get an isomorphism
of invertible $K$-vector spaces
$\Theta_{l}:K\rightarrow\mathcal{L},\lambda\mapsto\lambda.l$ and therefore an
isomorphism:

$\operatorname{ev}_{l}=(\Theta_{l}^{-1})_{*}:\operatorname{K}^{MW}_{*}(K,\mathcal{L})\rightarrow\operatorname{K}^{MW}_{*}(K,K)=\operatorname{K}^{MW}_{*}(K).$

According to definition, for any $u\in K^{*}$, one has:

$\operatorname{ev}_{ul}=\langle u\rangle.\operatorname{ev}_{l}.$

Given an element $\alpha\in\operatorname{K}^{MW}_{*}(K,\mathcal{L})$, one
obtains a function:

$\underline{\alpha}:\mathcal{L}^{\times}\rightarrow\operatorname{K}^{MW}_{*}K,l\mapsto\operatorname{ev}_{l}(\alpha)$

which is $K^{*}$-equivariant: $\underline{\alpha}(ul)=\langle
u\rangle.\underline{\alpha}(l)$. In other words, one further deduces the
following isomorphism272727To get the reciproc, choose an arbitrary
$l\in\mathcal{L}^{*}$, and consider $f\mapsto f(l)\otimes l$; of
$\mathbb{Z}$-graded rings:

$\begin{array}[]{rcl}\operatorname{K}^{MW}_{*}(K,\mathcal{L})&\rightarrow&\operatorname{Hom}_{K^{\times}}\big{(}\mathcal{L}^{\times},\operatorname{K}^{MW}_{*}(K)\big{)}\\\
\alpha&\mapsto&\underline{\alpha}.\\\ \end{array}$

######  Remark 2.4.3.

In particular, the twisted groups $\operatorname{K}^{MW}_{*}(K,\mathcal{L})$
are all abstractly isomorphic, but via a _non-canonical_ isomorphism.

In the theory of quadratic cycles, the invertible vector space $\mathcal{L}$
will be the space of local parameters (see eg. 3.1.8). Then one has two
interpretations of the elements of the twisted groups, in view of the
preceding isomorphism:

  * •

in the form $\alpha=\sigma\otimes l$, $\sigma$ is some coefficient, and $l$ is
a choice of a local parameter;

  * •

in the form
$\underline{\alpha}:\mathcal{L}^{\times}\rightarrow\operatorname{K}^{MW}_{*}(K)$,
we have a functional coefficient which to any choice of a local
parametrization associates some symbol in a $K^{\times}$-equivariant way.

Both point of views are useful.

######  Example 2.4.4.

Let $(K,\mathcal{L})$ be as above. Then for any $n\geq 0$, the isomorphism of
2.3.1 induces a _canonical_ isomorphism:

$\displaystyle\operatorname{GW}(K,\mathcal{L})$
$\displaystyle\xrightarrow{\simeq}\operatorname{K}^{MW}_{0}(K,\mathcal{L})$
$\displaystyle\operatorname{W}(K,\mathcal{L})$
$\displaystyle\xrightarrow{\simeq}\operatorname{K}^{MW}_{n}(K,\mathcal{L}),n<0.$

where the left hand-side was defined in 2.1.14. Indeed, it suffices to use the
isomorphism:

$\operatorname{GW}(K)\otimes_{\mathbb{Z}[K^{\times}]}\mathbb{Z}[\mathcal{L}^{\times}]\rightarrow\operatorname{GW}(K,\mathcal{L}),[\phi]\otimes
l\mapsto[\phi.l].$

######  Remark 2.4.5.

We consider again the situation of Remark 2.2.8, and assume that $A$ is
regular and semi-local (thus noetherian). Let $\mathcal{L}$ be an
invertible282828 _i.e._ locally free of rank $1$ $A$-module. As $A$ is regular
semi-local, $\mathcal{L}$ is trivializable: in fact, for any
$l\in\mathcal{L}^{\times}:=\mathcal{L}-\\{0\\}$, the map
$\Theta_{l}:A\rightarrow\mathcal{L},\lambda\mapsto\lambda.l$ is an
isomorphism.292929This comes from the isomorphism
$\operatorname{Pic}(A)=\mathbb{Z}$.

Moreover, the definition of $\langle a\rangle=1+\eta.[a]$ makes sense for any
unit $a\in A^{\times}$. Thus we can define:

$\operatorname{K}^{MW}_{n}(A,\mathcal{L})=\operatorname{K}^{MW}_{*}(A)\otimes_{\mathbb{Z}[A^{\times}]}\mathbb{Z}[\mathcal{L}^{\times}]$

######  2.4.6.

Basic operations on twisted Milnor-Witt K-theory. We have the following
structure on twisted Milnor-Witt K-theory:

  1. (1)

Products:

$\operatorname{K}^{MW}_{n}(K,\mathcal{L})\otimes\operatorname{K}^{MW}_{m}(K,\mathcal{L}^{\prime})\rightarrow\operatorname{K}^{MW}_{n+m}(K,\mathcal{L}\otimes\mathcal{L}^{\prime}),(\sigma\otimes
l,\tau\otimes l^{\prime})\mapsto(\sigma.\tau,l\otimes l^{\prime}).$

  2. (2)

First functoriality: given a morphism of field $\varphi:K\rightarrow L$, one
gets:

$\varphi_{*}:\operatorname{K}^{MW}_{n}(K,\mathcal{L})\rightarrow\operatorname{K}^{MW}_{n}(L,\mathcal{L}\otimes_{K}L),(\sigma,l)\mapsto(\varphi_{*}(\sigma),l\otimes_{K}1_{L}).$

  3. (3)

Second functoriality: given an isomorphism of $K$-vector spaces
$\theta:\mathcal{L}\rightarrow\mathcal{L}^{\prime}$ one gets:

$\theta_{*}:\operatorname{K}^{MW}_{n}(K,\mathcal{L})\rightarrow\operatorname{K}^{MW}_{n}(L,\mathcal{L}^{\prime}),(\sigma,l)\mapsto(\sigma,\Theta(l)).$

which is an isomorphism of abelian groups.

######  Remark 2.4.7.

It is possible to unite the first and second functorialities. One considers
the category of _twisted fields_ $\mathscr{TF}$ whose objects are pairs
$(K,\mathcal{L})$ where $K$ is a field and $\mathcal{L}$ of an invertible
$K$-vector space. Morphisms are given by

$(\varphi,\Theta):(K,\mathcal{L})\rightarrow(L,\mathcal{L}^{\prime})$

where $\varphi:K\rightarrow L$ is a morphism of fields, and
$\Theta:\mathcal{L}\otimes_{K}L\rightarrow\mathcal{L}^{\prime}$ is an
isomorphism. Composition is defined in the obvious way. Then
$\operatorname{K}^{MW}_{*}$ becomes a covariant functor from the category of
twisted fields to the category of graded abelian groups.

The category of twisted fields is _cofibred_ over the category of fields
$\mathscr{F}$. To interpret correctly the tensor product, via a _symmetric_
monoidal structure, one has to consider the graded category of twisted fields.
This is obtained via the Grothendieck construction applied to the graded
Picard category over fields (see [Del87] for this category and [Fas20] for the
monoidal structure).

######  Example 2.4.8.

Consider $(K,\mathcal{L})$ as above. Remark that the action of $K^{\times}$ on
$\operatorname{K}^{MW}_{*}(K)/\eta$ via the map $u\mapsto\langle u\rangle$ is
trivial: indeed, $\langle u\rangle=1\mod\eta$. This implies that
$\operatorname{K}^{MW}_{*}(K,\mathcal{L})/\eta$ is canonically isomorphic to
$\operatorname{K}^{MW}_{*}(K)/\eta=\operatorname{K}^{M}_{*}(K)$, which I
recall is just the Milnor K-theory of $K$.

We further extends 2.2.10 as follows.

######  Definition 2.4.9.

Let $(K,\mathcal{L})$ be a twisted field. Then one defines the twisted
forgetful (resp. hyperbolic) maps:

$\displaystyle\operatorname{F}$
$\displaystyle:\operatorname{K}^{MW}_{*}(K,\mathcal{L})\rightarrow\operatorname{K}^{M}_{*}(K),(\sigma\otimes
l)\mapsto\operatorname{F}(\sigma)$ $\displaystyle\operatorname{H}$
$\displaystyle:\operatorname{K}^{M}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(K,\mathcal{L}),\sigma\mapsto(h\sigma)\otimes
l$

where the last formula does not depend on the choice of
$l\in\mathcal{L}^{\times}$, given Equation 2.3.2.a.

Obviously, the two relations of 2.2.11 still holds with twists.

######  2.4.10.

Consider a twisted field $(K,\mathcal{L})$. There exists an action of
$K^{\times}$ on the graded algebra $\operatorname{I}^{*}(K)$ associated with
the fundamental ideal $\operatorname{I}(K)\subset\operatorname{W}(K)$ (see
2.2.4), via its $\operatorname{W}(K)$-module structure. This allows us to
define

$\operatorname{I}^{*}(K,\mathcal{L}):=\operatorname{I}^{*}(K)
\otimes_{\mathbb{Z}[K^{\times}]}\mathbb{Z}[\mathcal{L}^{\times}]$

as in 2.4.1. In fact, one also has
$\operatorname{I}^{*}(K,\mathcal{L})\subset\operatorname{W}(K,\mathcal{L})[t,t^{-1}]$
where $t$ is a formal variable as in 2.3.3. The isomorphism of 2.3.5 induces
an isomorphism (of $\mathbb{Z}$-graded $\operatorname{W}(K)$-algebras):

$\operatorname{K}^{W}_{*}(K,\mathcal{L})\xrightarrow{\psi}\operatorname{I}^{*}(K,\mathcal{L}).$

As remarked in 2.2.4, the action of $\operatorname{W}(K)$ on the quotient
algebra $\bar{\operatorname{I}}^{*}(K,\mathcal{L})$ is trivial. Therefore, one
deduces as in 2.4.8 a canonical identification:
$\bar{\operatorname{I}}^{*}(K,\mathcal{L})=\bar{\operatorname{I}}^{*}(K)$.

These considerations allow to extend 2.3.7 as follows:

######  Proposition 2.4.11.

The following commutative square of $\mathbb{Z}$-graded algebras is cartesian:

$\textstyle{\operatorname{K}^{MW}_{*}(K,\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\mu^{\prime}}$$\textstyle{\operatorname{K}^{M}_{*}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{\operatorname{I}^{*}(K,\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\bar{\operatorname{I}}^{*}(K).}$

Here $F$ is the twisted forgetful map (2.2.10), $\mu^{\prime}$ is the
$\mathcal{L}$-twisted version of the map defined in 2.3.6, and $\mu$ is the
map defined by Milnor (2.2.3).

###  2.5. Residues

######  2.5.1.

Residues are a famous part of the functoriality of Milnor K-theory (see [BT73,
§4]). A _discretely valued field_ will be a pair $(K,v)$ of a field $K$ with a
discrete valuation $v$. We let $\mathcal{O}_{v}$ be its ring of integers,
$\mathcal{M}_{v}$ the maximal ideal of $\mathcal{O}_{v}$ and
$\kappa_{v}=\mathcal{O}_{v}/\mathcal{M}_{v}$ its residue field.

Given a valuation $v:K^{\times}\rightarrow\mathbb{Z}$, with residue field
$\kappa_{v}$, one deduces for any $n>0$ a canonical morphism:

$\partial_{v}:\operatorname{K}^{M}_{n}(K)\rightarrow\operatorname{K}^{M}_{n-1}(\kappa_{v})$

uniquely characterized by the property:

$\partial_{v}(\\{u_{1},\ldots,u_{n}\\})=m.\\{\overline{u_{2}},\ldots,\overline{u_{n}}\\}$

for units $u_{i}\in K^{\times}$ such that $v(u_{1})=m$ and for $i>1$,
$v(u_{i})=0$, $\overline{u_{i}}$ being the residue class of $u_{i}$.

The analogous construction exists on Milnor-Witt K-theory, but the twists are
now necessary.

######  Theorem 2.5.2.

Consider as above a discretely valued field $(K,v)$. The $\kappa_{v}$-space
$\mathcal{C}_{v}:=\mathcal{M}_{v}/\mathcal{M}_{v}^{2}$ is the _conormal cone_
associated with $(K,v)$. It is an invertible $\kappa_{v}$-space (_i.e._ of
dimension $1$) and we let
$\omega_{v}:=(\mathcal{M}_{v}/\mathcal{M}_{v}^{2})^{\vee}$ be its
$\kappa_{v}$-dual – in other words, the _normal cone_ associated with $(K,v)$.

Then for any integer $n\in\mathbb{Z}$, there exists a unique morphism of
abelian groups:

$\partial_{v}:\operatorname{K}^{MW}_{n}(K)\rightarrow\operatorname{K}^{MW}_{n-1}(\kappa_{v},\omega_{v})$

satisfying the two following properties:

  1. (Res1)

$\partial_{v}(\eta.\sigma)=\eta.\partial_{v}(\sigma)$, for all
$\sigma\in\operatorname{K}^{MW}_{n+1}(K)$.

  2. (Res2)

For any uniformizer $\pi\in K$ and any units $u_{1},\ldots,u_{n}\in
K^{\times}$ such that $u_{1}=v_{1}\pi^{m}$, and $v(u_{i})=0$ for $i>1$, one
has:

$\partial_{v}([u_{1},u_{2},\ldots,u_{n}])=m_{\epsilon}\langle\overline{v_{1}}\rangle[\overline{u_{2}},\ldots,\overline{u_{n}}]\otimes\bar{\pi}^{*}.$

###### Proof.

(See [Mor12, 3.15, 3.21]) We first choose some uniformizing parameter
$\pi\in\mathcal{M}_{v}$ of $v$. Then we introduce the following quotient ring:

$A_{*}=\operatorname{K}^{MW}_{*}(\kappa_{v})[\xi]/(\xi-[-1].\xi)$

which we view as a graded ring by putting $\xi$ in degree $1$. Then the proof
reduces to show that the canonical map:

$K^{\times}\rightarrow
A_{*},(u=a\pi^{m})\mapsto[\bar{a}]+m_{\epsilon}\langle\bar{a}\rangle.\xi$

extends uniquely to a morphism of $\mathbb{Z}$-graded ring

$\Theta_{\pi}:\operatorname{K}^{MW}_{*}(K)\rightarrow A_{*}$

such that $\Theta_{\pi}(\eta)=\eta$.

Then given $\sigma\in\operatorname{K}^{MW}_{n}(K)$, one can write uniquely:

$\Theta_{\pi}(\sigma)=s_{v}^{\pi}(\sigma)+\partial_{v}^{\pi}(\sigma).\xi$

so that we get two maps

$\displaystyle s_{v}^{\pi}:\operatorname{K}^{MW}_{*}(K)$
$\displaystyle\rightarrow\operatorname{K}^{MW}_{*}(\kappa_{v})$
$\displaystyle\partial_{v}^{\pi}:\operatorname{K}^{MW}_{*}(K)$
$\displaystyle\rightarrow\operatorname{K}^{MW}_{*-1}(\kappa_{v})$

such that $s_{v}^{\pi}$ is a (homogeneous) morphism of $\mathbb{Z}$-graded
ring (obvious).

Both maps depend on the choice of $\pi$ in general. We then get the desired
canonical map by the formula:

(2.5.2.a)
$\partial_{v}(\sigma)=\partial_{v}^{\pi}(\sigma)\otimes\bar{\pi}^{*}.$

∎

######  Definition 2.5.3.

Consider the notation of the above theorem. The homogeneous morphism of
$\mathbb{Z}$-graded abelian groups

$\partial_{v}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(\kappa_{v},\omega_{v})$

of degree $-1$ is called the residue map associated with the valued field
$(K,v)$.

Given a prime $\pi$ of $(K,v)$, one also defines the residue map specialized
at $\pi$ as the map

$\partial_{v}^{\pi}=\operatorname{ev}_{\bar{\pi}^{*}}\circ\partial_{v}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(\kappa_{v})$

with the notation of 2.4.2. Equivalently, this is the unique homogeneous
morphism of $\mathbb{Z}$-graded abelian groups of degree $-1$ such that
relation (2.5.2.a) holds.

Finally, one defines the specialization map associated with of $(K,v,\pi)$ as
the morphism of $\mathbb{Z}$-graded rings

$s_{v}^{\pi}:\operatorname{K}^{MW}_{*}(K)\rightarrow\operatorname{K}^{MW}_{*}(\kappa_{v})$

defined in the above proof.

######  2.5.4.

Let $(K,v)$ be a valued field, $\pi$ a prime of $v$ and $u\in\mathcal{O}_{v}$
a unit. One can derive from the previous formula the following rule to compute
residues for any symbol $\sigma\in\operatorname{K}^{MW}_{*}(K)$:

$\displaystyle\partial_{v}^{u\pi}(\sigma)$ $\displaystyle=\langle
u\rangle\partial_{v}^{\pi}(\sigma)$ $\displaystyle\partial_{v}(\langle
u\rangle\sigma)$ $\displaystyle=\langle\bar{u}\rangle\partial_{v}(\sigma)$
$\displaystyle\partial_{v}([u]\sigma)$
$\displaystyle=\epsilon[\bar{u}]\partial_{v}(\sigma)$

The first statement follows from 2.4.2, and the other ones follows by using
the formula of the previous theorem and 2.2.13.

The specialization map can be computed by the following formulas (similar
proof):

$s_{v}^{\pi}(\sigma)=\partial_{v}^{\pi}([\pi].\sigma)-[-1]\partial_{v}^{\pi}(\sigma)=-\epsilon\partial_{v}^{\pi}([-\pi].\sigma).$

Consider finally a morphism of rings $R\rightarrow\mathcal{O}_{v}$, and let
$\varphi:R\rightarrow K$, $\bar{\varphi}:R\rightarrow\kappa_{v}$ be the
induced morphisms. Then, for any symbol
$\alpha\in\operatorname{K}^{MW}_{*}(R)$ (notation of 2.2.8), one deduces the
relation (use the same argument as for the previous relations):

$\partial_{v}(\varphi_{*}(\alpha)\sigma)=\bar{\varphi}_{*}(\alpha)\partial_{v}(\sigma).$

This implies that $\partial_{v}$ is
$\operatorname{K}^{MW}_{*}(\mathbb{Z})$-linear. In particular, it commutes
with multiplication not only by $\eta$ but also by $\epsilon$ and $h$.

######  2.5.5.

Consider the assumptions of the previous theorem. One can further define, for
any invertible $\mathcal{O}_{v}$-module $\mathcal{L}$, a twisted version:

$\partial^{\mathcal{L}}_{v}:\operatorname{K}^{MW}_{n}(K,\mathcal{L}_{K})\rightarrow\operatorname{K}^{MW}_{n-1}(K,\omega_{v}\otimes_{\kappa_{v}}\mathcal{L}_{\kappa_{v}})$

where $\mathcal{L}_{E}=\mathcal{L}\otimes_{\mathcal{O}_{v}}E$ for
$E=K,\kappa_{v}$. The procedure is a bit intricate: take an element
$\sigma\otimes l$ on the left hand-side:
$\sigma\in\operatorname{K}^{MW}_{n}(K)$ and $l\in(\mathcal{L}_{K})^{\times}$.
By definition, there exists elements $l_{0}\in(\mathcal{L}-\\{0\\})$ (recall
$\mathcal{L}$ is an $\mathcal{O}_{v}$-module) and $a\in K^{\times}$ such that
$l=l_{0}\otimes_{K}a$. Then one deduces by definition:

(2.5.5.a) $\sigma\otimes l=(\langle
a\rangle\sigma)\otimes(l_{0}\otimes_{K}1_{K}).$

One puts:

$\partial_{v}^{\mathcal{L}}(\sigma\otimes l)=\partial_{v}(\langle
a\rangle\sigma)\otimes(l_{0}\otimes_{K}1_{\kappa_{v}})$

or simply $\partial_{v}$ when $\mathcal{L}$ is clear from the context.

######  Remark 2.5.6.

The necessity to “renormalize” the parameter, as in (2.5.5.a), when
considering residues makes the computation in quadratic intersection theory
sometime quite cumbersome! Intuitively, we will be following a given
orientation from open subschemes to the complementary (reduced) closed
subscheme.

######  Example 2.5.7.

We can specialize the definition of the above residue map to negative degree.
Then according to 2.3.1, we get a canonical residue map:

$\partial_{v}:\operatorname{W}(K)\rightarrow\operatorname{W}(\kappa_{v},\omega_{v})$

such that

$\partial_{v}(\langle u\rangle)=\begin{cases}0&v(u)\text{ even},\\\ \langle
u\pi^{-v(u)}\rangle\otimes\bar{\pi}^{*}&v(u)\text{ odd},\pi\text{ any
prime.}\end{cases}$

(Use 2.2.19, point (3)!) Despite untwisted, this residue map is well-known in
Witt theory: after the choice of a prime $\pi$, one has
$\partial_{v}^{\pi}=\psi^{1}$ in the notation of [MH73, IV, §1], and it is
called the _second residue class morphism_.303030The first residue class
morphism is defined by the formula
$\psi^{0}=\psi^{1}\circ\gamma_{\langle\pi\rangle}$.

Note also that in degree $0$, we get a more regular formula:

$\partial_{v}:\operatorname{GW}(K)\rightarrow\operatorname{W}(\kappa_{v},\omega_{v}),\partial_{v}(\langle
u\rangle)=m_{\epsilon}\langle\bar{a}\rangle\otimes\bar{\pi}^{*},$

for $u=a\pi^{m}$, $v(a)=0$, $v(\pi)=1$.

######  Remark 2.5.8.

Comparing the formulas in 2.5.1 and 2.5.2, it is clear that the residue in
Milnor-Witt K-theory “modulo $\eta$” coincides with the residue map in Milnor
K-theory. One can be more precise using the maps of 2.4.9. Given a discretely
valued field $(K,v)$, and an invertible $\mathcal{O}_{v}$-module
$\mathcal{L}$, one gets a commutative diagram:

$\textstyle{\operatorname{K}^{M}_{*}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{H}}$$\scriptstyle{\partial^{M}_{v}}$$\textstyle{\operatorname{K}^{MW}_{*}(K,\mathcal{L}_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{F}}$$\scriptstyle{\partial_{v}}$$\textstyle{\operatorname{K}^{M}_{*}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial^{M}_{v}}$$\textstyle{\operatorname{K}^{M}_{*}(\kappa_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{H}}$$\textstyle{\operatorname{K}^{MW}_{*}(\kappa_{v},\omega_{v}\otimes\mathcal{L}_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{F}}$$\textstyle{\operatorname{K}^{M}_{*}(\kappa_{v})}$

where, for clarity, $\partial_{v}^{M}$ is the residue on Milnor K-theory. The
commutativity of the right-hand square was just explained, while the second
one follows from the formula
$\partial_{v}(h.\sigma)=h.\partial^{M}_{v}(\sigma)$ (indeed $h$ is unramified
with respect to $v$).

Similarly, the second residue morphism on (twisted) Witt $K$-theory of the
previous example obviously induces a canonical residue map:

$\partial_{v}^{\operatorname{I}}:\operatorname{I}^{n}(K,\mathcal{L}_{K})\rightarrow\operatorname{I}^{n+1}(\kappa_{v},\omega_{v}\otimes\mathcal{L}_{v}).$

On the quotient ring, we get a canonical untwisted residue map:
$\partial_{v}^{\bar{\operatorname{I}}}:\bar{\operatorname{I}}^{n}(K)\rightarrow\bar{\operatorname{I}}^{n+1}(\kappa_{v})$
(because of 2.4.10). It is now a routine check to prove that all the maps of
the square of 2.4.11 are compatible with the corresponding residue maps.

The following computation is an analogue of the Gersten exact sequence for
Milnor K-theory (see [Ker09]):

######  Theorem 2.5.9.

Let $(K,v)$ be a discretely valued field, and $\mathcal{L}$ be an invertible
$\mathcal{O}_{v}$-module.

  1. (1)

Then the following sequence (see 2.4.5 for the first term) is exact:

$\operatorname{K}^{MW}_{n}(\mathcal{O}_{v},\mathcal{L})\xrightarrow{\nu_{*}}\operatorname{K}^{MW}_{n}(K,\mathcal{L}_{K})\xrightarrow{\partial_{v}}\operatorname{K}^{MW}_{n-1}(\kappa_{v},\omega_{v}\otimes_{\kappa_{v}}\mathcal{L}_{\kappa_{v}})\rightarrow
0$

where $\nu:\mathcal{O}_{k}\rightarrow K$ is the obvious inclusion and
$\nu_{*}$ is defined as in 2.4.6(2).

  2. (2)

If moreover the ring $\mathcal{O}_{v}$ contains an infinite field of
characteristic not $2$, then the map $\nu_{*}$ is injective.

Idea of proof for (1): the surjectivity of $\partial_{v}$ is obvious: given
any (abelian) generator
$\sigma=[\eta^{r},v_{1},\ldots,v_{n-1+r}]\otimes\bar{\pi}^{*}$ of the right
hand-side group, $\pi\in\omega_{v}^{\times}$, $v_{i}\in\kappa_{v}^{\times}$
(see 2.2.13), there exists lifts $u_{i}\in\mathcal{O}_{v}^{\times}$ of
$v_{i}$, along the epimorphism $\mathcal{O}_{v}\rightarrow\kappa_{v}$. Then
formulas (Res1) and (Res2) implies that $[\eta^{n},\pi,v_{1},\ldots,v_{n}]$
lifts $\sigma$.

Also, (Res2) implies that $\partial_{v}\nu_{*}=0$. Therefore, one only needs
to prove that the induced map
$\operatorname{Im}(\nu_{*})\rightarrow\operatorname{Ker}(\partial_{v})$ is an
isomorphism. This is the serious part! We refer the reader to the proof of
[Mor12, Th. 3.22].

Point (2) is the _Gersten conjecture for Milnor-Witt K-theory_ and for the
local ring $\mathcal{O}_{v}$. This is due to Gille, Zhong and Scully: cf.
[GSZ16].

##  3\. A detour on Chow-Witt groups of Dedekind schemes

###  3.1. Chow-Witt groups, quadratic divisors and rational equivalence

######  3.1.1.

We let $X$ be a connected $1$-dimensional scheme which is assumed to be normal
(or equivalently regular). Let $\mathcal{L}$ be an invertible sheaf over $X$.
The main examples are smooth algebraic curves over a field and the spectrum of
a Dedekind ring.

Let $\kappa(X)$ be the function field of $X$ and $\mathcal{L}_{\eta}$ be the
pullback to $\operatorname{Spec}(\kappa(X))$ seen as an invertible
$\kappa(X)$-vector space.313131We will also use the notation
$\mathcal{L}_{\kappa(X)}=\mathcal{L}_{\eta}$ later. We let $X^{(1)}$ be the
set of points $x\in X$ which are closed (_i.e._ of codimension $1$). This
amounts to ask that the local ring $\mathcal{O}_{X,x}$ is $1$-dimensional, and
therefore a discrete valuation ring. In particular, $x$ uniquely corresponds
to a valuation $v_{x}$ on $\kappa(X)$ and we can consider the associated
residue map (2.5.2)

$\partial_{x}:\operatorname{K}^{MW}_{*}\big{(}\kappa(X),\mathcal{L}_{\eta}\big{)}\rightarrow\operatorname{K}^{MW}_{n-1}(\kappa_{x},\omega_{x/X}\otimes\mathcal{L}_{x})$

where $\mathcal{L}_{x}$ is the restriction of the invertible
$\mathcal{O}_{X}$-module $\mathcal{L}$ to $\kappa(x)$ and $\omega_{x/X}$ is
the normal sheaf of $(\kappa(X),v_{x})$.323232The notation $\omega_{x/X}$ will
take all its meaning in 4.1.5. See also 4.1.6. Explicitly:

$\omega_{x/X}:=\left(\mathcal{M}_{X,x}/\mathcal{M}_{X,x}^{2}\right)^{\vee}.$

Given an element $f\in\operatorname{K}^{MW}_{*}\big{(}\kappa(X)\big{)}$, we
will interpret $\partial_{x}(f)$ as the $\operatorname{K}^{MW}$-order of $f$
at $x$.

######  Lemma 3.1.2.

With the above notations, for any
$f\in\operatorname{K}^{MW}_{n}\big{(}\kappa(X)\big{)}$, the set:

$\\{x\in X\mid\partial_{x}(f)\neq 0\\}$

is finite.

Given the definition of the residue map, and 2.2.13, this directly follows
from the (more classical) fact:

######  Lemma 3.1.3.

Let $u\in\kappa(X)^{\times}$ be a unit. Then the set $\\{x\in X\mid
v_{x}(u)\neq 0\\}$ is finite.

Even in our generality, the finiteness is very classical. The alert reader
will have recognized the support of the divisor associated with the rational
function $u$ of $X$ appearing in the previous lemma!

######  Remark 3.1.4.

The fact the scheme $X$ is noetherian is essential here. However, in case one
withdraw this assumption, everything would still be fine as we will obtain a
locally finite subset of $X$. The theory of cycles, and quadratic cycles,
would be fine as we will consider locally finite sums. This fits particularly
well with the fact Chow groups (as well as Chow-Witt groups) are a kind of
_Borel-Moore homology_ in topology, and the latter is represented by the
complex of locally finite singular chains (for suitable topological spaces).

The following definition is a slight generalization of the known definition of
the classical definition of Chow-Witt groups. We refer the reader to [DFJ] for
further developments.

######  Definition 3.1.5.

Consider the previous notation. We define the _quadratic divisor class_ map as
the following sum:

$\operatorname{\widetilde{div}}_{X}=\sum_{x}\partial_{x}:\operatorname{K}^{MW}_{*}\big{(}\kappa(X),\mathcal{L}_{\eta}\big{)}\rightarrow\bigoplus_{x\in
X^{(1)}}\operatorname{K}^{MW}_{*-1}(\kappa_{x},\omega_{x/X}\otimes\mathcal{L}_{x})$

which is well-defined according to 3.1.2. This is a homogeneous morphism of
$\mathbb{Z}$-graded abelian groups of degree $-1$.

We then define the group $\widetilde{\mathrm{C}}{}^{p}(X,\mathcal{L})_{q}$ for
$p=0$ (resp. $p=1$) as respectively the source (resp. target) of
$\operatorname{\widetilde{div}}_{X}$ with $*=q$ (resp. $*=q+1$), and as $0$
otherwise. Therefore we have obtained a complex
$\widetilde{\mathrm{C}}{}^{*}(X,\mathcal{L})_{q}$, concentrated in
cohomological degree $0$ and $1$. We call it the _(cohomological) Rost-Schmid
complex_ of $X$.

We define the _Chow-Witt group_
$\widetilde{\mathrm{CH}}{}^{p}(X,\mathcal{L})_{q}$ of codimension $p$ and
$\mathbb{G}_{m}$-degree $q$ as the cohomology in degree $p$ of this complex.
When $q=0$, we call it simply the Chow-Witt group, written
$\widetilde{\mathrm{CH}}{}^{p}(X,\mathcal{L})$.

######  Remark 3.1.6.

  1. (1)

Beware that the differentials of the Rost-Schmid complex are homogeneous of
degree $-1$ with respect to the $\mathbb{G}_{m}$-grading. There are other
possible conventions for the bigrading of
$\widetilde{\mathrm{C}}{}^{*}(X,\mathcal{L})_{*}$ but we will not use them
here.

  2. (2)

Even if we are mainly interested in the Chow-Witt groups, the other
$\mathbb{G}_{m}$-degrees for $q\neq 0$ will be crucial for computations. See
Section 3.3.

  3. (3)

The groups $\widetilde{\mathrm{CH}}{}^{p}(X,\mathcal{L})_{q}$ are analogues of
the higher Chow groups. However, they do not deserve the name higher Chow-Witt
groups as they only contribute to some part of the latter (that one can
interpret as the Milnor-Witt motivic Borel-Moore homology; see [BY20,
BCD+22]). In fact, while the latter are represented by a full ring spectrum
$\mathrm{H}_{\mathrm{MW}}\mathbb{Z}$, the former are represented by a
truncation of $\mathrm{H}_{\mathrm{MW}}\mathbb{Z}$. On the other hand, the
groups just defined satisfy the same formalism than higher Chow groups.

  4. (4)

If one replaces Milnor-Witt K-theory by the usual K-theory, one obtains Rost’s
($\mathbb{G}_{m}$-)graded Chow groups $\operatorname{CH}^{p}(X)_{q}$ defined
in [Ros96]. This was in fact the model for the previous definition. We refer
the reader to 3.1.14 for more discussion.

######  Example 3.1.7.

In codimension $0$, $\widetilde{\mathrm{CH}}{}^{0}(X,\mathcal{L})$ is the
kernel of the map in degree $0$:

$\operatorname{GW}\big{(}\kappa(X),\mathcal{L}_{\eta}\big{)}\rightarrow\bigoplus_{x\in
X^{(1)}}\operatorname{W}(\kappa_{x},\omega_{x/X}\otimes\mathcal{L}_{x}).$

A virtual inner $\mathcal{L}_{\eta}$-space over the function field $\kappa(X)$
which is in the kernel of this map is said to be _unramified_ (with respect to
the curve $X$).

######  3.1.8.

Quadratic divisors. Let us explicit the above definition when $q=0$. The
abelian group $\widetilde{\mathrm{CH}}{}^{1}(X,\mathcal{L})$ is the cokernel
of the map in degree $1$:

$\operatorname{\widetilde{div}}:\operatorname{K}^{MW}_{1}\big{(}\kappa(X),\mathcal{L}_{\eta}\big{)}\rightarrow\bigoplus_{x\in
X^{(1)}}\operatorname{GW}(\kappa_{x},\omega_{x/X}\otimes\mathcal{L}_{x}).$

The abelian group at the target will be called the group of _quadratic
divisors_ (or $1$-codimensional cycles) of $(X,\mathcal{L})$. These are formal
sums of the form

(3.1.8.a) $\sum_{i\in I}(\sigma_{i}\otimes\bar{\pi}_{i}^{*}\otimes
l_{i}).x_{i}$

where:

  * •

$x_{i}\in X$ is a closed point,

  * •

$\sigma_{i}\in\operatorname{GW}(\kappa_{x_{i}})$ is the class of an inner
space over $\kappa_{x_{i}}$,

  * •

$\pi_{i}$ is a uniformizing parameter of the valuation ring
$\mathcal{O}_{X,x_{i}}$, (equivalently a _local parameter_ of the closed
subscheme $\\{x_{i}\\}\subset X$),333333the notation $\bar{\pi}_{i}^{*}$
reminds the reader that we consider the element in
$\omega_{x/X}=(\mathcal{M}_{X,x_{i}}/\mathcal{M}_{X,x_{i}}^{2})^{\vee}$
corresponding to $\pi_{i}$

  * •

$l_{i}\in\mathcal{L}$ is a non-zero element.

In practice, one can also view the coefficient
$(\sigma_{i}\otimes\bar{\pi}_{i}^{*}\otimes l_{i})$ as a virtual inner
$(\omega_{x_{i}/X}\otimes\mathcal{L}_{x_{i}})$-space over $\kappa(x_{i})$.
Recall also from 2.4.2 the interpretation of this latter element as a
$\kappa(x_{i})$-equivariant map from the space of non zero elements
$(\omega_{x_{i}/X}\otimes\mathcal{L}_{x_{i}})^{\times}$ to the Grothendieck-
Witt group $\operatorname{GW}(\kappa_{x_{i}})$.

As in the classical case, quadratic divisors which are in the image of
$\operatorname{\widetilde{div}}$ are said to be _principal_. Two quadratic
divisors are _rationally equivalent_ is there difference is principal.

######  Example 3.1.9.

In the case $X$ is in addition local, thus the spectrum of a discrete
valuation ring $\mathcal{O}_{v}$, 2.5.9 implies in particular:

$\widetilde{\mathrm{CH}}{}^{p}(\mathcal{O}_{v})=\begin{cases}\operatorname{GW}(\mathcal{O}_{v})&p=0,\mathcal{O}_{v}\supset
k_{0}\\\ 0&p=1.\end{cases}$

where $k_{0}$ is an infinite field of characteristic not $2$. The vanishing of
$\widetilde{\mathrm{CH}}{}^{1}(\mathcal{O}_{v})$ can be interpreted by saying
that every quadratic divisor of $X$ is _principal_.

######  3.1.10.

Quadratic order of vanishing. One can also make explicit the definition of the
quadratic divisor class map. Let us fix a point $x\in X^{(1)}$, and $v_{x}$
the corresponding discrete valuation $v_{x}$ on $\kappa(X)$. We know that the
abelian group $\operatorname{K}^{MW}_{1}(\kappa(X))$ is generated by elements
$[f]$ for a unit $f\in\kappa(X)^{\times}$ (see 2.2.14 and 2.2.15). Given a
rational function $f\in\kappa(X)^{\times}$ on $X$, we get with the notation of
the above definition:

(3.1.10.a)
$\partial_{x}([f])=m_{\epsilon}.\langle\bar{u}\rangle\otimes\bar{\pi}_{x}\in\operatorname{GW}(\kappa_{x},\omega_{x/X})$

where we have chosen a local parameter $\pi_{x}$ of $x$ in $X$ (_i.e._ a
uniformizing parameter of the valuation ring $\mathcal{O}_{X,x}$),
$m=v_{x}(f)$ is the classical order of vanishing of $f$ at $x$, and
$u=f.\pi_{x}^{-m}$ and $\bar{u}$ is its class in
$\kappa_{x}=\mathcal{O}_{X,x}/\mathcal{M}_{X,x}$. The formula, as well as the
fact this element does not depend on the particular choice of $\pi_{x}$,
directly follows from 2.5.2.

######  Definition 3.1.11.

Consider the above assumptions. We define the quadratic order of vanishing of
a rational function $f\in\kappa(X)$ as the element
$\operatorname{\widetilde{ord}}_{x}(f)=\partial_{x}([f])$ in
$\operatorname{GW}(\kappa_{x},\omega_{x/X})$.

One can rewrite the definition of the divisor class map when $q=0$ in more
classical terms:

$\operatorname{\widetilde{div}}([f])=\sum_{x\in
X^{(1)}}\operatorname{\widetilde{ord}}_{x}(f).x.$

######  Remark 3.1.12.

One should be careful that the quadratic order of vanishing, as well as the
quadratic divisor class map, is only additive in $f$ with respect to the
addition of $\operatorname{K}^{MW}_{1}(\kappa(X))$, which in general differs
from the group law of $K^{\times}$ (see 2.2.15).

######  3.1.13.

Let us consider the previous definitions modulo $\eta$. Then we get in degree
$0,1$ a map, independent of $\mathcal{L}$:

$\kappa(X)^{\times}=\operatorname{K}^{M}_{1}\big{(}\kappa(X),\mathcal{L}_{\eta}\big{)}\xrightarrow{\operatorname{\widetilde{div}}_{X}\text{
mod }\eta}\bigoplus_{x\in
X^{(1)}}\operatorname{K}^{M}_{0}(\kappa_{x},\omega_{x/X}\otimes\mathcal{L}_{x})=Z^{1}(X)$

where the right hand-side is the groupe of (ordinary!) $0$-cycles of $X$. This
is precisely the divisor class map: in fact, one obviously have the formula

$\operatorname{\widetilde{ord}}_{x}([f])=\operatorname{ord}_{x}(f)\text{ mod
}\eta,$

which amounts to say that the rank of the underlying inner form of
$\operatorname{\widetilde{ord}}_{x}([f])$ is the classical order of vanishing
$\operatorname{ord}_{x}(f)$ of $f$ at $x$ (use Formula Equation 3.1.10.a). In
particular, we get:

$\widetilde{\mathrm{CH}}{}^{p}(X,\mathcal{L})/(\eta)=\begin{cases}\mathbb{Z}^{\pi_{0}(X)}&p=0\\\
\operatorname{Pic}(X)&p=1.\end{cases}$

Moreover, one can describe explicitly the image of the map:

$\widetilde{\mathrm{CH}}{}^{p}(X,\mathcal{L})\rightarrow\widetilde{\mathrm{CH}}{}^{p}(X,\mathcal{L})/(\eta)\simeq\operatorname{CH}^{p}(X)$

It is just induced by the rank map: in degree $0$, it sends an unramified
inner $\mathcal{L}$-space $\sigma$ over $\kappa(X)$ to its rank
$\operatorname{rk}(\sigma)$. In degree $1$, it sends a quadratic $0$-cycle

$\sigma:\sum_{i\in I}\sigma_{i}\otimes\bar{\pi}_{i}^{*}\otimes l_{i}.x_{i}$

to the $0$-cycle:

$\operatorname{rk}(\sigma)=\sum_{i\in I}\operatorname{rk}(\sigma_{i}).x_{i}.$

######  3.1.14.

We can be more precise about the relation between Chow and Chow-Witt groups,
using the definitions of 2.4.9. Indeed, 2.5.8 implies that the following
diagram is commutative:

$\textstyle{\operatorname{K}^{M}_{*}\big{(}\kappa(X)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H_{\eta}}$$\scriptstyle{\operatorname{div}_{X}}$$\textstyle{\operatorname{K}^{MW}_{*}\big{(}\kappa(X),\mathcal{L}_{\eta}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\eta}}$$\scriptstyle{\operatorname{\widetilde{div}}_{X}}$$\textstyle{\operatorname{K}^{M}_{*}\big{(}\kappa(X)\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{div}_{X}}$$\textstyle{Z^{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{x}H_{x}}$$\textstyle{\widetilde{\mathrm{C}}{}^{1}(X,\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{x}f_{x}}$$\textstyle{Z^{1}(X)}$

where $Z^{1}(X)$ denotes the group of codimension $1$ algebraic cycles (_i.e._
the Weil divisors) of $X$. Taking cokernel, one gets well defined maps:

$\operatorname{CH}^{1}(X)\xrightarrow{\operatorname{H}}\widetilde{\mathrm{CH}}{}^{1}(X,\mathcal{L})\xrightarrow{\operatorname{F}}\operatorname{CH}^{1}(X)$

whose composite is multiplication by $2$. This relation obviously extends to
We still call them respectively the hyperbolic and forgetful maps.

###  3.2. Homotopy invariance over a field

Our next result was first proved for Milnor K-theory by Milnor: see [Mil70,
Th. 2.3] (and also [BT73, 5.2]). It was generalized by Morel in [Mor12, Th.
3.24].

######  Theorem 3.2.1 (Morel).

Let $k$ be an arbitrary field, $X=\mathbb{A}^{1}_{k}$ with function field
$k(t)=\kappa(X)$. Let $\varphi:k\rightarrow k(t)$ be the obvious inclusion.

Then the quadratic divisor class map fits of $X$ into the following sequence

$0\rightarrow\operatorname{K}^{MW}_{q}(k)\xrightarrow{\ \varphi_{*}\
}\operatorname{K}^{MW}_{q}\big{(}k(t)\big{)}\xrightarrow{\
\operatorname{\widetilde{div}}_{X}\ }\bigoplus_{x\in
X^{(1)}}\operatorname{K}^{MW}_{q-1}(\kappa_{x},\omega_{x/X})\rightarrow 0$

which is split exact.

In particular,

$\widetilde{\mathrm{CH}}{}^{p}(\mathbb{A}^{1}_{k})_{q}=\begin{cases}\operatorname{K}^{MW}_{q}(k)&p=0\\\
0&p=1.\end{cases}$

Note that a splitting is easy to get: considering the valuation $v=\deg$ on
$k(t)$, the specialization map $s_{v}^{t}$ (2.5.3) gives a splitting. More
generally, any valuation $v$ on $k(t)$ trivial on $k$ with uniformizing
parameter $\pi$ will give a splitting $s_{v}^{\pi}$.

The proof of this proposition uses the same trick as in Milnor’s proof, and
argue inductively on the degree in $t$. The principle is to filter
$\operatorname{K}^{MW}_{*}(k(t))$ by the subring $L_{d}$ generated by $\eta$
and symbols of the form $[P(t)]$ where $P(t)$ is a polynomial of degree less
than $t$. We can then argue inductively on the $\mathbb{Z}$-graded
$\operatorname{K}^{MW}_{*}(k)$-rings $L_{d}$ using an explicit presentation of
the $\mathbb{Z}$-graded $\operatorname{K}^{MW}_{*}(k)$-module $L_{d}/L_{d-1}$.

As an example, the reader is encouraged to work out for himself the case of
$L_{1}$. The hint is to use the (obvious!) exact sequence:

$0\rightarrow k^{\times}\xrightarrow{\ \varphi_{*}\
}k(t)^{\times}\xrightarrow{\sum_{x}v_{x}}Z^{1}(\mathbb{A}^{1}_{k})\rightarrow
0$

Given that invertible sheaves on $\mathbb{A}^{1}_{k}$ are trivializable, one
inmmediately deduces the twisted version of the previous theorem.

######  Corollary 3.2.2.

Consider the notation of the previous theorem, and let $\mathcal{L}$ be an
invertible sheaf on $\mathbb{A}^{1}_{k}$. Then the following sequence of
abelian groups is exact:

$0\rightarrow\operatorname{K}^{MW}_{q}(k,\mathcal{L}_{0})\xrightarrow{\
\varphi_{*}\
}\operatorname{K}^{MW}_{q}\big{(}k(t),\mathcal{L}_{k(t)}\big{)}\xrightarrow{\
\operatorname{\widetilde{div}}_{X}\ }\bigoplus_{x\in
X^{(1)}}\operatorname{K}^{MW}_{q-1}(\kappa_{x},\omega_{x/X}\otimes\mathcal{L}_{x})\rightarrow
0$

where $\mathcal{L}_{0}$ (resp. $\mathcal{L}_{x}$) is the fiber of
$\mathcal{L}$ over the point $0$ (resp. a closed point $x$). In particular,
$\mathcal{L}_{0}=\mathcal{L}\otimes_{k[t]}k$ and the morphism $\varphi_{*}$ is
defined on twist by the canonical isomorphism:

$\mathcal{L}\rightarrow\mathcal{L}_{0}\otimes_{k}k[t],l\mapsto(l\otimes_{k[t]}1)\otimes_{k}1.$

###  3.3. Localization exact sequences

In this section, we will illustrate the usefulness of considering the
$\mathbb{G}_{m}$-graduation of the Rost-Schmid complex (3.1.5). The aim is to
compute the Chow-Witt groups of the projective line.

######  3.3.1.

Let again $X$ be a normal connected $1$-dimensional scheme, $\mathcal{L}$ an
invertible sheaf on $X$.

Consider in addition a finite subset $Z\subset X$ of closed points of $X$,
seen as reduced closed subscheme, $i:Z\rightarrow X$. Let
$\omega_{Z/X}=(\mathcal{I}(Z)/\mathcal{I}(Z)^{2})^{\vee}$ be the normal sheaf
of $i$, where $\mathcal{I}(Z)\subset\mathcal{O}_{X}$ is the ideal sheaf. Let
$U=X-Z$, and $j:U\rightarrow X$ the open immersion.

There is an obvious split epimorphism:

$j^{*}:\widetilde{\mathrm{C}}{}^{1}(X,\mathcal{L})_{q}\rightarrow\widetilde{\mathrm{C}}{}^{1}(U,\mathcal{L})_{q}$

whose kernel is the finite sum:

$\widetilde{\mathrm{CH}}{}^{0}(Z,\omega_{Z/X}\otimes\mathcal{L}_{Z})_{q}:=\oplus_{z\in
Z}\operatorname{K}^{MW}_{q}\big{(}\kappa_{z},\omega_{z/X}\otimes\mathcal{L}_{z}\big{)}.$

Remark that this notation fits in with the previous considerations as for any
point $z\in Z$, we have a _canonical_ isomorphism (this can be checked
directly, or see (4.1.7.a)) of invertible $\kappa_{z}$-vector spaces:

$\omega_{z/X}\otimes\mathcal{L}_{z}\simeq(\omega_{z/Z}\otimes\omega_{Z/X}|_{z})\otimes\mathcal{L}_{z}\simeq\omega_{z/Z}\otimes(\omega_{Z/X}\otimes\mathcal{L}_{Z})_{x}.$

Assembling all this, we get a commutative diagram whose lines are exact:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathrm{C}}{}^{0}(X,\mathcal{L})_{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\widetilde{div}}_{X}}$$\textstyle{\widetilde{\mathrm{C}}{}^{0}(U,\mathcal{L})_{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\widetilde{div}}_{U}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathrm{CH}}{}^{0}(Z,\omega_{Z/X}\otimes\mathcal{L}_{Z})_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{\widetilde{\mathrm{C}}{}^{1}(X,\mathcal{L})_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{\widetilde{\mathrm{C}}{}^{1}(U,\mathcal{L})_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

######  Definition 3.3.2.

Consider the previous notation. The exact sequence obtained by applying the
snake lemma to the preceding commutative diagram:

$\displaystyle 0\rightarrow\widetilde{\mathrm{CH}}{}^{0}(X,\mathcal{L})_{q+1}$
$\displaystyle\xrightarrow{j^{*}}\widetilde{\mathrm{CH}}{}^{0}(U,\mathcal{L}_{U})_{q+1}\xrightarrow{\partial_{Z/X}}\widetilde{\mathrm{CH}}{}^{0}(Z,\omega_{Z/X}\otimes\mathcal{L}_{Z})_{q}$
$\displaystyle\xrightarrow{\ i_{*}\
}\widetilde{\mathrm{CH}}{}^{1}(X,\mathcal{L})_{q}\xrightarrow{j^{*}}\widetilde{\mathrm{CH}}{}^{1}(U,\mathcal{L}_{U})_{q}\rightarrow
0$

is called the _localization exact sequence_ associated with $i$.

The connecting map $\partial_{Z/X}$ is called the _residue map_ associated
with $i$. It is induced by the following restriction and corestriction of the
quadratic divisor class map $d_{X}$:

$\sum_{z\in
Z}\partial_{z}:\operatorname{K}^{MW}_{q+1}\big{(}\kappa(X)\big{)}\longrightarrow\oplus_{z\in
Z}\operatorname{K}^{MW}_{q}\big{(}\kappa_{z},\omega_{z}\otimes\mathcal{L}_{z}\big{)}.$

###  3.4. Twisted Chow-Witt groups of the projective line

######  3.4.1.

We now illustrate the usage of the localization exact sequence devined in the
previous section. Let $k$ be an arbitrary field.

Let $\mathbb{P}^{1}_{k}=\operatorname{Proj}(k[x,y])$ be the projective line,
$\infty=[1:0]$ be point at infinity with complementary open subscheme
$\mathbb{A}^{1}_{k}=\operatorname{Spec}(k[x])$. We let
$i^{\infty}:\\{\infty\\}\rightarrow\mathbb{P}^{1}_{k}$ be the natural closed
immersion, and $j:\mathbb{A}^{1}_{k}\rightarrow\mathbb{P}^{1}_{k}$ the
complementary open immersion. We fix a line bundle $\mathcal{L}$ over
$\mathbb{P}^{1}_{k}$, which is therefore determined up to isomorphism by its
degree, $\deg(\mathcal{L})$. We let $\mathcal{L}^{\prime}$ be the restriction
of $\mathcal{L}$ to $\mathbb{A}^{1}_{k}$.

Then the localization exact sequence of $i^{\infty}$ together with Morel’s
homotopy invariance theorem (see 3.2.2) gives us the following exact sequence:

$\displaystyle
0\rightarrow\widetilde{\mathrm{CH}}{}^{0}(\mathbb{P}^{1}_{k},\mathcal{L})_{q+1}$
$\displaystyle\xrightarrow{j^{*}}\operatorname{K}^{MW}_{q+1}(k,\mathcal{L}_{0})\xrightarrow{\partial_{Z/X}}\operatorname{K}^{MW}_{q}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})$
$\displaystyle\xrightarrow{\ i^{\infty}_{*}\
}\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\mathcal{L})_{q}\rightarrow
0$

where we have denoted by $\mathcal{L}_{0}$ the restriction of
$\mathcal{L}^{\prime}$ to the point $0$ in $\mathbb{A}^{1}_{k}$.

The main problem is to determine the residue map $\partial_{Z/X}$.

######  Lemma 3.4.2.

Consider the above assumptions and notations.

Then if $\deg(\mathcal{L})$ is even, $\partial_{Z/X}=0$. If
$d=\deg(\mathcal{L})$ is odd, after choosing an isomorphism
$\mathcal{L}\simeq\mathcal{O}(d)$, and using the isomorphism
$\omega_{\infty}\simeq k$ given by the uniformizing parameter $1/x$, we get
the following commutative diagram:

$\textstyle{\operatorname{K}^{MW}_{q+1}(k,\mathcal{L}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{Z/X}}$$\scriptstyle{\sim}$$\textstyle{\operatorname{K}^{MW}_{q}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{K}^{MW}_{q+1}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{\eta}}$$\textstyle{\operatorname{K}^{MW}_{q}(k).}$

###### Proof.

One reduces to the case $\mathcal{L}=\mathcal{O}(d)$. We consider
$U=U_{\infty}$ (resp. $U_{0}$) the open complement of $\infty$ (resp. $0$) in
$\mathbb{P}^{1}_{k}$, so that $U_{\infty}=\operatorname{Spec}(k[x])$ and
$U_{0}=\operatorname{Spec}(k[y])$. The glueing map $U_{0}\cap
U_{\infty}\rightarrow U_{\infty}\cap U_{0}$ is given by mapping $x$ to
$y^{-1}$. Then the line bundle $\mathcal{L}=\mathcal{O}(d)$ is given on
$U_{\infty}$ (resp. $U_{0}$) by a free module
$\mathcal{L}^{\prime}_{\infty}=k[x].u$ (resp.
$\mathcal{L}^{\prime}_{0}=k[y].v$) with a glueing map $u\mapsto y^{-d}.v$.

Note therefore that one has prefered chosen isomorphisms:
$\mathcal{L}_{0}\simeq_{u}k$ and $\mathcal{L}_{\infty}\simeq_{v}k$. Therefore
we deduce a canonical map

$\operatorname{K}^{MW}_{q+1}(k)\simeq_{u^{-1}_{*}}\operatorname{K}^{MW}_{q+1}(k,\mathcal{L}_{0})\xrightarrow{\partial_{Z/X}}\operatorname{K}^{MW}_{q}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})\simeq_{y_{*}\otimes
v_{*}}\operatorname{K}^{MW}_{q}(k)$

denoted by $\partial^{\prime}_{Z/X}$.

We compute the image of $\sigma\in\operatorname{K}^{MW}_{q}(k)$ under
$\partial^{\prime}_{Z/X}$. First, $u^{-1}_{*}(\sigma)=\sigma\otimes u$. Then
we need to use the map $\varphi_{*}$ of 3.2.2, which sends the latter to

$\sigma\otimes(u\otimes
1)\in\operatorname{K}^{MW}_{q}{k(t),\mathcal{L}_{0}\otimes_{k}k(t)}.$

In order to compute its residue at $\infty$, one needs to write it as an
element of
$\operatorname{K}^{MW}_{q}\big{(}k(t),\mathcal{L}_{\infty}\otimes_{k}k(t)\big{)}$.
Therefore, one uses the above change of variables:

$\sigma\otimes(y^{-d}v\otimes 1)=(\langle y^{-d}\rangle\sigma)\otimes(v\otimes
1).$

Now if $d$ is even, $\langle y^{-d}\rangle=1$ and we get:
$\partial_{\infty}^{y}(\langle y^{-d}\rangle\sigma)=0$ as $\sigma$ comes from
$\operatorname{K}^{MW}_{*}(k)$. Thus $\partial_{X/Z}(\sigma)=0$.

If on the contrary, $d$ is odd, $\langle y^{-d}\rangle=\langle y\rangle$.
Therefore

$\partial_{\infty}^{y}(\langle
y^{-d}\rangle\sigma)=\partial_{\infty}^{y}(\langle
y\rangle\sigma)=\eta.\sigma$

and one deduces that $\partial^{\prime}_{Z/X}(\sigma)=\eta.\sigma$ as
expected. ∎

######  3.4.3.

Let $d=\deg(\mathcal{L})$. The lemma and the localization exact sequence gives
the following possibilities:

  1. (1)

if $d$ is even, one gets isomorphisms:

$\displaystyle j^{*}$
$\displaystyle:\widetilde{\mathrm{CH}}{}^{0}(\mathbb{P}^{1}_{k},\mathcal{L})_{q}\xrightarrow{\
\sim\
}\widetilde{\mathrm{CH}}{}^{0}(\mathbb{A}^{1}_{k},\mathcal{L})_{q}\simeq\operatorname{K}^{MW}_{q}(k,\mathcal{L}_{0})$
$\displaystyle i^{\infty}_{*}$
$\displaystyle:\operatorname{K}^{MW}_{q}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})\xrightarrow{\
\sim\ }\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\mathcal{L})_{q}.$

  2. (2)

If $d$ is odd, and after the choices indicated in the above lemma, we get an
exact sequence:

$\displaystyle
0\rightarrow\widetilde{\mathrm{CH}}{}^{0}(\mathbb{P}^{1}_{k},\mathcal{L})_{q+1}$
$\displaystyle\xrightarrow{j^{*}}\operatorname{K}^{MW}_{q+1}(k)\xrightarrow{\gamma_{\eta}}\operatorname{K}^{MW}_{q}(k)\xrightarrow{\
i^{\infty}_{*}\
}\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\mathcal{L})_{q}\rightarrow
0$

Recall that the cokernel of $\gamma_{\eta}$ is $\operatorname{K}^{M}_{q}(K)$
(see 2.2.9), and its kernel is $2.\operatorname{K}^{M}_{q}(K)$, that is the
$q$-th graded part of the ideal generated by $2$ in the ring
$\operatorname{K}^{M}_{*}(K)$ (see 2.3.10). To summarize, we have obtained the
following computation of (graded) Chow-Witt groups, first proved by Jean Fasel
for a perfect base field of characteristic not $2$ (see [Fas13]):

######  Theorem 3.4.4.

Consider the above assumption: $k$ is an arbitrary field, $\mathcal{L}$ an
invertible sheaf over $\mathbb{P}^{1}_{k}$ of degree $d$. Then

$\displaystyle\widetilde{\mathrm{CH}}{}^{0}(\mathbb{P}^{1}_{k},\mathcal{L})_{q}$
$\displaystyle\simeq\begin{cases}\operatorname{K}^{MW}_{q}(k,\mathcal{L}_{0})&\qquad\quad
d\text{ even}\\\ 2.\operatorname{K}^{M}_{q}(K)&\qquad\quad d\text{
odd}\end{cases}$
$\displaystyle\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\mathcal{L})_{q}$
$\displaystyle\simeq\begin{cases}\operatorname{K}^{MW}_{q}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})&d\text{
even}\\\ \operatorname{K}^{M}_{q}(k)&d\text{ odd}\end{cases}$

Recall finally from 2.2.19 that:
$\operatorname{Ker}(\eta)=2_{\epsilon}.\operatorname{K}^{M}_{q}(K)$ when
$\operatorname{char}(k)\neq 2$.

Let us draw the picture for Chow-Witt groups:

$\widetilde{\mathrm{CH}}{}^{p}(\mathbb{P}^{1}_{k},\mathcal{L})=\begin{cases}\operatorname{GW}(k,\mathcal{L}_{0})&p=0,\deg(\mathcal{L})=2r,\\\
\operatorname{GW}(k,\omega_{\infty}\otimes\mathcal{L}_{\infty})&p=1,\deg(\mathcal{L})=2r,\\\
\mathbb{Z}&p=0,1,\deg(\mathcal{L})=2r+1.\end{cases}$

In particular, these groups depend on the twist $\mathcal{L}$ when
$\operatorname{GW}(k)$ is non-trivial !

##  4\. Transfers

###  4.1. Cotangent complexes and canonical sheaves

Recall for convenience (and completeness) the following definition.

######  Definition 4.1.1.

Let $f:X\rightarrow S$ be a morphism of schemes.

  1. (1)

$f$ is _smoothable_ if there exists a factorization

$f:X\xrightarrow{i}P\xrightarrow{p}S$

such that $p$ is smooth and $i$ is a closed immersion.

  2. (2)

$f$ is a _complete intersection_ if there exists a factorization

$f:X\xrightarrow{i}P\xrightarrow{p}S$

such that $p$ is smooth and $i$ is a regular closed immersion.

  3. (3)

$f$ is a _local complete intersection_ if any point $x\in X$ admits an open
neighborhood $V$ such that the restriction $f|_{V}$ is a complete
intersection.
Following a classical abuse, we will simply say that $f$ is lci.

######  Remark 4.1.2.

  1. (1)

The second definition first appeared in [BGI71, VIII, 1.1] (see also [Stack,
37.59.2], especially in the non-noetherian case).

  2. (2)

A morphism $f$ is a complete intersection if and only if it is smoothable and
lci (see [BGI71, VIII, 1.2]).

######  4.1.3.

Cotangent Complexes. For a scheme $X$, we let
$\operatorname{\mathrm{D}}(\mathcal{O}_{X})$ be the derived category of
$\mathcal{O}_{X}$-modules.

Let $f:X\rightarrow S$ be a morphism of schemes. Recall that one can associate
to $f$ its cotangent complex $\mathcal{L}_{X/S}$ (see [Ill71, III, 1.2.3]), a
canonically defined object of $\operatorname{\mathrm{D}}(\mathcal{O}_{X})$ —
it is the derived functor of the Kälher differential functor evaluated at
$\mathcal{O}_{X}/f^{-1}\mathcal{O}_{S}$.

If $f$ is a complete intersection, choosing a factorization as in 4.1.1(2),
one can explicitly compute its cotangent complex $\mathcal{L}_{X/S}$. It is
quasi-isomorphic to a complex concentrated in two degrees

$\mathcal{C}_{X/P}\rightarrow\Omega_{P/S}|_{X}$

where $\mathcal{C}_{X/P}=\mathcal{I}_{i}/\mathcal{I}_{i}^{2}$ is the _conormal
sheaf_ associated with $i$, placed in homological degree $+1$, and
$\Omega_{P/S}$ is the _cotangent sheaf_ of $P/S$ (the conormal sheaf of the
diagonal of $P/S$) placed in degree $0$ (see [Ill71, VIII, 3.2.7]). This
obviously implies that if $f:X\rightarrow S$ is only assumed to be lci, then
its cotangent complex is Zariski locally in $X$ quasi-isomorphic to a complex
concentrated in degree $0$ and $1$ and whose terms are free.343434One says
that $\mathcal{L}_{X/Z}$ has perfect homological amplitude in $[0,1]$. In
particular, $\mathcal{L}_{X/S}$ is perfect.353535A complex $\mathcal{K}$ of
$\mathcal{O}_{X}$-module is perfect if any point of $X$ admits an open
neighborhood $U$ such that $\mathcal{K}|_{U}$ is quasi-isomorphic to a bounded
complex $\mathcal{L}$ such that for all integers $n$ the coherent sheaf
$\mathcal{L}^{n}$ is a direct factor of a finite free
$\mathcal{O}_{U}$-module. See [Stack, Def. 20.44.1/08C4].

The interest of the cotangent complex is to be compatible with composition in
the following sense. Consider a commutative diagram

(4.1.3.a)
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\Delta}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{S}$

of morphisms of schemes. Then one has a canonical homotopy exact sequence of
$\operatorname{\mathrm{D}}(\mathcal{O}_{X})$:

(4.1.3.b)
$(f^{*}\mathcal{L}_{Y/S})\rightarrow\mathcal{L}_{X/S}\rightarrow\mathcal{L}_{X/Y}.$

######  4.1.4.

Recall from [Del87, Ex. 4.13] that one associates to a perfect complex $C$ of
$\mathcal{O}_{X}$-modules its _rank_ $\operatorname{rk}(C)$ which is a locally
constant function $X\rightarrow\mathbb{Z}$ and its determinant $\det(C)$ which
is a well-defined invertible sheaf over $X$. 363636The couple
$(\det,\operatorname{rk})$ is actually the left Kan extension, as an
$\infty$-functor, from the $\infty$-category of perfect complexes to the
$\infty$-groupoid of graded line bundles,
$\operatorname{\mathscr{P}erf}(X)\rightarrow\operatorname{\mathscr{P}ic^{\mathbb{Z}}}(X)$
of the functor sending a locally free $\mathcal{O}_{X}$-module to its rank and
its maximal exterior power (see [LO21, §5]). It is also obtained by
restriction of the canonical functor from the Thomason-Trobaugh $K$-theory
space $K(X)$ to $\operatorname{\mathscr{P}ic^{\mathbb{Z}}}(X)$ (see [BS17] in
the affine case).

######  Definition 4.1.5.

Let $f:X\rightarrow S$ be a morphism whose cotangent complex is perfect (eg:
lci). One associates to $f$ its _canonical sheaf_ :

$\omega_{X/S}=\det(\mathcal{L}_{X/S}).$

We will also say that $f$ is of (virtual) relative dimension
$d=\operatorname{rk}(\mathcal{L}_{X/S})$.

When $X/S$ is the spectrum of a ring extension $B/A$, the canonical sheaf
$\omega_{X/S}$ is determined by its global sections. We will denote by
$\omega_{B/A}$ the $B$-module of its global sections, and call it the
_canonical module_ associated with $B/A$.

######  Example 4.1.6.

  1. (1)

If $f:X\rightarrow S$ is smooth, the above definition coincides with the
classical definition of the canonical sheaf: the cotangent sheaf of $f$ is
locally free $\Omega_{X/S}$, and $\omega_{X/S}$ is the maximal exterior power
of $\Omega_{X/S}$ as a $\mathcal{O}_{X}$-module.

Note that in particular that when $f$ is étale, one canonical has:
$\omega_{X/S}=\mathcal{O}_{X}$. This is really an identity, and not just an
isomorphism.

  2. (2)

if $f=i:Z\rightarrow X$ is a regular closed immersion of pure codimension $1$,
then $\omega_{Z/S}=\mathcal{C}_{Z/X}^{\vee}$, the dual of the conormal sheaf.

  3. (3)

A morphism $f:X\rightarrow S$ of schemes which is flat, of finite presentation
and lci is called _syntomic_ after Fontaine and Messing. Syntomic morphisms
are stable under base change and composition ([Stack, 29.30.3, 29.30.4]). In
this case the virtual relative dimension of $X/S$ equals the dimension of
fibers functions, which to a point $s\in S$ associates the dimension of
$X_{s}=f^{-1}(\\{s\\})$. This can be seen by reducing to the case where $S$ is
the spectrum of a field as the cotangent complex of $f$ is stable under
(naive) pullbacks.

######  4.1.7.

Let us consider a commutative diagram (4.1.3.a) such that the cotangent
complexes of all three morphisms are perfect (for example, $f$ and $g$ are
lci). Then the above homotopy exact sequence translates into a canonical
isomorphism of invertible sheaves over $X$:

(4.1.7.a)
$\psi_{\Delta}:\omega_{X/S}\simeq\omega_{X/Y}\otimes(f^{*}\omega_{Y/S})$

######  Remark 4.1.8.

It is also useful to consider commutative squares:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{q}$$\scriptstyle{\Theta}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{S.}$

Dividing the square into two commutative triangles, and applying the preceding
isomorphism for both triangles, one gets a canonical isomorphism:

$\psi_{\Theta}:\omega_{Y/X}\otimes(g^{*}\omega_{X/S})\simeq\omega_{Y/T}\otimes(q^{*}\omega_{T/S}).$

When the preceding square is affine corresponding to a commutative square of
rings:

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

one gets the following simpler form, an isomorphism of invertible $D$-modules:

$\psi_{\Theta}:\omega_{D/C}\otimes_{C}\omega_{C/A}\simeq\omega_{D/B}\otimes_{B}\omega_{B/A}$

where the tensor product on the left (resp. right) is taken with respect to
the induced structure of $C$-module on $\omega_{D/C}$ (resp. $B$-module on
$\omega_{D/B}$).

######  Example 4.1.9.

Let us consider a finitely generated lci $A$-algebra $B$. We assume that there
exists a smooth $A$-algebra $R$ and a regular ideal $I\subset R$ such that
$B\simeq R/I$ as an $A$-algebra so that we get a surjection
$\varphi:B\rightarrow R$.373737By Noether normalization lemma, this will
automatically be the case if $A$ is a field; moreover we can choose $R$ to be
a polynomial $k$-algebra.

Assume $\operatorname{Spec}(A)$ and $\operatorname{Spec}(R)$ are irreducible
and let $n$ be the rank of the $A$-algebra $R$, $m$ be the height of $I$. Then
one can compute the canonical module of $B/A$ as:

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

Indeed, as $I$ is regular, $I/I^{2}$ is a locally free $B$-module of constant
rank $m$.

In general, we have an exact sequence of $B$-modules:

(4.1.9.a) $0\rightarrow N\rightarrow
I/I^{2}\xrightarrow{\psi}\Omega_{R/A}\otimes_{R}B\xrightarrow{\varphi_{*}}\Omega_{B/A}\rightarrow
0$

where $N=\operatorname{Ker}(\psi)$, $\psi$ is induced by the composition

$I\hookrightarrow
R\xrightarrow{d_{R}}\Omega_{R/A}\rightarrow\Omega_{R/A}\otimes_{R}B$

and the last map is induced by $\varphi:R\rightarrow B$. As recalled in 4.1.3,
the cotangent complex $\mathcal{L}_{B/A}$, that we view as a complex of
$B$-modules, is concentrated in homological degree $[0,1]$ and one deduces
from the above exact sequence and the homotopy exact sequence (4.1.3.b) the
following isomorphisms:

$\displaystyle H_{0}(\mathcal{L}_{B/A})$ $\displaystyle\simeq\Omega_{B/A},$
$\displaystyle H_{1}(\mathcal{L}_{B/A})$ $\displaystyle\simeq N.$

When $B/A$ is étale, one gets $\Omega_{B/A}=N=0$, and $n=m$. So
$\omega_{B/A}=B$ (this is really an identity), and the isomorphism
$\Theta:B\simeq\Lambda^{n}(I/I^{2})^{\vee}\otimes_{B}\Lambda^{n}(\Omega_{R/A}\otimes_{P}B)$
is obtained by transposing the isomorphism $\psi$.

######  Example 4.1.10.

We consider a particular case of the preceding example, that of a finite field
extension $L/K$. We can choose a set of generators
$(\alpha_{1},...,\alpha_{n})$, $L=K[\alpha_{1},...,\alpha_{n}]$. If we
consider the polynomial $K$-algebra $R=K[t_{1},...,t_{n}]$, then one can write
$L=R/I$, and $I=(f_{1},...,f_{n})$ where $f_{i}$ is a polynomial in the
variables $t_{1},...,t_{i}$, monogenic in $t_{i}$, which is a lift of the
characteristic polynomial of the algebraic element $\alpha_{i}$ of
$L/K[\alpha_{1},...,\alpha_{i-1}]$. Thus, $I$ is regular. Then we get from the
previous example a canonical isomorphism:

(4.1.10.a)
$\Theta:\omega_{L/K}\simeq\Lambda^{n}_{L}(I/I^{2})^{\vee}\otimes_{R}\Omega^{n}_{R/K}.$

We then get an explicit basis of the invertible $B$-module $\omega_{B/A}$,
given by the element:

(4.1.10.b)
$(\bar{f}_{1}\wedge\ldots\wedge\bar{f}_{n})^{*}\otimes(dt_{1}\wedge\ldots
dt_{n}).$

If $L/K$ is separable, as explained in the end of the previous example,
$\Omega^{1}_{L/K}=0$, so $\omega_{L/K}=L$. According to the description of
$\psi$, one obtains that the element (4.1.10.b) goes under $\Theta^{-1}$ to
the unit:

$\big{(}\pi^{\prime}_{1}(\alpha_{1})\pi^{\prime}_{2}(\alpha_{1},\alpha_{2})\ldots\pi^{\prime}_{n}(\alpha_{1},\ldots,\alpha_{n})\big{)}^{-1}\in
L^{\times}.$

We have to take the inverse of the obvious element as $\Theta$ is obtained
after transposition, as seen in the end of the previous example.

Let us assume on the contrary that $L/K$ is totally inseparable. Let $p>0$ be
the characteristic of $K$. Then $\alpha_{i}=(a_{i})^{1/q_{i}}$, $a_{i}\in
K-K^{p}$. Moreover, in the sequence (4.1.9.a) with $B/A=L/K$, one obtains that
$\psi=0$. In other words, one gets isomorphisms:

$\displaystyle\varphi_{*}:\Omega_{R/K}\otimes_{R}L\xrightarrow{\sim}\Omega_{L/K},$
$\displaystyle N\simeq I/I^{2}.$

In particular, $(d\alpha_{1},...,d\alpha_{n})$, which is the image of
$(dt_{1},...,dt_{n})$ by the isomorphism $\varphi_{*}$, is an $L$-basis of
$\Omega_{L/K}$, which can be identified to $H_{0}(\mathcal{L}_{L/K})$.
Similarly, $N\simeq I/I^{2}$ can be identified with
$H_{1}(\mathcal{L}_{L/K})$, and an $L$-basis is given by
$(\bar{f}_{1},...,\bar{f}_{n})$ — each $\bar{f}_{i}$ goes to zero in
$\Omega_{R/K}\otimes_{R}L$.

######  Remark 4.1.11.

In the case of a totally inseparable extension $L/K$, $\mathbb{F}_{p}\subset
K$, one defines the _imperfection module_ $\Gamma_{L/K}$ of $L/K$ by the
following exact sequence:

$0\rightarrow\Gamma_{L/K}\rightarrow\Omega_{K/\mathbb{F}_{p}}\otimes_{K}L\rightarrow\Omega_{L/\mathbb{F}_{p}}\rightarrow\Omega_{K/\mathbb{F}_{p}}\rightarrow
0.$

One deduces that $H_{1}(\mathcal{L}_{L/K})\simeq\Gamma_{L/K}$. In particular,
with the notations of the previous paragraph, the imperfection module
$\Gamma_{L/K}$ is an $n$-dimensional $L$-vector space which is isomorphic to
$I/I^{2}$.

###  4.2. The quadratic degree map

######  4.2.1.

We will now come back to 3.4.4 and give its fundamental application to build
transfers on Milnor-Witt K-theory.

Let $k$ be an arbitrary field, and $\omega=\omega_{\mathbb{P}^{1}_{k}/k}$ be
the canonical sheaf on $\mathbb{P}^{1}_{k}$ (4.1.5), and let $\infty$ (resp.
$\eta$) be the point at infinity (resp. generic point) of
$\mathbb{P}^{1}_{k}$. We first rewrite the quadratic divisor class map in
_homological_ conventions (see [Dég23] more generally). Consider a point
$x\in\mathbb{P}^{1}_{k}$ with residue field $\kappa_{x}$. Note that
$\kappa_{x}/k$ is not necessarily separable so the canonical sheaf
$\omega_{\kappa_{x}/k}$ can be non-trivial. Nevertheless, the commutative
diagram

$\textstyle{\operatorname{Spec}(\kappa_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{\mathbb{P}^{1}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Spec}(k)}$

gives a canonical isomorphism
$\psi^{x}:\omega_{L/k}\simeq\omega_{x/X}\otimes\omega|_{x}$ — see (4.1.7.a).

In particular, the quadratic divisor class map for $\mathbb{P}^{1}_{k}/k$ in
$\mathbb{G}_{m}$-degree $q\in\mathbb{Z}$ and with twists $\omega$ can be
rewritten as:

$\operatorname{\widetilde{div}}:\operatorname{K}^{MW}_{q+1}\big{(}k(t),\omega_{k(t)/k}\big{)}\longrightarrow\bigoplus_{x\in\mathbb{P}^{1}_{k,(0)}}\operatorname{K}^{MW}_{q}(\kappa_{x},\omega_{\kappa_{x}/k})=:\tilde{C}_{0}(\mathbb{P}^{1}_{k})_{q}.$

Recall that $\operatorname{\widetilde{div}}$ is the sum of the residue maps
$\partial_{x}:\operatorname{K}^{MW}_{q+1}\big{(}k(t),\omega_{k(t)/k}\big{)}\rightarrow\operatorname{K}^{MW}_{q}(\kappa_{x},\omega_{\kappa_{x}/k})$
for $x$ a closed point in $\mathbb{P}^{1}_{k}$, corresponding to a valuation
$v_{x}$ on $k(t)$ with residue field $\kappa_{x}$.

The cokernel of $\operatorname{\widetilde{div}}$ equals the Chow-Witt group
$\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\omega)$ and, as $\omega$ is
even, 3.4.4 and paragraph 3.4.3 tells us that the pushforward map

$i^{\infty}_{*}:\operatorname{K}^{MW}_{q}(k)\rightarrow\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\omega)_{q}$

is an isomorphism. Let us introduce the following definition.383838This is a
the mother case of the degree map on Chow-Witt groups. See [Dég23].

######  Definition 4.2.2.

Using the above notation, we denote by

$\operatorname{\widetilde{deg}}_{q}:\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\omega)_{q}\rightarrow\operatorname{K}^{MW}_{q}(k)$

the inverse of the isomorphism $i^{\infty}_{*}$ and call it the _quadratic
degree map_ in $\mathbb{G}_{m}$-degree $q$ (associated with
$\mathbb{P}^{1}_{k}$).

In degree $q=0$, we therefore get a map:

$\operatorname{\widetilde{deg}}:\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\omega)\rightarrow\operatorname{GW}(k).$

Following Bass and Tate ([BT73, I.5.4]) and Morel ([Mor12, §4.2]), we can be
more precise about this notion of quadratic degree.

######  Proposition 4.2.3.

Consider the above assumptions and notation. Then there exists a _unique_
family of maps

$\operatorname{Tr}^{MW}_{\kappa_{x}/k}:\operatorname{K}^{MW}_{q}(\kappa_{x},\omega_{\kappa_{x}/k})\rightarrow\operatorname{K}^{MW}_{q}(k),x\in\mathbb{P}^{1}_{k,(0)}$

which fits into the following commutative diagram

$\textstyle{\operatorname{K}^{MW}_{q}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\infty}_{*}}$$\textstyle{\operatorname{K}^{MW}_{q+1}\big{(}k(t),\omega_{k(t)/k}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{\widetilde{div}}}$$\textstyle{\tilde{C}_{0}(\mathbb{P}^{1}_{k})_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{x}\operatorname{Tr}^{MW}_{\kappa_{x}/k}}$$\textstyle{\operatorname{K}^{MW}_{q}(k)}$

in such a way that composition of the horizontal maps is zero.

In particular, the quadratic degree map is defined at the level of cycles:

$\operatorname{\widetilde{deg}}_{q}:=\sum_{x\in\mathbb{P}^{1}_{k,(0)}}\operatorname{Tr}^{MW}_{\kappa_{x}/k}:\tilde{C}_{0}(\mathbb{P}^{1}_{k})_{q}\rightarrow\operatorname{K}^{MW}_{q}(k).$

The last condition in the above statement can be translated by saying that the
quadratic $0$-cycles of degree $0$ on $\mathbb{P}^{1}_{k}$ are exactly the
principal (_i.e._ rationally trivial) quadratic divisors (using the
terminology of 3.1.8).

######  Remark 4.2.4.

  1. (1)

Note that the commutative triangle corresponds to the normalization property:
$\operatorname{Tr}^{MW}_{\kappa_{\infty}/k}=\operatorname{Id}$.

  2. (2)

The formula
$\operatorname{\widetilde{deg}}\circ\operatorname{\widetilde{div}}=0$ is the
quadratic analogue of the _Weil reciprocity formula_. Given the preceding
normalization property, it can be restated as the following equation:

$\sum_{x\in(\mathbb{A}^{1}_{k})_{(0)}}\operatorname{Tr}^{MW}_{\kappa_{x}/k}\circ\partial_{x}=-\partial_{\infty}.$

######  4.2.5.

Quadratic and classical degree. As the pushforward morphism $i^{\infty}_{*}$
is compatible with the forgetful and hyperbolic maps of 3.1.14, we get by
definition the following commutative diagram:

$\textstyle{\operatorname{CH}^{1}(\mathbb{P}^{1}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{H}}$$\scriptstyle{\deg}$$\textstyle{\widetilde{\mathrm{CH}}{}^{1}(\mathbb{P}^{1}_{k},\omega)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{F}}$$\scriptstyle{\operatorname{\widetilde{deg}}}$$\textstyle{\operatorname{CH}^{1}(\mathbb{P}^{1}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\deg}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{H}}$$\textstyle{\operatorname{GW}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{rk}}$$\textstyle{\mathbb{Z}.}$

Specializing at a point $x\in\mathbb{P}^{1}_{k}$ with residue field
$\kappa_{x}$ as before, one gets the following computation:

(4.2.5.a) $\displaystyle\forall
n\in\mathbb{Z},\operatorname{Tr}^{MW}_{\kappa_{x}/k}(n.h)=(d_{x}n).h,$
(4.2.5.b)