# Milnor-Witt cycle modules over an excellent DVR

Chetan Balwe  Department of Mathematical Sciences, Indian Institute of Science
Education and Research Mohali, Knowledge City, Sector-81, Mohali 140306,
India<EMAIL_ADDRESS>,  Amit Hogadi  Department of Mathematical
Sciences, Indian Institute of Science Education and Research Pune, Dr. Homi
Bhabha Road, Pashan, Pune 411008, India<EMAIL_ADDRESS>and  Rakesh
Pawar  Department of Mathematical Sciences, Indian Institute of Science
Education and Research Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India.
<EMAIL_ADDRESS>
###### Abstract.

The definition of Milnor-Witt cycle modules in [5] can easily be adapted over
general regular base schemes. However, there are simple examples (see (2.9))
to show that Gersten complex fails to be exact for cycle modules in general if
the base is not a field. The goal of this article is to show that, for a
restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying
an extra axiom, called here as R5, the expected properties of exactness of
Gersten complex and $\mathbb{A}^{1}$-invariance hold. Moreover R5 is vacuously
satisfied when the base is a field and it is also satisfied by ${K}^{\rm MW}$
over any base. As a corollary, we obtain the strict
$\mathbb{A}^{1}$-invariance and the exactness of Gersten complex for ${K}^{\rm
MW}$ over an excellent DVR.

###### Key words and phrases:

Cycle modules; Milnor-Witt K-theory

###### 2010 Mathematics Subject Classification:

14C17; 14C35; 14F20 (Primary)

The second author acknowledges the support of India DST-DFG Project on Motivic
Algebraic Topology DST/IBCD/GERMANY/DFG/2021/1.

##  1\. Introduction

Rost’s theory of cycle modules was generalized to Milnor-Witt cycle modules
(or MW-cycle modules) by N. Feld in [5]. With minor modifications and
verifications, this definition makes sense as it is, over a regular excellent
base scheme.

Over a field, an MW-cycle module $M$ satisfies two additional properties (see
[5, 8.1, 9.1]) implied by the axioms, viz. local acylicity and homotopy
invariance. The definition of a cycle module makes it possible to write a
Gersten complex for it over any scheme equipped with a graded line bundle.
Local acyclicity is a consequence of exactness of this Gersten complex in
positive degrees. For a MW-cycle module over a positive dimensional base
scheme, the Gersten complex may not be exact (see Example 2.9) in positive
degrees. We introduce an additional axiom, called R5, to remedy this
situation. The axiom R5 says (see Definition 2.13) that for a field $F$ and
finitely many horizontal valuations (see Definition 2.1) $v_{1},..,v_{n}$, the
sum of residue morphisms

$M(F,*)\xrightarrow{\sum_{i=1}^{n}\partial_{v_{i}}}\bigoplus_{i=1}^{n}M(\kappa(v_{i}),*)$

is surjective.

Given a cycle module $M$ and a graded line bundle $\mathcal{L}$ on a scheme
$X$, one has a naturally defined Nisnevich sheaf, $M(X,\mathcal{L})$ (see
(2.6)). The following is one of the main results of this paper:

######  Theorem 1.1 (Local acyclicity).

Let $S$ be the spectrum of an excellent DVR. Let $M$ be a MW-cycle module on
$S$ satisfying the property R5 (see Definition 2.13). Let $X$ be an
essentially smooth semi-local $S$-scheme and $\mathcal{L}$ be a graded line
bundle on $X$, then the Gersten complex $C^{\bullet}(X,M,\mathcal{L})$ is
exact in degrees $\geq 1$. In particular,

$H^{i}_{Nis}(X,M(X,\mathcal{L}))=0\ \forall\ i>0.$

The requirement for this additional axiom R5 can be justified as follows.
Whenever the field $F$ appearing in the condition R5 is the function field of
a smooth $S$-scheme, R5 is exactly equivalent to the exactness of Gersten
complex for a suitable semi-local one dimensional scheme. In particular, R5 is
motivated from a property always satisfied by MW-cycle modules over a perfect
field (see Proposition 2.11 for an elementary argument). Theorem 1.1 may be
loosely paraphrased by saying that exactness of Gersten complex for general
semi-local essentially smooth schemes over a DVR follows from local acyclicity
in the one-dimensional case. However, this should be taken with the caveat
that R5 only refers to horizontal valuations and it is applied to fields $F$
which are not necessarily function fields of smooth schemes. The strategy of
the proof is motivated by [13, Theorem 3.1], which is similar to the original
argument in [15, §6], and is also used in [5, 8.1].

######  Remark 1.2.

Note that the vanishing in Theorem 1.1 implies that the Zariski and the
Nisnevich cohomology groups of $M(X,\mathcal{L})$ agree.

In the last section, we show (see Theorem 1.3) that strict
$\mathbb{A}^{1}$-invariance holds for MW-cycle modules satisfying R5.

######  Theorem 1.3.

Let $X$ be a smooth $S$-scheme. Then for all $i\geq 0$,

$H^{i}_{Nis}(X,M(X,\mathcal{L}))\simeq
H^{i}_{Nis}(\mathbb{A}^{1}_{X},M(\mathbb{A}^{1}_{X},\mathcal{L}\otimes(\omega_{\mathbb{A}^{1}_{X}/X}^{\vee},-1))).$

The above theorem is deduced from local-acyclicity and
$\mathbb{A}^{1}$-invariance in the field case using a spectral sequence
argument. The main example of a MW-cycle module is ${K}^{\rm MW}$ itself.

######  Theorem 1.4.

Let $S$ be a regular excellent scheme. Then ${K}^{\rm MW}$ is a MW-cycle
module over $S$ satisfying the property R5.

As a corollary of the above results, we get

######  Corollary 1.5.

Over an excellent DVR, the Milnor-Witt sheaves $K_{n}^{MW}$ are strictly
$\mathbb{A}^{1}$-invariant.

######  Remark 1.6.

The entire argument of (1.5) and also analogues of Theorems 1.1 and 1.3 also
work with MW-cycle modules replaced by cycle modules as defined by Rost using
Milnor K-theory. In particular the arguments in this paper also show that the
Milnor K-theory sheaves $K^{M}_{n}$, are strictly $\mathbb{A}^{1}$-invariant.
When $S$ is the spectrum of a field $k$, (not necessarily perfect), the strict
$\mathbb{A}^{1}$-invariance for $K_{n}^{M}$ is proved in [15, Proposition
8.6]. When $k$ is a perfect field, strict $\mathbb{A}^{1}$-invariance of
${K}^{\rm MW}_{n}$ was shown in [14, Theorem 5.38].

Acknowledgement: We thank Anand Sawant for useful comments and suggestions.
The third author acknowledges the support of NBHM Postdoctoral Fellowship and
IISER Pune.

##  2\. Preliminaries on cycle modules over a base

Let $S$ be a regular excellent scheme (see [17, Tag 07QT]). Let
$\mathcal{F}ield/S$ be the category of finitely generated $S$-fields. The
assumption on $S$, allows us to define, for every $F/S\in\mathcal{F}ield/S$,
the canonical invertible sheaf $\omega_{F/S}$ following [12, 6, Definition
4.3] (see Definition 5.1). We first recall the definition of Milnor-Witt cycle
modules over $S$, exactly as in [5, Section 3]. The only minor difference is
that we use graded line bundles instead of virtual vector bundles for
coefficients. This results in minor straightforward changes, for e.g. in the
definition of pull back via smooth map (see Definition 2.14(2)) we use
canonical sheaf instead of sheaf of differentials. By [7, Lemma 3.1.3], using
graded line bundles does not amount to loss of generality while at the same
time this enables us to talk about cycle modules over imperfect fields.

######  Definition 2.1.

For an $S$-field $F$, a valuation $v$ on $F$ is called an $S$-valuation if
${\rm Im}(\mathcal{O}(S)\to F)\subset\mathcal{O}_{v}$. One can show that since
$S$ is universally Japanese, this condition is equivalent to $\mathcal{O}_{v}$
being a localization of a finite type normal $S$-scheme. We say that the
valuation $v$ is horizontal if the images of ${\rm Spec\,}(F)$ and ${\rm
Spec\,}(\kappa(v))$ in $S$ are distinct.

We first consider the category $\mathfrak{F}_{S}$, defined as follows:

  1. (1)

The objects of $\mathfrak{F}_{S}$ are pairs $(F,\mathcal{L}_{F})$ where $F$ is
a finitely generated field over $S$ and $\mathcal{L}_{F}=(L,n)$ is a graded
line bundle on $F$, i.e. a pair consisting of a one-dimensional $F$-vector
space $L$ and an integer $n$ (see Appendix).

  2. (2)

A morphism $\phi:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$ is a pair
consisting of a homomorphism $E\to F$ and an isomorphism
$\mathcal{L}_{F}\cong\mathcal{L}_{E}\otimes_{E}F$. Such a morphism is said to
be a _finite extension_ if the homomorphism $E\to F$ is a finite extension.

We denote, as in [5]:

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

where $\mathcal{L}_{E}=(L,n).$ To recall the definition of Milnor-Witt cycle
modules following [1, 1.1] and [5, 3.1] we first construct the category
$\widetilde{\mathfrak{F}}_{S}$, having the same objects as $\mathfrak{F}_{S}$,
but the morphisms of which are defined via the following generators and
relations. We freely use notation from [5, 2.2].

Generators:

  * D1

$\phi_{*}:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$ for any
$\phi:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$ in $\mathfrak{F}_{S}$.

  * D2

$\phi^{*}:(F,\omega_{F/S}\otimes\mathcal{L}_{F})\to(E,\omega_{E/S}\otimes\mathcal{L}_{E})$
for any finite extension $\phi:(E,\mathcal{L}_{F})\to(F,\mathcal{L}_{F})$ in
$\mathfrak{F}_{S}$.

  * D3

$\gamma_{x}:(E,\mathcal{L}_{E})\to(E,\mathcal{L}^{\prime}_{E}\otimes\mathcal{L}_{E})$
for $x\in{K}^{\rm MW}(E,\mathcal{L}^{\prime}_{E})$.

  * D4

$\partial_{v}:(F,\mathcal{L}_{F})\to(\kappa(v),\mathcal{N}_{v}^{\vee}\otimes\mathcal{L}_{\kappa(v)})$
where

    1. (i)

$F\in\mathcal{F}ield/S$ and $v$ is a discrete $S$-valuation on $F$ over $S$.

    2. (ii)

$\mathcal{L}$ is a graded line bundle on $\mathcal{O}_{v}$, such that
$\mathcal{L}_{F}=\mathcal{L}\otimes_{\mathcal{O}_{v}}F$ and
$\mathcal{L}_{\kappa(v)}=\kappa(v)\otimes_{\mathcal{O}_{v}}\mathcal{L}$.

    3. (iii)

$\mathcal{N}_{v}:=(\mathfrak{m}_{v}/\mathfrak{m}_{v}^{2},1)$.

Relations:

  1. R0

For $x\in{K}^{\rm MW}(E,\mathcal{L}_{E})$ and $y\in{K}^{\rm
MW}(E,\mathcal{L}^{\prime}_{E})$, $\gamma_{x}\circ\gamma_{y}=\gamma_{x\cdot
y}$.

  2. R1a

$(\psi\circ\phi)_{*}=\psi_{*}\circ\phi_{*}$ for morphisms $\phi$ and $\psi$ in
$\mathfrak{F}_{S}$.

  3. R1b

$(\psi\circ\phi)^{*}=\phi^{*}\circ\psi^{*}$ for morphisms $\phi$ and $\psi$ in
$\mathfrak{F}_{S}$.

  4. R1c

Let $\phi:(F,\mathcal{L}_{F})\to(E,\mathcal{L}_{E})$ and
$\psi:(E,\mathcal{L}_{E})\to(L,\mathcal{L}_{L})$ be in $\mathfrak{F}_{S}$ such
that with $E\to F$ is finite and $E\to L$ is separable. Let $R=E\otimes_{L}F$.
For each $p\in{\rm Spec\,}R$, let
$\phi_{p}:(L,\mathcal{L}_{L})\to(R/p,\mathcal{L}_{R/p})$ and
$\psi_{p}:(F,\mathcal{L}_{F})\to(R/p,\mathcal{L}_{R/p})$ be morphisms in
$\mathfrak{F}_{S}$ induced by $\phi$ and $\psi$. Then

$\psi_{*}\circ\phi^{*}=\sum_{p\in{\rm
Spec\,}R}\phi_{p}^{*}\circ(\psi_{p})_{*}.$

  5. R2

Let $\phi:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$ be in $\mathfrak{F}_{S}$,
$x\in{K}^{\rm MW}(F,\omega_{F/S}\otimes\mathcal{L}_{F})$ and $y\in{K}^{\rm
MW}(E,\mathcal{L}^{\prime}_{E})$. Then the following relations hold:

    1. R2a

$\phi_{*}\circ\gamma_{x}=\gamma_{\phi_{*}(x)}\circ\phi_{*}$.

    2. R2b

For $\phi$ finite,
$\phi^{*}\circ\gamma_{\phi_{*}(x)}=\gamma_{x}\circ\phi^{*}$.

    3. R2c

For $\phi$ finite,
$\phi^{*}\circ\gamma_{y}\circ\phi_{*}=\gamma_{\phi^{*}(y)}$.

  6. R3a

Let $\phi:E\to F$ be a field extension and $w$ be a valuation on $F$ which
restricts to a non trivial valuation $v$ on $E$ with ramification index $e$.
Let $\mathcal{L}$ be a graded line bundle on $\mathcal{O}_{v}$, so that we
have a morphism $\phi:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$ which induces
a morphism
$\overline{\phi}:(\kappa(v),-\mathcal{N}_{v}\otimes\mathcal{L}_{\kappa(v)})\to(\kappa(w),-\mathcal{N}_{w}\otimes\mathcal{L}_{\kappa(w)})$.
Then

$\partial_{w}\circ\phi_{*}=\gamma_{e_{\epsilon}}\circ\overline{\phi}_{*}\circ\partial_{v}.$

  7. R3b

Let $E\to F$ be a finite extension of fields, let $v$ be a valuation on $E$
and let $\mathcal{L}$ be a graded line bundle on $\mathcal{O}_{v}$. For each
extension $w$ of $v$, we denote by
$\phi_{w}:(\kappa(v),\mathcal{L}_{\kappa(v)})\to(\kappa(w),\mathcal{L}_{\kappa(w)})$
the morphism induced by $\phi:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$. We
have

$\partial_{v}\circ\phi^{*}=\sum_{w}(\phi_{w})^{*}\circ\partial_{w}.$

  8. R3c

Let $\phi:(E,\mathcal{L}_{E})\to(F,\mathcal{L}_{F})$ be a morphism in
$\mathfrak{F}_{S}$. Let $w$ be a valuation on $F$ that is trivial on $E$. Then

$\partial_{w}\circ\phi_{*}=0.$

  9. R3d

Let $\phi$ and $w$ be as in R3c, and let
$\overline{\phi}:(E,\mathcal{L}_{E})\to(\kappa(w),\mathcal{L}_{\kappa(w)})$ be
the induced morphism. For any uniformizer $\pi$ of $v$, we have

$\partial_{w}\circ\gamma_{[-\pi]}\circ\phi_{*}=\overline{\phi}_{*}.$

  10. R3e

Let $E$ be a field over $S$, $v$ be a valuation on $E$ and
$u\in\mathcal{O}_{v}^{\times}.$ Then

$\partial_{v}\circ\gamma_{[u]}=\gamma_{\epsilon[\overline{u}]}\circ\partial_{v}$

and

$\partial_{v}\circ\gamma_{\eta}=\gamma_{\eta}\circ\partial_{v}.$

  11. R4a

Let $(E,\mathcal{L}_{E})$ and let $\Theta$ be an endomorphism of
$(E,\mathcal{L}_{E})$ given by an automorphism of $\mathcal{L}_{E}$. Denote by
$\Delta$ the canonical map from the group of automorphisms of
$\mathcal{L}_{E}$ to the group ${K}^{\rm MW}(E,0)$. Then
$\Theta_{*}=\gamma_{\Delta(\Theta)}:(E,\mathcal{L}_{E})\to(E,\mathcal{L}_{E})$.

######  Definition 2.2.

A MW-pre-cycle module $M$ over $S$ is a covariant functor
$M:\widetilde{\mathfrak{F}}_{S}\to{\rm Ab}$, to the category ${\rm Ab}$ of
abelian groups. A morphism of pre-cycle modules is a natural transformation
between the corresponding functors. We will denote the maps
$\partial_{v},\gamma_{x},\phi^{*},\phi_{*}$ etc. for
$M(\partial_{v}),M(\gamma_{x}),M(\phi^{*}),M(\phi_{*})$ respectively
throughout the paper to ease the notation.

######  Definition 2.3.

Let $M$ be a pre-cycle module over $S$, $X$ a scheme over $S$ and
$\mathcal{L}$ a graded line bundle $X$.

  1. (1)

For a point $x\in X$,
$M(x,\mathcal{L}_{X})=M(\kappa(x),\Lambda_{x}\otimes\mathcal{L}_{X})$ where
$\Lambda_{x}:=(\omega_{x/S},\dim\overline{\\{x\\}})$ and $\omega_{x/S}$ is the
canonical bundle (see 5.1).

  2. (2)

Suppose $X$ is normal, $\xi_{X}$ is the generic point of $X$ and $z$ is a
codimension 1 point of $X$, then

$\partial^{\xi_{X}}_{z}:=\partial_{v_{z}}:M(\xi_{X},\mathcal{L}_{X})\to
M(\kappa(z),\mathcal{L}_{X}).$

  3. (3)

Let $x\in X$ and let $Z$ be the normalization of the reduced closed subscheme
$\overline{\\{x\\}}_{red}$. Let $\\{z_{1},\cdots,z_{r}\\}\subset Z^{(1)}$,
where each $z_{i}\mapsto y$ under the map $Z\to X$ and

$\phi_{z_{i}}:(\kappa(y),\mathcal{L}_{X})\to(\kappa(z_{i}),\mathcal{L}_{X}).$

We define $\partial^{x}_{y}:M(x,\mathcal{L}_{X})\to M(y,\mathcal{L}_{X})$ as
follows

(2.1)
$\displaystyle\partial^{x}_{y}:=\begin{cases}\sum_{z_{i}}\phi_{z_{i}}^{*}\circ\partial^{x}_{z_{i}}&\text{if
$y\in\overline{\\{x\\}}$,}\\\ 0&\text{if
$y\notin\overline{\\{x\\}}$.}\end{cases}$

######  Definition 2.4.

[5, Definition 4.2] A MW-pre-cycle module $M$ over $S$ is a MW-cycle module if
it satisfies the following two properties:

  1. (FD)

Finite support of divisors: Let $X$ be a normal scheme, $\mathcal{L}_{X}$ be a
graded line bundle over $X$ and $\rho$ be an element of
$M(\xi_{X},\mathcal{L}_{X})$. Then $\partial_{x}(\rho)=0$ for all but finitely
many $x\in X^{(1)}$.

  2. (C)

Closedness: Let $X$ be an integral $S$-scheme which is local of dimension 2
and $\mathcal{L}_{X}$ be a graded line bundle over $X$. Then

$0=d\circ
d=\sum\partial^{x}_{x_{0}}\circ\partial^{\xi}_{x}:M(\xi,\mathcal{L}_{X})\to
M(x_{0},\mathcal{L}_{X})$

where $\xi$ is the generic point and $x_{0}$ is the closed point of $X$.

######  Definition 2.5.

[5, 7.1] Let $M$ be a MW-cycle module $M$ on $S$. Let $X/S$ be any scheme
which is equi-dimensional of dimension $d$. Let $\mathcal{L}$ be a graded line
bundle on $X$. One defines a Gersten complex $C^{\bullet}(X,M,\mathcal{L})$ by

$C^{p}(X,M,\mathcal{L})=\bigoplus_{x\in X^{(p)}}M(x,\mathcal{L})$

where $X^{(p)}$ denotes the set of points of codimension $p$ in $X$. As in [5,
7.1], when $X$ is not equi-dimensional one can also define a homological
version of the Gersten complex $C_{\bullet}(X,M,\mathcal{L})$ indexed by the
dimension of the points instead of the codimension. However, throughout this
paper we only use the cohomological indexing. We let $A^{p}(X,M,\mathcal{L})$
denote the $p$-th cohomology of $C^{\bullet}(X,M,\mathcal{L})$.

Let $Z$ be a closed subset of $X$ and $U:=X-Z\xrightarrow{j}X$, the open
immersion of $U$ in $X$, then the Gersten complex for $M$ on $X$ with support
on $Z$ is given by

$C^{\bullet}_{Z}(X,M,\mathcal{L}):={\rm Ker\
}\bigg{(}C^{\bullet}(X,M,\mathcal{L})\xrightarrow{j^{*}}C^{\bullet}(U,M,\mathcal{L})\bigg{)}.$

$A^{p}_{Z}(X,M,\mathcal{L})$ denotes the $p$-th cohomology of the complex
$C^{\bullet}_{Z}(X,M,\mathcal{L}).$

######  Definition 2.6.

Given a MW-cycle module $M$ over $S$ and a graded line bundle $\mathcal{L}$ on
a smooth $S$-scheme $X$, we have a naturally defined Zariski sheaf (see [5,
8.5]) $M(X,\mathcal{L})$ on $X$ given by

$U\mapsto A^{0}(U,M,\mathcal{L}).$

It is left to the reader to check that R3a implies that this is in fact a
Nisnevich sheaf on $X$.

######  Remark 2.7.

Given MW-cycle module $M$ on $S$ and a morphism $f:Y\to X$ of smooth
$S$-schemes one shows that for a graded line bundle $\mathcal{L}$ on $S$, we
have using R3a a well defined map $A^{0}(X,M,\mathcal{L}_{X})\to
A^{0}(Y,M,\mathcal{L}_{Y})$. As in [7, Theorem 5.3.1], one can easily show
that $A^{0}(-,M,\mathcal{L})$ is a birational invariant for smooth proper
$S$-schemes.

######  Proposition 2.8.

Let $M$ be a MW-cycle module on $S$ and $\mathcal{L}$ a graded line bundle on
$S$. Then $M$ satisfies the following two properties:

  1. (WR)

Weak reciprocity: For every $S$-field $F$, let $\partial_{\infty}$ be the
valuation on $F(u)$ at infinity, then

$\partial_{\infty}(A^{0}(\mathbb{A}^{1}_{F},M,\mathcal{L}))=0.$

  2. (H)

For every $S$-field $F$, the complex

$0\to
M(F,\mathcal{L})\xrightarrow{r}M(\xi,\mathcal{L})\xrightarrow{d}\bigoplus_{x\in(\mathbb{A}^{1}_{F})^{(1)}}M(x,\mathcal{L})\to
0$

is exact. Here $\xi$ denotes the generic point of $\mathbb{A}^{1}_{F}$.

###### Proof.

The proof is as in [15, Proposition 2.2, Theorem 2.3], with necessary
modifications as in the identities ${ab}=\\{a\\}+\\{b\\}$ replaced by
$[ab]=[a]+[b]+\eta[a][b]$, etc. ∎

######  Example 2.9.

Let $S={\rm Spec\,}(R)$ where $R$ is an excellent DVR. Let $\eta$ and $s$
denote its generic and closed points respectively. Let

$\displaystyle M(F,\mathcal{L}_{F}):=\begin{cases}{K}^{\rm
MW}(F,\mathcal{L}_{F})&\text{if $F$ lies over }s,\\\
0&\text{otherwise}.\end{cases}$

It is easy to see that $M$ has the structure of a cycle module, using the
cycle module structure on ${K}^{\rm MW}$ on the $\kappa(s)$. The Gersten
complex of $M$ on $S$ is

$0\to M(\eta,\mathcal{L}_{S})\xrightarrow{\partial}M(s,\mathcal{L}_{S})\to 0$

which is clearly not exact in degree 1.

First, we make some conventions applicable throughout the paper, unless
mentioned otherwise.

######  Notation 2.10.

For a given Milnor-Witt pre-cycle module $M$ over $S$, we simply write
$M(E,*)$ when the coefficient is implicitly determined by the context or when
it can be any arbitrary coefficient. For example, given a field $E$ and an
$S$-discrete valuation $v$ instead of writing

$M(E,\mathcal{L}_{E})\xrightarrow{\partial_{v}}M(\kappa(v),\mathcal{N}^{\vee}_{v}\otimes\mathcal{L}_{\kappa(v)})$

we sometimes simply write

$M(E,\mathcal{L}_{E})\xrightarrow{\partial_{v}}M(\kappa(v),*)$

The axioms canonically tell us what coefficient to fill in place of $*$ above.
Moreover, if the coefficient $\mathcal{L}_{E}$ is not of any particular
interest and can be arbitrary, we simply write the above map as

$M(E,*)\xrightarrow{\partial_{v}}M(\kappa(v),*)$

This abuse of notation, which will only be required in this section, is to
facilitate the focus on the idea, instead of coefficients, whenever they are
anyway canonically determined or are arbitrary.

In order to find a good class of Milnor-Witt cycle modules over a DVR which
satisfy local acylicity, we first observe the following.

######  Proposition 2.11.

Let $M$ be a Milnor-Witt cycle module over a perfect field $k$. Let
$F\in\mathcal{F}ield/k$ with distinct discrete $k$-valuations
$v_{1},..,v_{n}$. Let $\mathcal{L}$ be a graded line bundle on the semi-local
ring $\mathcal{O}=\bigcap_{i}\mathcal{O}_{v_{i}}$.Then

$\
M(F,*)\xrightarrow{\sum_{i=1}^{n}\partial_{v_{i}}}\bigoplus_{i=1}^{n}M(\kappa(v_{i}),*)$

is surjective.

###### Proof.

The statement of this proposition simply says that local acyclicity holds for
the semi-local ring $\mathcal{O}$. Thus, the claim for essentially smooth
semilocal ring $\mathcal{O}$, directly follows from Feld (see [5, Theorem
8.1]). However, we give a direct proof which is shorter. To show this, it is
enough to show that for any $1\leq j\leq n$, and an $\alpha\in
M(\kappa(v_{j}),*)$, there exists a $\beta\in M(F,*)$ such that

$\partial_{v_{j}}(\beta)=\alpha\ \ \ \text{and}\ \ \
\partial_{v_{i}}(\beta)=0\ \forall\ i\neq j.\ \ \ ({\star})$

Without loss of generality, we may assume $j=1$.

Step 1: Choose a finite extension $E/F$ such that there exists a valuation
$w_{1}$ of $E$ extending the valuation $v_{1}$ on $F$ satisfying

  1. (1)

$k(w_{1})/k(v_{1})$ is a trivial extension.

  2. (2)

There exists a field $l\subset E$ which maps isomorphically onto $k(w_{1})$.

To see this, let $F^{h}$ be the fraction field of the Henselization
$\mathcal{O}^{h}_{v_{1}}=\widehat{\mathcal{O}}_{v_{1}}\cap F^{sep}$ of
$\mathcal{O}_{v_{1}}$ which has the residue field $k(v_{1})$. Existence of a
finite subextension $E/F$ of $F^{h}/F$ follows from the fact that $k(v_{1})/k$
is a separably finitely generated field extension.

Let $\Sigma$ be the set of all valuations of $E$ except $w_{1}$, which lie
over one of the $v_{i}$ for $1\leq i\leq n$. Choose a uniformizer $\pi$ of
$w_{1}$ such that $\pi\in F$ and $\pi\equiv-1\ {\rm mod}\ {\mathfrak{m}}_{w}$
for all $w\in\Sigma$.

Step 2: First observe that (using the fact that $E$ contains the residue field
$k(w_{1})$)

$\partial_{w_{1}}\circ\gamma_{[-\pi]}:M(E,*)\to M(k(w_{1}),*)$

is surjective. We leave it to the reader to check that this follows directly
from R3d. Thus, there exists a $\widetilde{\beta}\in M(E,\mathcal{L}_{E})$
such that its image via the above map followed by the isomorphism

$M(k(w_{1}),*)\to M(k(v_{1}),*)$

is $\alpha$. Let $\beta=\phi^{*}\widetilde{\beta}$. where $\phi$ is the map
from $(F,\mathcal{L}_{F})\to(E,\mathcal{L}_{E})$ We will need to show that
$\beta$ satisfies the required property mentioned in equation $(\star)$. This
follows from R3b and Lemma 2.12. Details are left to the reader. ∎

######  Lemma 2.12.

Let $M$ be a cycle module over a base $S$. Let $F$ be an $S$-field and $v$ be
a $S$-valuation on $F$. Let $c\in F$ such that $c\equiv 1\ {\rm mod}\
{\mathfrak{m}}_{v}$. Then $\partial_{v}\circ\gamma_{c}=0$.

###### Proof.

Since $\epsilon[\overline{c}]=0$, this is a direct consequence of R3e. ∎

To avoid the non-acyclic example 2.9 above, we consider the following class of
Milnor-Witt cycle modules.

######  Definition 2.13 (R5).

Let $M$ be a MW-cycle module over $S$. $M$ is said to satisfy property R5 if
for every field $F\in\mathcal{F}ield/S$ and horizontal $S$-discrete valuations
$v_{1},..,v_{n}$, let $\mathcal{L}$ be a graded line bundle on the semi-local
ring $\mathcal{O}=\bigcap\mathcal{O}_{v_{i}}$. Then

$\
M(F,\mathcal{L}_{F})\xrightarrow{\sum_{i=1}^{n}\partial_{v_{i}}}\bigoplus_{i=1}^{n}M(\kappa(v_{i}),*)$

is surjective.

We now prove Theorem 1.4 to show that ${K}^{\rm MW}$ is a cycle module
satisfying R5.

###### Proof of Theorem 1.4.

We leave it to the reader to check that ${K}^{\rm MW}$ is MW-pre-cycle module
as the proofs are exactly the same as in the field case. It thus remains to
check closedness property (C) in the Definition 2.4 and property R5.

Step 1: In this step we show that ${K}^{\rm MW}$ satisfies R5. To see this,
let $E$ be an $S$-field and $v_{1},v_{2},..,v_{n}$ be finitely many distinct
valuations. As in the proof of Theorem 2.11 it is enough to show that for any
$\alpha\in{K}^{\rm MW}(k(v_{1}),*)$ there exists a $\beta$ such that
$\partial_{v_{1}}(\beta)=\alpha$ and $\partial_{v_{i}}(\beta)=0$ for all
$i>1$.

We first claim that for any uniformizer $\pi$ of $v_{1}$, the specialization
map

$s^{\pi}_{v}=\partial_{v}\circ\gamma_{[-\pi]}:{K}^{\rm
MW}(F,\mathcal{L}_{F})\to{K}^{\rm MW}(\kappa(v),*)$

is surjective. This follows from the fact that if $\alpha\in{K}^{\rm
MW}_{n}(k(v_{1}),(({\mathfrak{m}}_{1}/{\mathfrak{m}}_{1}^{2})^{\vee},-1))$ is
of the form $[u_{1},\cdots,u_{n}]\otimes\overline{\pi}^{\vee}$ then for lifts
$\widetilde{u}_{i}$ of $u$ in $\mathcal{O}_{v_{1}}$, we have

$\partial_{v_{1}}([-\pi][\widetilde{u}_{1},\cdots,\widetilde{u}_{n}])=\alpha$

Moreover if we choose $\pi\equiv-1\ {\rm mod}\ {\mathfrak{m}}_{j}$ for $j>1$,
then clearly

$\partial_{v_{j}}([-\pi][\widetilde{u}_{1},\cdots,\widetilde{u}_{n}])=0\
\forall\ j>1$

thus showing R5.

Step 2: In the remaining steps we verify the closedness property (2.4)(C)
following analogous proofs for Milnor K-theory in [11, Proposition 1] and [3,
Proposition 49.30]. As in [11, Proposition 1], it suffices to check (2.4)(C)
when $X={\rm Spec\,}(A)$ where $(A,\mathfrak{m})$ is a complete local domain
of dimension 2. By Cohen’s structure theorem (see [17, Tag 032D]), there is a
complete DVR $\Lambda\subset A$ such that $A$ is finite over $\Lambda[[t]]$
for an indeterminate $t$. Let $\pi$ be the uniformizer of $\Lambda$. Let
$A/\mathfrak{m}=\kappa$. Let $K={\rm Frac}(A).$ It remains to prove the claim
that

${K}^{\rm MW}_{n+2}(K)\xrightarrow{\sum_{x\in X_{(1)}}d_{x}}\bigoplus_{x\in
X_{(1)}}{K}^{\rm MW}_{n+1}(\kappa(x),\Lambda_{x})\xrightarrow{\sum_{x\in
X_{(1)}}d^{\prime}_{x}}{K}^{\rm
MW}_{n}(\kappa,(\det\mathfrak{m}/\mathfrak{m}^{2})^{\vee})$

is a complex.

Step 3: We will first prove the claim in Step 2 when $m:=n+2\geq 1$. In this
case, ${K}^{\rm MW}_{m}(K)$ is generated by $[u_{1}]\cdots[u_{m}]$ for
$u_{i}\in K^{\times}.$ Moreover, $K^{\times}$ is generated by
$A^{\times},H,\pi,t$ where $H$ is the subgroup of $K^{\times}$ generated by
elements of the form

$1+a_{1}t^{-1}+\cdots+a_{r}t^{-r}\ \text{where}\ a_{i}\in(\pi)\ \forall\ i\
\text{and}\ r\geq 0.$

Let $H^{\prime}$ be the subgroup generated by $H,\pi,t$. Using this fact, and
the results in [14, 3.17 (2), (3), 3.5 (1)], we leave it for the reader to
verify that proving the claim is equivalent to showing the following

(2.2) $\sum_{x\in X_{(1)}}d^{\prime}_{x}\circ d_{x}([\pi][t])=0$ (2.3)
$\sum_{x\in X_{(1)}}d^{\prime}_{x}\circ d_{x}([f][g_{1}]\cdots[g_{n+1}])=0$

for all $f\in H$ and $g_{i}\in H^{\prime}$.

By Weierstrass preparation theorem, the height 1 prime ideals in $A$ are given
by either $(\pi)$ or $(f)$ where $f\in\Lambda[[t]]$ given by

$f=t^{r}+a_{r-1}t^{r-1}+\cdots+a_{0}\ \text{where}\ a_{i}\in(\pi)\ \forall\
i.$

Step 4: In this step, we show that (2.2) of Step 3 holds. Let $x_{1}$ (resp.
$x_{2}$) be the codimension 1 point in $X$ corresponding to the prime ideal
$(t)$ (resp. $(\pi)$). Then, for $x\in X_{(1)}-\\{x_{1},x_{2}\\}$,
$d_{x}([\pi][t])=0$, since both $\pi$ and $t$ are units at $x$. Also, since
$t$ is a uniformizer of $X$ at $x_{1}$ and $\pi$ is a uniformizer of $x_{1}$
at the closed point,

$\displaystyle d^{\prime}_{x_{1}}\circ d_{x_{1}}([\pi][t])$
$\displaystyle=-1\otimes\overline{t}^{\vee}\wedge\overline{\pi}^{\vee}.$

Similarly, since $\pi$ is a uniformizer of $X$ at $x_{2}$ and $t$ is a
uniformizer of $x_{2}$ at the closed point,

$d^{\prime}_{x_{2}}\circ
d_{x_{2}}([\pi][t])=d^{\prime}_{x_{2}}([t]\otimes\overline{\pi}^{\vee})=1\otimes\overline{t}^{\vee}\wedge\overline{\pi}^{\vee}.$

Thus,

$\sum_{x\in X_{(1)}}d^{\prime}_{x}\circ
d_{x}([\pi][t])=-1\otimes\overline{t}^{\vee}\wedge\overline{\pi}^{\vee}+1\otimes\overline{t}^{\vee}\wedge\overline{\pi}^{\vee}=0.$

Step 5: We now verify (2.3) of Step 3.

Let $L={\rm Frac}(\Lambda)$ and

$h:X^{\prime}={\rm Spec\,}A[\pi^{-1}]\to{\rm Spec\,}L[t]=\mathbb{A}^{1}_{L}.$

Then, $X^{\prime}_{(0)}=X_{(1)}-\\{x_{2}\\}$ and $h(X^{\prime}_{(0)})$ is the
set of closed points of $\mathbb{A}^{1}_{L}$ associated to the irreducible
polynomials of the form $t^{r}+a_{r-1}t^{r-1}+\cdots+a_{0}$ where
$a_{i}\in(\pi)$ for all $i$. For each $x\in
X^{\prime}_{(0)}=X_{(1)}-\\{x_{2}\\}$, $k(x)$ is a finite field extension of
$L$ via $h$. We have $d^{\prime}_{x}=\partial\circ Nm_{k(x)/L}$, where

$\partial:{K}^{\rm MW}_{n+1}(L,((t)/(t)^{2})^{\vee})\to{K}^{\rm
MW}_{n}(\kappa,(\det\mathfrak{m}/\mathfrak{m}^{2})^{\vee})$

is the residue morphism associated to the discrete valuation on $L$. To see
this, we observe from [14, Remark 5.20, page 127] and an analogue of [9, Cor.
7.4.3 page 205] for Milnor-Witt K theory of fields proved in [6, Corollary
2.2.23] that the following diagram

$\textstyle{{K}^{\rm
MW}_{n+1}(\kappa(x),\Lambda_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}\sum_{w}\partial_{w}}$$\scriptstyle{Nm_{\kappa(x)/L}}$$\textstyle{\bigoplus_{w}{K}^{\rm
MW}_{n}(L(w),\Lambda_{w})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{w}Nm_{L(w)/\kappa}}$$\textstyle{{K}^{\rm
MW}_{n+1}(L,((t)/(t)^{2})^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{{K}^{\rm
MW}_{n}(\kappa,\det\mathfrak{m}/\mathfrak{m}^{2})}$

commutes. Here $w$ denotes a discrete valuation of $\kappa(x)$ which restricts
to the discrete valuation on $L$ and $L(w)$ is the residue field of $w$. Let
$i:{K}^{\rm MW}_{n+1}(L(t))\to{K}^{\rm MW}_{n+1}(K)$ induced by the inclusion
$L(t)\to K$.

Thus,

(2.4) $\displaystyle\sum_{x\in X_{(1)}}d^{\prime}_{x}\circ d_{x}\circ i$
$\displaystyle=d^{\prime}_{x_{2}}\circ d_{x_{2}}\circ i+\sum_{x\in
X_{(1)}-\\{x_{2}\\}}d^{\prime}_{x}\circ d_{x}\circ i$ (2.5)
$\displaystyle=d^{\prime}_{x_{2}}\circ d_{x_{2}}\circ
i+\partial\Big{(}\sum_{x\in X_{(1)}-\\{x_{2}\\}}Nm_{\kappa(x)/L}\circ
d_{x}\circ i\Big{)}.$

Let $\alpha=[f][g_{1}]\cdots[g_{n+1}]\in{K}^{\rm MW}_{n+2}(L(t)).$ We observe
that $div(f),div(g_{i})$ are supported in the image
$h(X^{\prime}_{(0)})\subset(\mathbb{A}^{1}_{L})_{(0)}$. Thus, for
$x\in(\mathbb{A}^{1}_{L})_{(0)}-h(X^{\prime}_{(0)}),d_{x}(\alpha)=0.$

Also, at $\infty\in(\mathbb{P}^{1}_{L})_{(0)}$, $f(\infty)=1$, hence
$d_{\infty}\circ i(\alpha)=0$.

By the reciprocity law [14, Chapter 4, (4.8), page 98],

$\sum_{x\in X_{(1)}-\\{x_{2}\\}}Nm_{\kappa(x)/L}\circ d_{x}\circ
i(\alpha)=\sum_{x\in(\mathbb{P}^{1}_{L})_{(0)}}Nm_{\kappa(x)/L}\circ
d_{x}\circ i(\alpha)=0.$

At $x_{2}$, $f(x_{2})=1$, hence $d_{x_{2}}\circ i(\alpha)=0.$ Thus, (2.3)
holds.

Step 6: It now remains to prove the claim of Step 2 in the case $m:=n+2\leq
0$. In this case ${K}^{\rm MW}_{m}(K)$ is generated by $\eta^{m}\langle
u\rangle$, for $u\in K^{\times}$. Since the differentials $d$ commute with
$\eta$, it is enough to show that $d\circ d(\langle u\rangle)=0$ for $u\in
K^{\times}$. We note that $\langle u\rangle=1+\eta[u]$, so it suffices to show
:

(2.6) $\displaystyle d\circ d([\pi])$ $\displaystyle=0$ (2.7) $\displaystyle
d\circ d([t])$ $\displaystyle=0$ (2.8) $\displaystyle d\circ d([h])$
$\displaystyle=0\ \ \ \text{where}\ h\in A^{\times}\ \text{or}\ h\in H.$

The above are explicit and straightforward calculations along the lines of
arguments used in Steps 4 and 5. We leave the details to the reader. ∎

We now recall some definitions of morphisms between Gersten complexes. These
are exactly as in [5, Section 5]

######  Definition 2.14.

  1. (1)

Let $f:X\to Y$ be a finite type morphism of equi-dimensional $S$-schemes with
$\dim X=d$ and $\dim Y=e$. Let $\mathcal{L}_{Y}$ be a graded line bundle on
$Y$. Let $\mathcal{L}_{X}$ be the pullback of $\mathcal{L}_{Y}$ on $X$. Then
there is a map

$f_{*}:C^{q}(X,M,\mathcal{L}_{X})\to C^{e-d+q}(Y,M,\mathcal{L}_{Y})$

defined for $\alpha\in M(x,\mathcal{L}_{X})$ as

$\displaystyle({f_{*}})_{xy}(\alpha)=\begin{cases}\phi^{*}(\alpha)&\text{if
$f(x)=y$ and $\phi:\kappa(y)\to\kappa(x)$ is a finite field extension}\\\
0&\text{otherwise}\end{cases}$

where $\phi^{*}:M(x,\mathcal{L}_{X})\to M(y,\mathcal{L}_{X})$. When $\phi$ is
finite morphism, $f_{*}$ is a morphism of complexes (Lemma 2.15).

  2. (2)

Let $g:Y\to X$ be a smooth morphism of equi-dimensional $S$-schemes with
constant relative dimension $s$. Let $\mathcal{L}_{X}$ be a graded line bundle
on $X$. Let $\mathcal{L}_{Y}$ be the pullback of $\mathcal{L}_{X}$ on $Y$.
Then there is a map

$g^{*}:C^{q}(X,M,\mathcal{L}_{X})\to
C^{q}(Y,M,(\omega_{Y/X}^{\vee},-s)\otimes\mathcal{L}_{Y})$

defined for $\alpha\in M(x,\mathcal{L}_{X})$ as

$\displaystyle({g^{*}})_{xy}(\alpha)=\begin{cases}\Theta\circ
i_{*}(\alpha)&\text{if $g(y)=x$ and $y$ is a generic point of $Y_{x}$}\\\
0&\text{otherwise}\end{cases}$

where
$i_{*}:M(x,\mathcal{L}_{X})=M(\kappa(x),\Lambda_{x}\otimes\mathcal{L}_{X})\to
M(\kappa(y),\Lambda_{x}\otimes\mathcal{L}_{Y})$ and

$\Theta:M(\kappa(y),\Lambda_{x}\otimes\mathcal{L}_{Y})\to
M(\kappa(y),\Lambda_{y}\otimes(\omega_{Y/X}^{\vee},-s)\otimes\mathcal{L}_{Y})=M(y,(\omega_{Y/X}^{\vee},-s)\otimes\mathcal{L}_{Y})$

is the isomorphism induced by the canonical isomorphism

$\Lambda_{y}\simeq\Lambda_{x}\otimes(\omega_{Y/X},s)$

of graded line bundles on ${\rm Spec\,}\kappa(y)$.

  3. (3)

Let $X$ be an equi-dimensional $S$-scheme of dimension $d$. For a unit
$u\in\mathcal{O}(X)$, there is a map

$[a]:C^{q}(X,M,\mathcal{L}_{X})\to
C^{q}(X,M,\mathbb{A}^{1}\otimes\mathcal{L}_{X})$

defined by

$\displaystyle[a]_{xy}(\alpha)=\begin{cases}\langle-1\rangle^{d-q}[a]\cdot\alpha&\text{if
$x=y$ }\\\ 0&\text{otherwise}.\end{cases}$

  4. (4)

Let $i:Z\hookrightarrow X$ be a closed subscheme of $X$ and
$U:=X-Z\xrightarrow{j}X$ be the open complement of $Z$ in $X$. Let
$\mathcal{L}$ be a graded line bundle on $X$. There are term-wise maps
$j_{*}:C^{q}(U,M,\mathcal{L})\to C^{q}(X,M,\mathcal{L})$ and
$i^{*}:C^{q}(X,M,\mathcal{L})\to C^{q}_{Z}(X,M,\mathcal{L})$. There is a
boundary map

$\partial^{U}_{Z}:C^{q}(U,M,\mathcal{L})\to C^{q+1}_{Z}(X,M,\mathcal{L})$

defined as

$\partial^{U}_{Z}:=i^{*}\circ d_{X}\circ j_{*}.$

We recall some properties satisfied by the above maps which will be used in
the next section.

######  Lemma 2.15.

  1. (1)

For a finite morphism $f:X\to Y$ of $S$-schemes, $f_{*}\circ d_{X}=d_{Y}\circ
f_{*}$.

  2. (2)

Let $g:Y\to X$ be a smooth morphism of equi-dimensional $S$-schemes with
constant relative dimension $s$. Then $g^{*}\circ d_{X}=d_{Y}\circ g^{*}$.

  3. (3)

For a unit $u\in\mathcal{O}(X)$, $\gamma_{\epsilon\cdot[\overline{u}]}\circ
d_{X}=d_{X}\circ\gamma_{[u]}$.

###### Proof.

  1. (1)

The proof in [3, Proposition 49.9] works for $M$ as is.

  2. (2)

The proof follows as in [15, Proposition 4.6] using R1c and R3c.

  3. (3)

This follows from R3e.

∎

##  3\. Local acyclicity

In this section, we assume $S={\rm Spec\,}R$, where $R$ is an excellent DVR
$(R,\mathfrak{m})$ with residue field $\kappa=\kappa(s)$ and the generic point
$\eta$. The goal of this section is to prove Theorem 1.1. This proof is an
adaptation of the proofs of [15, Theorem 6.1] and [13, Theorem 3.1].

######  Lemma 3.1.

Let $M$ be a MW-cycle module on $S$. Let $g:Y\to X$ be a smooth $S$-morphism
of finite type $S$-schemes of relative dimension 1. Let $\sigma:X\to Y$ be a
section to g and let $t\in\mathcal{O}_{Y}$ be a global parameter defining the
subscheme $\sigma(X)$. Moreover, let $\widetilde{g}:U=Y-\sigma(X)\to X$ be the
restriction of g and let $\partial$ be the boundary map associated to the
tuple $(X\xrightarrow{\sigma}Y\xleftarrow{}U)$, then

$\partial\circ\widetilde{g}^{*}=0$

and

$\Theta\circ\partial\circ[t]\circ\widetilde{g}^{*}=(id_{X})_{*},$

where $\Theta$ is the isomorphism from the Gersten complex for $M$ on $Y$
supported at $X$ to the Gersten complex for $M$ on $X$.

###### Proof.

As in Rost’s original proof given in [15, Lemma 4.5], the proof of this
theorem reduces to the case when $X$ and $S$ are fields. We refer the reader
to [5] for the proof in this case, which is a minor modification of Rost’s
original proof.

We add a remark about $\Theta$: $\partial$ maps into
$C^{p+1}_{X}(Y,M,\mathbb{A}^{1}\otimes(\omega_{Y/X}^{\vee},-1)\otimes\mathcal{L}_{Y})$.
For $x\in X$ and $y\in U$ the generic point of the fiber
$\widetilde{g}^{-1}(x)$, the canonical isomorphisms

$\Lambda_{y}\simeq\Lambda_{x}\otimes(\omega_{Y/X},1)\otimes\kappa(y)$

and

$(\omega_{Y/X}^{\vee},-1)\otimes(\mathbb{A}^{1}_{\kappa(y)},1)=(\kappa(y),0)$

of graded line bundles on ${\rm Spec\,}\kappa(y)$ induces the isomorphism

$\Theta:C^{p+1}_{X}(Y,M,\mathbb{A}^{1}\otimes(\omega_{Y/X},-1)\otimes\mathcal{L}_{Y})\simeq
C^{p}(X,M,\mathcal{L}_{X}).$

∎

We recall a geometric presentation lemma from Gillet-Levine [10], that will be
used in the proof of Theorem 1.1, following [15, 6.1].

######  Lemma 3.2.

([10, Lemma 1]) Let $X$ be an affine scheme flat of finite type over a
discrete valuation ring $R$. Let $T\subset X$ be a finite set of points such
that $X$ is smooth at $T$ over $R$. Let $Y$ be a principle effective divisor
which is flat over $R$. Then there exists an affine open neighbourhood
$X^{\prime}\subset X$ of $T$ and a morphism
$\pi:X^{\prime}\to\mathbb{A}^{d-1}_{S}$, where d is the relative dimension of
$X$ over $S$, such that

  1. (1)

$\pi$ is smooth of relative dimension 1 at the points of $T$.

  2. (2)

$\pi|_{Y^{\prime}}:Y^{\prime}\to\mathbb{A}^{d-1}_{S}$ restricted to
$Y^{\prime}=X^{\prime}\cap Y$ is quasi-finite.

######  Proposition 3.3.

Let $X$ be a smooth $S$-scheme of relative dimension $d$. Let $Y\subset X$ be
a closed subscheme of codim $\geq 1$ such that $Y$ is flat over $S$. Let $y$
be a point in $Y$ lying over the closed point $s\in S$. Then there is a
Zariski neighbourhood $X^{\prime}$ of $y$ in $X$ such that the map

$A^{p}_{Y\times_{X}X^{\prime}}(X^{\prime},M,\mathcal{L}_{X^{\prime}})\to
A^{p}(X^{\prime},M,\mathcal{L}_{X^{\prime}})$

is trivial.

###### Proof.

We follow Rost’s proof of [15, Proposition 6.4]. By Lemma 3.2, we have a
Zariski neighbourhood $X^{\prime}$ of $y$ and a morphism
$\pi:X^{\prime}\to\mathbb{A}^{d-1}_{S}$ of relative dimension 1 which is
smooth at $y$ such that $\pi|_{Y^{\prime}}:Y^{\prime}\to\mathbb{A}^{d-1}_{S}$
is quasi-finite where $Y^{\prime}:=Y\times_{X}X^{\prime}$ . Let
$Z:=Y\times_{\mathbb{A}^{d-1}_{S}}X^{\prime}$ and $\sigma:Y^{\prime}\to Z$ be
the section of $Z\to Y^{\prime}$ induced by the morphism $Y^{\prime}\to
X^{\prime}$. We have the commutative diagram

(3.1)
---
$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{~{}1_{Y}}$$\scriptstyle{i}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}p}$$\scriptstyle{q}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi|_{Y^{\prime}}}$$\textstyle{\mathbb{A}^{d-1}_{S}}$

By Zariski’s main theorem, we factorize the quasi-finite map
$Z\xrightarrow{p}X^{\prime}$ as an open immersion $Z\to\overline{Z}$ followed
by a finite map $\overline{p}:\overline{Z}\to X^{\prime}$, so that we have the
following commutative diagram

(3.2)
---
$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{~{}1_{Y}}$$\scriptstyle{\overline{\sigma}}$$\textstyle{\overline{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{p}}$$\textstyle{D:=\overline{Z}-Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}p}$$\scriptstyle{q}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi|_{Y^{\prime}}}$$\textstyle{\mathbb{A}^{d-1}_{S}}$

By passing to a smaller open subscheme of $X^{\prime}$, we can assume that
$\sigma(Y^{\prime})\subset Z$ is given by a section $t\in\mathcal{O}_{Z}(Z)$
and $D\cap\overline{\sigma}(Y^{\prime})=\emptyset$. As in the proof of [8,
Theorem 4.1], by Chinese remainder theorem, by further shrinking to a smaller
open subscheme, we can find $\overline{t}\in\mathcal{O}(\overline{Z})$ such
that $\overline{t}|_{Z}=t$ and $\overline{t}|_{D}=1$.

Let $Q:=Z-\sigma(Y^{\prime})$,
$\overline{Q}:=\overline{Z}-\overline{\sigma}(Y^{\prime})$,
$j:Q\to\overline{Q}$ and $\overline{j}:\overline{Q}\to\overline{Z}$. Consider
the correspondence

$\textstyle{H:Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{*}}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\overline{t}]}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\textstyle{\overline{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{j}_{*}}$$\textstyle{\overline{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{p}_{*}}$$\textstyle{X^{\prime}}$

giving a homomorphism

$H:C^{p}_{Y^{\prime}}(X^{\prime},M,\mathcal{L}_{X^{\prime}})\to
C^{p}(X^{\prime},M,\mathcal{L}_{X^{\prime}}).$

We claim that $H$ satisfies the following relation

(3.3) $d_{X^{\prime}}\circ H-\epsilon H\circ d_{Y^{\prime}}=i_{*}$

where $i:Y^{\prime}\hookrightarrow X^{\prime}$ is the closed immersion and
$\epsilon=-\langle-1\rangle\in{K}^{\rm MW}_{0}(S)$.

$\displaystyle d_{X^{\prime}}\circ H$
$\displaystyle=d_{X^{\prime}}\circ\overline{p}_{*}\circ\overline{j}_{*}\circ
j_{*}\circ[\overline{t}]\circ q^{*}$
$\displaystyle\overset{(1)}{=}\overline{p}_{*}\circ
d_{\overline{Z}}\circ\overline{j}_{*}\circ j_{*}\circ[\overline{t}]\circ
q^{*}$
$\displaystyle\overset{(2)}{=}\overline{p}_{*}\circ(\sigma_{*}\circ\Theta\circ\partial^{Q}_{Y^{\prime}}+{i_{\overline{Z}-Z}}_{*}\circ\partial^{Q}_{\overline{Z}-Z}+\overline{j}_{*}\circ
j_{*}\circ d_{Q})\circ[\overline{t}]\circ q^{*}$
$\displaystyle\overset{(3)}{=}\overline{p}_{*}\circ\sigma_{*}\circ\Theta\circ\partial^{Q}_{Y^{\prime}}\circ[\overline{t}]\circ
q^{*}+\overline{p}_{*}\circ j_{*}\circ d_{Q}\circ[\overline{t}]\circ q^{*}$
$\displaystyle\overset{(4)}{=}\overline{p}_{*}\circ\sigma_{*}\circ{1_{Y^{\prime}}}_{*}+\overline{p}_{*}\circ\overline{j}_{*}\circ
j_{*}\circ(\epsilon[\overline{t}]\circ d_{Q})\circ q^{*}$
$\displaystyle\overset{(5)}{=}i_{*}+\overline{p}_{*}\circ\overline{j}_{*}\circ
j_{*}\circ(\epsilon[\overline{t}])\circ q^{*}\circ d_{Y^{\prime}}$
$\displaystyle\overset{(6)}{=}i_{*}+\epsilon\overline{p}_{*}\circ\overline{j}_{*}\circ
j_{*}\circ[\overline{t}]\circ q^{*}\circ d_{Y^{\prime}}$
$\displaystyle=i_{*}+\epsilon H\circ d_{Y^{\prime}}.$

(1) follows since $\overline{p}$ is finite, hence
$d_{X^{\prime}}\circ\overline{p}_{*}=\overline{p}_{*}\circ d_{\overline{Z}}$
[see Lemma 2.15(1)].
(2) follows since
$\sigma_{*}\circ\sigma^{*}+{i_{\overline{Z}-Z}}_{*}\circ{i_{\overline{Z}-Z}}^{*}+\overline{j}_{*}\circ\overline{j}^{*}=1_{\overline{Z}}$,
by the argument similar to the one in [15, (3.10), page 350]. Thus

$\sigma_{*}\circ\sigma^{*}\circ d_{\overline{Z}}\circ\overline{j}_{*}\circ
j_{*}+{i_{\overline{Z}-Z}}_{*}\circ{i_{\overline{Z}-Z}}^{*}\circ
d_{\overline{Z}}\circ\overline{j}_{*}\circ
j_{*}+\overline{j}_{*}\circ\overline{j}^{*}\circ
d_{\overline{Z}}\circ\overline{j}_{*}\circ
j_{*}=d_{\overline{Z}}\circ\overline{j}_{*}\circ j_{*}.$

Since $\Theta\circ\partial^{Q}_{Y}=\sigma^{*}\circ
d_{\overline{Z}}\circ\overline{j}_{*}\circ j_{*}$ (recall from Lemma 3.1 the
definition of $\Theta$), we have

$d_{Z}\circ\overline{j}_{*}\circ
j_{*}=\sigma_{*}\circ\Theta\circ\partial^{Q}_{Y^{\prime}}+{i_{\overline{Z}-Z}}_{*}\circ\partial^{Q}_{\overline{Z}-Z}+\overline{j}_{*}\circ
j_{*}\circ d_{Q}.$

(3) follows since $\overline{t}|_{\overline{Z}-Z}=1$ implies
$\partial^{Q}_{\overline{Z}-Z}\circ[\overline{t}]=0$.
(4) First term in (4) follows by Lemma 3.1 applied to triple
$(Y^{\prime}\xrightarrow{\sigma}Z\xleftarrow{j}Q).$ The second term follows
since $d_{Q}\circ[t]=\epsilon[t]\circ d_{Q}$ [5, Prop. 6.6 (3) page 22].
(5) follows since $\overline{p}\circ\sigma=i$ and because $d_{Q}\circ
q^{*}=q^{*}\circ d_{Y}^{\prime}$ (see Lemma 2.15(2)).
(6) follows since $\epsilon$ commutes with $p_{*}$ by projection formula R2b
and also commutes with $j_{*}$ and $\overline{j}_{*}$.
The relation (3.3) implies that
$i_{*}:A^{p}_{Y\times_{X}X^{\prime}}(X^{\prime},M,\mathcal{L}_{X^{\prime}})\to
A^{p}(X^{\prime},M,\mathcal{L}_{X^{\prime}})$ is zero. ∎

We are now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

We need to show that

$A^{p}(X,M,\mathcal{L}_{X})=0\ \forall\ p\geq 1.$

The proof is an adaptation of the ideas in [15, Proposition 6.1] and [13,
Theorem 3.1]) with input from Proposition 3.3. For simplicity, we prove the
theorem when $X$ is local. The proof of the semi-local case is very similar
with just more notation.

Let $X=U_{u}:={\rm Spec\,}\mathcal{O}_{U,u}$ where $U$ is a smooth,
irreducible, equi-dimensional $S$-scheme of dimension $d$ and a point $u$ in
$U$ lying over the closed point $s$ in $S$. We have

$C^{p}(X,M,\mathcal{L}_{X})=\mathop{\varinjlim}\limits_{(U^{\prime},u)}C^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$

where $(U^{\prime},u)$ runs over Zariski neighbourhoods $U^{\prime}$ of $u$ in
$U$ and

$C^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})=\mathop{\varinjlim}\limits_{Y\subset
U^{\prime}}C^{p}_{Y}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$

where $Y$ runs over closed subschemes of $U^{\prime}$ of dimension $d-p$.

We have

$A^{p}(X,M,\mathcal{L}_{X})=\mathop{\varinjlim}\limits_{(U^{\prime},u)}A^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$

where $(U^{\prime},u)$ runs over Zariski neighbourhoods of $u$ in $U$. Thus,
every element of $A^{p}(U^{\prime},{K}^{\rm MW}_{n})$ is represented by an
element in $C^{p}_{Y}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$ for some closed
subscheme $Y$ of $U^{\prime}$ of dimension $d-p$.

We claim that, every element of $A^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$
is represented by an element of
$C^{p}_{Y}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$ where $Y$ runs over all
closed subschemes of $U^{\prime}$ of dimension $d-p$ and $Y$ is flat over $S$.

Let $y\in Y^{(0)}$ such that $y\in U^{\prime}_{s}$, the special fiber of
$U^{\prime}$ over the closed point $s$ of $S$. By [16, Lemma 7.2, page 1622]
there exists an integral closed subscheme $Z\subset U^{\prime}$ of dimension
$d-p+1$ such that

  1. (1)

$Z\cap(U^{\prime}-U_{s})\neq\emptyset$

  2. (2)

$y\in Z$

  3. (3)

$Z$ is regular at $y$.

Let $\alpha\in M(\kappa(y),\mathcal{L}_{U^{\prime}}).$ The set of all
codimension 1 points of $Z$ is given by $Z^{(1)}=Z_{s}^{(1)}\coprod
Z_{\eta}^{(1)}$ where
$Z_{s}^{(1)}:=\bigg{\\{}\overline{\\{y\\}}\bigg{\\}}\cup\bigg{\\{}Z_{1},Z_{2},\cdots
Z_{m}\bigg{\\}}$ where $Z_{i}$’s are pairwise distinct irreducible components
of the closed fiber $Z_{s}$ of $Z$ over the closed point $s$ of $S$ and not
equal to $\overline{\\{y\\}}$ and $Z_{\eta}^{(1)}$ is the collection of
codimension 1 points of $Z$ not in $Z_{s}^{(1)}.$

Let $z$ be the generic point of $Z$. Let $z_{i}$ be the generic points of
$Z_{i}$ for $i=1,\cdots,m$. Let $J_{i}$ be the collection of discrete
valuations on $\kappa(z)$ with center $z_{i}$ for all $i$. Let us denote the
valuation on $\kappa(z)$ associated to $y$ by $v$. Observe that
$\\{v\\}\cup\bigcup_{i}J_{i}$ is a finite collection of distinct non-
equivalent discrete valuations on $\kappa(z)$.

By Chinese remainder theorem, we can choose a uniformizer $\pi$ for $v$ such
that $\pi+1\in\mathfrak{m}_{w}$ for all $w\in\bigcup_{i}J_{i}$, where
$\mathfrak{m}_{w}$ is the maximal ideal for a discrete valuation $w$. Now,
since $M$ satisfies the property R5, there is $\beta\in
M(z,\mathcal{L}_{U^{\prime}})$ such that $\partial_{v}(\beta)=\alpha$ and for
all $w\in\bigcup_{i}J_{i}$, $\partial_{w}(\beta)=0$. Hence, $\forall\ i,$ the
map

$\partial^{z}_{z_{i}}=\sum_{w\in
J_{i}}(\phi^{*})_{\kappa(w)/\kappa(z_{i})}\circ\partial_{w}:M(z,\mathcal{L}_{U^{\prime}})\to
M(z_{i},\mathcal{L}_{U^{\prime}})$

takes $\beta\mapsto 0.$ Then under the differential

$d=\sum_{w\in
Z^{(1)}}\partial^{z}_{w}:M(z,\mathcal{L}_{U^{\prime}})\to\Bigg{(}\bigoplus_{i=1}^{m}M(z_{i},\mathcal{L}_{U^{\prime}})\Bigg{)}\oplus
M(y,\mathcal{L}_{U^{\prime}})\oplus\Bigg{(}\bigoplus_{w\in
Z_{\eta}^{(1)}}M(w,\mathcal{L}_{U^{\prime}})\Bigg{)}$
$d(\beta)=0+\cdots+0+\alpha+\sum_{w\in
Z_{\eta}^{(1)}}\partial^{z}_{w}(\beta).$

We observe that $\sum_{w\in Z_{\eta}^{(1)}}\partial^{z}_{w}(\beta)\in
C^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$ is supported on $w$, where
$\overline{\\{w\\}}$ is a dimension $d-p$ closed subscheme and flat over $S$,
since $w\in Z_{\eta}^{(1)}$. Thus, we have that, any element of
$A^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$ is represented by an element in
$A^{p}_{Y}(U^{\prime},M,\mathcal{L}_{U^{\prime}})$ for some closed subscheme
$Y\subset U^{\prime}$ of codimension $p$ such that $Y$ is flat over $S$. Now

$A^{p}_{Y}(U^{\prime},M,\mathcal{L}_{U^{\prime}})\to
A^{p}(U^{\prime},M,\mathcal{L}_{U^{\prime}})\to
A^{p}(U^{\prime\prime},M,\mathcal{L}_{U^{\prime\prime}})$

is trivial for some $(U^{\prime\prime},u)$ Zariski neighbourhood of $u$ in
$U^{\prime}$, since it factors through

$A^{p}_{Y}(U^{\prime},M,\mathcal{L}_{U^{\prime}})\to
A^{p}_{Y\times_{U^{\prime}}U^{\prime\prime}}(U^{\prime\prime},M,\mathcal{L}_{U^{\prime\prime}})\to
A^{p}(U^{\prime\prime},M,\mathcal{L}_{U^{\prime\prime}})$

which is trivial by Proposition 3.3. ∎

##  4\. Strict $\mathbb{A}^{1}$-invariance

Throughout this section, we let $S$ be as in Theorem 1.1. The goal of this
section is to give a slightly modified construction of a spectral sequence
constructed in [5, 9.2]. We show that this spectral sequence can easily be
constructed using a filtration on the Gersten complex which makes it clear
that the perfectness of the field is not a necessary hypothesis as might
appear while reading the proof of [5, 9.3]. In fact, the spectral sequence
exists over any regular excellent base scheme $S$. As in [5, 9.1], this
spectral sequence will deduce the proof of Theorem 1.3 from the field case.

We first recall the definition of the sheaf theoretic version of the Gersten
complex in Definition 2.5. For a smooth scheme $f:X\to S$ and a graded line
bundle $\mathcal{L}$ on $X$, we let

$\mathcal{C}^{q}(X,M,\mathcal{L})=\bigoplus_{x\in
X^{(q)}}{i_{x}}_{*}M(x,\mathcal{L}).$

where $i_{x}:{\rm Spec\,}(\kappa(x))\to X$ is the inclusion of the point $x\in
X$ and the group $M(x,\mathcal{L}_{X})$ is considered as a Nisnevich sheaf on
${\rm Spec\,}(\kappa(x))$. Note that the Gersten complex
$C^{\bullet}(X,M,\mathcal{L})$ in Definition 2.5 is obtained by applying
global sections functor to $\mathcal{C}^{\bullet}(X,M,\mathcal{L})$. Moreover,
Theorem 1.1 implies that $\mathcal{C}^{\bullet}(X,M,\mathcal{L})$ is an
acyclic resolution of the sheaf $M(X,\mathcal{L})$ in the category of
Nisnevich sheaves on $X$. In particular, we have

$H^{q}_{Nis}(X,M(X,\mathcal{L}))\cong A^{q}(X,M,\mathcal{L})\ \ \forall\ q\geq
0.$

One now defines a decreasing filtration on the complex
$\mathcal{C}^{\bullet}(X,M,\mathcal{L})$ by setting

$F^{p}\mathcal{C}^{q}(X,M,\mathcal{L})=\bigoplus_{\begin{subarray}{c}x\in
X^{(q)}\\\ f(x)\in S^{(\geq p)}\end{subarray}}M(x,\mathcal{L}).$

######  Lemma 4.1.

We have an isomorphism of complexes

$F^{p}\mathcal{C}^{\bullet}(X,M,\mathcal{L})/F^{p+1}\mathcal{C}^{\bullet}(X,M,\mathcal{L})=\bigoplus_{s\in
S^{(p)}}C^{\bullet-p}(X_{s},M,\mathcal{L}_{X_{s}}).$

Hence, we obtain the spectral sequence

$E_{1}^{pq}=\bigoplus_{s\in
S^{(p)}}A^{q}(X_{s},M,\mathcal{L}_{X_{s}})\Longrightarrow
A^{p+q}(X,M,\mathcal{L}).$

###### Proof.

Let $X\xrightarrow{f}S$ denote the structure map. It is elementary to check
that as groups we have an isomorphism of groups

$F^{p}\mathcal{C}^{\bullet}(X,M,\mathcal{L})/F^{p+1}\mathcal{C}^{\bullet}(X,M,\mathcal{L})=\bigoplus_{s\in
S^{(p)}}C^{q-p}(X_{s},M,\mathcal{L}_{X_{s}})$

since both are isomorphic to

$\bigoplus_{s\in S^{(p)}}\Bigg{(}\bigoplus_{x\in
X_{s}^{(q-p)}}M(x,\mathcal{L}_{X})\Bigg{)}$

To see that, this is indeed an isomorphism of complexes, we only need to check
that there are no non-trivial boundary maps from $M(x,\mathcal{L})$ to
$M(y,\mathcal{L})$ in $F^{p}(C^{\bullet})/F^{p+1}(C^{\bullet})$ unless $x$ and
$y$ belong to the same fiber, i.e. we need to show that if $x$ specializes to
$y$ then either $f(x)=f(y)$ or $f(y)\in S^{(\geq p+1)}$. This is clear,
because $f(x)$ speciallizes to $f(y)$ and both cannot have the same
codimension unless $f(x)=f(y)$. ∎

###### Proof of Theorem 1.3.

This is deduced from the spectral sequence (4.1) and the Proposition 2.8(H) as
in [5, Theorem 9.4]. ∎

##  5\. Appendix

In this appendix, we quickly recall facts of canonical sheaves [12] and fix
some notation required in this paper. Nothing in this appendix is new.

###  5.1. Canonical invertible sheaf

Let $S$ be a J-2 [17, Tag 07P7], Noetherian regular scheme of finite Krull
dimension, in particular $S$ could be an excellent scheme as in the previous
sections. In [12, Chapter 6, Definition 4.3] canonical sheaf is defined for
all l.c.i morphisms. For a point $x\in X$, where $X/S$ is of finite type, we
wish to define $\omega_{x/S}$.

######  Definition 5.1.

Consider $\overline{\\{x\\}}$ the closure of $x$ in $X$ with the reduced
induced structure. Now $\overline{\\{x\\}}$ being a reduced, finite type
scheme over $S$, the regular locus $U_{x}:={\rm Reg}(\overline{\\{x\\}})$ is
open and non-empty. The morphism $U_{x}\to S$ is l.c.i. over $S$, since it is
of locally finite type morphism of regular schemes. There is a canonical
invertible sheaf $\omega_{{U_{x}}/S}$ on $U_{x}$ as defined in [12, Chapter 6,
Definition 4.3]. We define

$\omega_{x/S}:=(\omega_{{U_{x}}/S})_{x}\otimes_{\mathcal{O}_{x}}\kappa(x).$

We leave it to the reader to check that this $\omega_{x/S}$ depends only on
the $S$-field $\kappa(x)$. Thus, for any finitely generated $S$-field $F$, we
have a well defined canonical sheaf $\omega_{F/S}$.

######  Remark 5.2.

We note various properties of the canonical sheaf.

  1. (1)

If the given morphism $X\to S$ is l.c.i., then for any regular point $x\in X$
we have

$\omega_{x/S}\simeq(\det\mathfrak{m}_{x}/\mathfrak{m}_{x}^{2})^{\vee}\otimes_{\kappa(x)}((\omega_{X/S})_{x}\otimes_{\mathcal{O}_{X,x}}\kappa(x)).$

  2. (2)

When $S={\rm Spec\,}(k)$ is a field, then for any finitely generated field
extension $F/k$, $\omega_{F/k}\cong\det\Omega_{F/k}$.

  3. (3)

Given field extensions $F\supset K$ of finitely generated $S$-fields, there is
a canonical isomorphism

$\omega_{F/S}\simeq\omega_{F/K}\otimes\omega_{K/S}.$

###  5.2. Graded line bundles

We recall the notation from [4, section 1.3]. Let $X$ be a connected scheme.
The category $\mathcal{G}(X)$ of graded line bundles on $X$ is defined as
follows. The objects are a pair $\mathcal{L}:=(L,a)$ for a line bundle $L$ on
$X$ and an integer $a$. The morphisms are given as follows

(5.1) $\displaystyle{\rm
Hom}_{\mathcal{G}(X)}((L,a),(L^{\prime},a^{\prime}))=\begin{cases}\emptyset&\text{if
$a\neq a^{\prime}$}\\\ {\rm Hom}_{\mathcal{O}_{X}}(L,L^{\prime})&\text{if
$a=a^{\prime}$}\end{cases}$

$\mathcal{G}(X)$ is a symmetric monoidal category with monoidal structure
given as follows:

$(L,a)\otimes(L^{\prime},a^{\prime})=(L\otimes L^{\prime},a+a^{\prime}).$

$\mathcal{G}(X)$ is a Picard category in the sense of [2, §4].

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# StRDAN: Synthetic-to-Real Domain Adaptation Network for Vehicle Re-
identification

Sangrok Lee  MODULABS  Eunsoo Park  MODULABS  Hongsuk Yi  Korea Institute of
Science and Technology Information  Sang Hun Lee Corresponding author Kookmin
University

###### Abstract

Vehicle re-identification aims to obtain the same vehicles from vehicle
images. This is challenging but essential for analyzing and predicting traffic
flow in the city. Although deep learning methods have achieved enormous
progress for this task, their large data requirement is a critical
shortcoming. Therefore, we propose a synthetic-to-real domain adaptation
network (StRDAN) framework, which can be trained with inexpensive large-scale
synthetic and real data to improve performance. The StRDAN training method
combines domain adaptation and semi-supervised learning methods and their
associated losses. StRDAN offers significant improvement over the baseline
model, which can only be trained using real data, for VeRi and CityFlow-ReID
datasets, achieving 3.1% and 12.9% improved mean average precision,
respectively.

##  1 Introduction

Vehicle re-identification (Re-ID) aims to identify the same vehicles that are
captured by various cameras. It is an essential technology for analyzing and
predicting traffic flow in smart cities and uses visual appearance based Re-ID
methods in general. However, vehicle Re-ID is challenging for two reasons.

  * •

Different lighting and complex environments create difficulties with
appearance based vehicle Re-ID, and large apparent variations can be generated
using different cameras.

  * •

Different vehicles can be visually very similar when they are in the same type
category.

Figure 1: Proposed synthetic-to-real domain adaptation method to improve
vehicle re-identification performance. It can be difficult to obtain
meaningful labels for real data, but it is relatively simple for synthetic
data.

Deep learning methods [23, 10, 17] are commonly employed to tackle this
complex vehicle Re-ID task with significant progress. These models extract
features using deep learning networks and distinguish vehicles by comparing
feature distances. However, they require large datasets for training and
improved performance, which rapidly becomes a drawback. Many studies [30] have
confirmed that more training data provides better model performance.
Therefore, data from real environments require considerable annotation
workload. On the other hand, domain adaptation approaches employ inexpensive
synthetic data to replace real data.

This paper explores how to improve model performance using inexpensive
synthetic data (see Fig. 1). We adopted an adversarial domain adaptation
approach [4] where an artificial neural network (ANN) learns the best
discriminating features for classification using real data, while
simultaneously learning indistinguishable features between real and synthetic
data [1] [5]. To implement this concept, we introduce a domain discrimination
layer and associated cross-entropy loss to train the whole network to be
indiscriminative for both domains. We also adopted a semi-supervised learning
method to better exploit specific synthetic data labels, such as color, type,
and orientation. Since these labels only exist for synthetic data, a semi-
supervised learning approach that can handle unlabeled data is appropriate to
improve performance. In training, classification losses for the exclusive
labels are selectively applied depending on the data domain [30]. The proposed
model trained on real and synthetic data from the AI City Challenge using
domain adaptation and semi-supervised learning approaches achieved 12.9%
improvement over the baseline model, which was trained with only real data.

This work proposes a novel synthetic-to-real domain adaptation network StRDAN
framework, with major contributions as follows.

  * •

StRDAN can be successfully trained with inexpensive large-scale synthetic as
well as real data to improve performance.

  * •

We propose a new training approach for StRDAN, combining domain adaptation and
semi-supervised learning methods and corresponding losses.

  * •

StRDAN shows significant improvement over the baseline model for two
significant data sets: VeRi [15] and CityFlow-ReID [25].

##  2 Related Work

This section reviews relevant prior studies regarding vehicle Re-ID and domain
adaptation methods with synthetic data.

Vehicle Re-ID:  Vehicle Re-ID methods generally incorporate contrastive loss
and spatio-temporal features. Previous studies [14, 15, 16] have proposed
several contrastive loss based methods, such as siamese networks, triplet
loss, and metric learning. Liu _et al_. [15] introduced the VeRi dataset,
being the first large scale vehicle Re-ID benchmark. Spatio-temporal features
are critical for performance improvement and have helped vehicle Re-ID studies
to achieve huge progress. Tan _et al_. [23] used spatial-temporal features for
multi-camera vehicle tracking and vehicle Re-ID to win the AI City Challenge
in 2019 [18]. Shen _et al_. [22] proposed a two-stage framework to match
visual appearance based on an long-short term memory (LSTM) based path
inference mechanism.

Domain Adaptation with Synthetic Data:  To overcome the lack of data, Zhou _et
al_. [34, 35] proposed a method to improve Re-ID performance by augmenting
various viewpoint vehicle images with generative adversarial networks (GANs).
Performance significantly reduces when deploying trained models onto new
datasets due to differences among the datasets, commonly called domain bias.
Peng _et al_. [19] proposed a domain adaptation framework to address this
problem, incorporating an image-to-image translation network and an attention
based feature learning network. The VehicleX [29] simulator also leverages
synthetic data and domain randomization to overcome the reality gap [26, 27].
Although Liu _et al_. [12] proposed a domain adaptation method, they only
considered real-to-real domain adaptation. The recently proposed PAMTRI
approach [24] uses synthetic data to improve model performance and has similar
architecture to the proposed StRDAN framework. However, PAMTRI requires
considerable effort to obtain vehicle pose and labels for real data, whereas
StRDAN uses domain adaptation to utilize synthetic data and semi-supervised
learning does not require additional annotation workload. Thus, StRDAN is
somewhat simpler and easier to train.

Figure 2: Proposed synthetic-to-real domain adaptation network architecture
comprising a ResNet-50 backbone for feature extraction and five fully
connected softmax layers for classification, trained using joint and disjoint
losses between synthetic and real data.

##  3 Proposed Synthetic-to-Real Domain Adaptation Network

###  3.1 Datasets

We developed an ANN using real and synthetic vehicle datasets provided for
Track 2 of the 2020 AI City Challenge. The real dataset was the CityFlow-reID
dataset, a subset of CityFlow made available for the Track 2 challenge
comprising 56,277 images for 666 unique vehicles collected from 40 cameras,
with 36,935 images from 333 vehicle identities for training, and 18,290 images
from the other 333 identities for testing. The remaining 1052 images in the
test set were provided as query data.

The synthetic vehicle dataset comprised 192,150 images from 1,362 distinct
vehicles created using the VehicleX [29] synthetic dataset generator, forming
an augmented training set. The synthetic dataset includes vehicle ID, color,
type, and object orientation; whereas the real dataset includes only vehicle
ID. Vehicles were distinguished into 12 colors and 11 types, and orientation
was represented by rotation angle [0, 360) on the horizontal plane.

We trained and evaluated the proposed StRDAN model using the VeRi real dataset
[8] and City Challenge synthetic data to examine validity and robustness for
the approach. The Veri dataset contains over 50,000 images for 776 vehicles
captured by 20 cameras, with the training set containing 37,781 images for 576
vehicles, and the testing set containing 11,579 images for 200 vehicles. The
VeRi dataset includes labels for vehicle color and type.

###  3.2 Overall Architecture

Figure 2 shows the proposed overall StRDAN architecture. The model comprises a
backbone network for feature extraction and multiple fully connected (FC)
softmax layers for classification. Input images are batch sampled in equal
numbers from the real and synthetic datasets. For a mini-batch, n different
vehicle identities are chosen from the real and synthetic datasets,
respectively, then m samples are randomly selected from these chosen images.
Therefore, each batch contains $2\times n\times m$ images.

The backbone network extracts a highly abstracted feature vector (dim = 2048)
from the input image. In principle, any convolution neural network (CNN)
designed for image classification can be used as the backbone network, and a
variety of CNNs have been employed in previous studies, including VGG-
CNN-M1024 [3], MobileNet [9], and ResNet [7], as vehicle Re-ID model backbone.
We selectedResNet-50 as the backbone network for StRDAN. Feature maps
extracted by the backbone network are flattened and fed into various FC
softmax layers to classify vehicle id, real or synthetic, color, type, and
orientation. Outputs are then fed into five cross-entropy loss functions and
one triplet loss function. StRDAN was end-to-end trained by updating the
network parameters to reduce total loss, combining cross-entropy and triplet
losses.

###  3.3 Key Features

Adversarial Domain Adaptation. An annotated dataset is essential for deep
neural network supervised learning. However, collecting and manually
annotating large datasets is time consuming and expensive. Therefore, the
VehicleX approach was introduced in the AI City Challenge to generate
automatically labeled data using a graphic simulator and hence overcome the
dearth of real data. However, synthetic data has similar but different
distributions than real data. Therefore, it is necessary to train the ANN to
predict the classification, regardless of the input domain.

We adopted the adversarial domain adaptation approach, where the ANN learns
features that are most discriminative for classification on the real domain
and simultaneously as indistinguishable as possible between the real and
synthetic domains [1] [5]. To implement this approach, we introduced a domain
discrimination layer and its associated cross-entropy loss train the network
to be indiscriminate to the domains. We also introduced a vehicle-id
classification layer and its associated cross-entropy loss along with triplet
loss to train the network to better discriminate vehicle identities and shape
signatures.

Semi-supervised Learning. In contrast with the real data, synthetic data
includes vehicle type, color, and orientation labels, and we use these labels
under multitask learning to improve generalization performance for all tasks
[31]. Many semi-supervised learning approaches improve learning accuracy by
combining a small amount of labeled data with a large amount of unlabeled data
during training. Zhai _et al_.? s work [30] created artificial labels for
unlabeled and labeled data and utilize them in training, and this approach
inspired us to use joint and disjoint labels between real and synthetic data
to improving performance. Joint labels attached to real and synthetic data are
vehicle ID and domain (real or synthetic), whereas disjoint labels were
attached to only synthetic data and includes vehicle type, color, and
orientation. Losses were classified as joint or disjoint losses, associated
with joint and disjoint labels, respectively, as shown in Fig. 2. Triplet loss
is classified as a joint loss because the vehicle id contributes to
distinguishing batch images into anchor, positive, or negative images.

The semi-supervised learning approach considered here has learning objective

$\operatorname*{min}_{\theta}\;\mathcal{L}_{joint}(\theta)+w\mathcal{L}_{disjoint}(\theta),$
(1)

where $\mathcal{L}_{joint}$ is the joint loss defined in both real and
synthetic domains and $\mathcal{L}_{disjoint}$ is the disjoint loss defined in
the synthetic domain. $\theta$ is parameters of the network. In the next
section, we will describe the losses in more detail.

##  4 Loss Function

###  4.1 Joint Losses

Vehicle ID. Cross-entropy following the softmax function is the most commonly
employed loss for image classification, and can be represented for vehicle ID
classifier, Lid, as follows

$L_{id}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{C}y_{ij}\log(\hat{y}_{ij})$ (2)

where $N$ denotes the number of images in a mini-batch, $C$ is the number of
classes, $y_{ij}$ is the $j^{\mathit{th}}$ element of an one hot encoded
vector for the ground-truth of the $i^{\mathit{th}}$ sample in a mini-batch,
and $\hat{y}_{ij}$ is the $j^{\mathit{th}}$ element of the softmax FC layer
for the $i^{\mathit{th}}$ image.

Domain. We adopted the adversarial domain adaptation approach, with real and
synthetic domains. A softmax FC layer was added to the backbone network for
domain discrimination, with loss function to train the network to be
indiscriminate to the domains,

$L_{domain}=\frac{1}{N}\sum_{i=1}^{N}y_{i}\log(\hat{y}_{i})+(1-y_{i})\log(1-\hat{y}_{i}).$
(3)

Domain discrimination loss is defined as the negative of binary cross-entropy
loss. Since cross-entropy loss trains the network to discriminative between
the domains, the negative loss trains the model to be less discriminating.
Thus, if a vehicle captured by a camera is drawn by a graphic simulator in the
same orientation, features extracted from a synthetic image would be similar
to that from a real image since domain dependent features are suppressed. The
negative cross-entropy loss function was implemented by a gradient reversal
layer [4].

Triplet Loss. In a mini-batch that contains $P$ identities and $Q$ images for
each identity, each image (anchor) has $Q-1$ images of the same identity
(positives) and $(P-1)\times Q$ images of different identities (negatives).
Triplet loss pulls the positive pair ($a$, $p$) together while pushing the
negative pair ($a$, $n$) away by some margin. Thus, triplet loss trains the
network to minimize the distance between features from the same image classes
and simultaneously maximizes distance between features from different image
classes. Triplet loss is defined as [8]

$\begin{split}L_{tri}=\sum_{i=1}^{P}\sum_{a=1}^{Q}\Bigg{[}m&+\operatorname*{max}_{\begin{subarray}{c}p=1\dots
Q\\\ p\neq a\end{subarray}}D(v_{a,i},v_{p,i})\\\
&-\operatorname*{min}_{\begin{subarray}{c}j=1\dots P\\\ n=1\dots Q\\\ j\neq
i\end{subarray}}D(v_{a,i},v_{n,j})\Bigg{]}_{+}\end{split}$ (4)

where $v_{a,i}$ is the prediction vector for the $a^{th}$ image of the
$i^{th}$ identity group, and $m$ is the margin to control the difference
between positive and negative pair distances and helps cluster the
distribution more densely.

###  4.2 Disjoint Losses

Color, Type, and Orientation. Softmax cross-entropy loss was applied for these
three targets. Orientation is continuous and numerical, whereas color and type
are categorical and nominal. Therefore, it seems reasonable to use regression
to predict orientation, but orientation is a difficult problem for regression
due to the wide range of the regression target. Indeed, experiments showed
that optimization would not converge for regression. Therefore, we predicted
orientation as direct classification into n discrete bins, with softmax cross-
entropy loss [33] or [6], dividing the 360 degrees of orientation space into
six bins of 60 degrees. Cross-entropy losses for color, type, and orientation
were only applied to synthetic images and set to zero for real images. The
loss function can be expressed as

$L_{x}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{C}\delta_{i}y_{ij}\log(\hat{y}_{ij}),\;\delta_{i}\in\\{1,0\\},$
(5)

where $x$ is one of color, type, and orientation, and $\delta_{i}$ is a mask
value that is set to 1 if the $i^{th}$ data in a mini-batch has $x$, and 0
otherwise.

##  5 Experiments

###  5.1 Evaluation Metric

We used rank-K mean Average Precision (mAP), the official AI City Challenge
evaluation metric, to evaluate model performances. mAP measures the mean
average precision for each query considering only the top K matches, where we
chose K = 100. Average precision was computed for each query image from the
area under the precision-recall curve, and then the mean of the average
precision over all queries was computed.

###  5.2 Implementation

The chosen backbone network, ResNet-50, was initialized with weights pre-
trained on ImageNet [21] to accelerate training. We trained the model end-to-
end with an AMSGrad optimizer [20] for 60 epochs. Initial learning rate =
0.0003, reduced by 0.1 after 20 and 40 epochs. Weight decay factor for L2
regulation was set = 0.0005, and batch size = 64. For each mini-batch, two
different vehicle-IDs were selected from each of the real and synthetic
datasets, and four images with the same ID were sampled. Therefore, 16
different images with four different IDs from real and synthetic datasets were
sampled. Input images were resized to 128 x 256 pixel, and we employed
horizontal flip and random erasure augmentations. Postprocessing used the re-
ranking algorithm [32], to order the distance matrix between features with
Jaccard and original output distance.

Table 1: Evaluation results of the StRDAN trained with CityFlow-reID and VehicleX datasets for the 2020 AI City Challenge, Track 2. The results are from the official evaluation leaderboard of the challenge.  Case | O | C | T | V | D | Dataset | mAP
---|---|---|---|---|---|---|---
1 |  |  |  | ✓ |  | R | 25.5
2 | ✓ |  |  | ✓ | ✓ | R+S | No Conv.
3 |  | ✓ |  | ✓ | ✓ | R+S | 35.2
4 |  |  | ✓ | ✓ | ✓ | R+S | 38.4
5 | ✓ | ✓ |  | ✓ | ✓ | R+S | 34.1
6 | ✓ |  | ✓ | ✓ | ✓ | R+S | 37.5
7 |  | ✓ | ✓ | ✓ | ✓ | R+S | 35.3
8 | ✓ | ✓ | ✓ | ✓ | ✓ | R+S | 34.0
Table 2: Evaluation results of the StRDAN trained with VeRi and VehicleX
datasets.

Case | O | C | T | V | D | Dataset | mAP
---|---|---|---|---|---|---|---
1 |  |  |  | ✓ |  | R | 73.0
2 | ✓ |  |  | ✓ | ✓ | R+S | 76.1
3 |  | ✓ |  | ✓ | ✓ | R+S | 74.2
4 |  |  | ✓ | ✓ | ✓ | R+S | 74.9
5 | ✓ | ✓ |  | ✓ | ✓ | R+S | 74.7
6 | ✓ |  | ✓ | ✓ | ✓ | R+S | 75.3
7 |  | ✓ | ✓ | ✓ | ✓ | R+S | 74.8
8 | ✓ | ✓ | ✓ | ✓ | ✓ | R+S | 74.6

Notes: O, C, T, V, D = orientation, color, type, vehicle ID, and domain,
respectively.
R, S = real and synthetic data, respectively.
No Conv. = no convergence
mAP = mean average precision
Boxes are checked if target loss was included.
Best result is shown in bold.

Table 3: Comparing Deep learning methods on the VeRi dataset. Method | mAP
---|---
FACT[13] | 18.7
ABLN[35] | 24.9
OIFE[28] | 48.0
PROVID[16] | 48.5
PathLSTM[22] | 58.3
GSTE[2] | 59.5
VAMI[36] | 61.3
BA[11] | 66.9
BS[11] | 67.6
PAMTRI[24] | 71.9
StRDAN (R, baseline) | 73.0
StRDAN (R+S, best) | 76.1

###  5.3 Results and Discussion

We trained and evaluated our models using the CityFlow-reID and VeRi real
datasets along with synthetic data generated by VehicleX, with selected
disjoint losses as shown in Table 1 and Table 2.

Performance on AI City Dataset. The baseline (Case 1) model comprised the
backbone network and vehicle ID classifier. The baseline was trained with the
real dataset using vehicle-ID cross-entropy and triplet losses. Table 1 shows
that the proposed domain adaptation and semi-supervised learning approaches
significantly improved model performance compared with the baseline by at
least 8.5% (Case 8) up to 12.9% (Case 4). The proposed model exhibited best
performance for Case 4, where only vehicle type was considered, whereas Case 8
considered all three labels and exhibited the worst performance.

Performance on VeRi Dataset. Table 2 also shows that domain adaptation and
semi-supervised learning approaches with synthetic dataset and additional
losses helped improve performance by at least 1.2% (Case 3) up to 3.1% (Case
2). In contrast to the AI City dataset, the best performance was when
considering only orientation (Case 2). However, the model could not converge
with the AI City dataset. Veri data model performances were much superior than
those for AI City data. Table 3 compares the proposed StRDAN approach with
other methods. All models were trained using only the VeRi dataset, aside from
PAMTRI and StRDAN (R+S). The proposed StRDAN (R+S) model outperformed all
other considered methods.

Domain Adaptation and Semi-supervised Learning. The experimental results
verified that domain adaptation and semi-supervised learning approaches help
extract more important semantic features for vehicle Re-ID. However, further
research is required regarding unexpected phenomena as follows.

  * •

The best model was trained with only one loss of three disjoint losses.

  * •

Performance degraded with including more disjoint losses.

  * •

Best performance depended on the real dataset.

##  6 Conclusions

This paper proposes using domain adaptation and semi-supervised learning to
fully utilize synthetic data. Experiment results confirmed that increasing
training data via with domain adaptation improved performance. We also showed
that using semi-supervised learning with labels only available for synthetic
data helped the model extract more semantic features.

Future work will investigate the following issues.

  * •

Synergy between disjoint losses and real-world data dependency on disjoint
losses, as discussed above.

  * •

Reality effects on synthetic data. Image data synthesized by VehicleX was far
from realistic and hence easily distinguishable from real image data. More
realistic synthetic data from more sophisticated simulation software could
further improve performance.

  * •

Orientation prediction. We converted orientation regression to six bin
classification, but we have not optimized bin count. Since orientation is a
key feature to identify vehicles captured at various camera angles, proper
orientation representation will also help to improve performance.

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