# Twisted K-theory in motivic homotopy theory

Elden Elmanto  IMJ-PRG and CNRS
SU 4 place Jussieu
Case 247
75252 Paris
France<EMAIL_ADDRESS>https://www.eldenelmanto.com/ ,  Denis Nardin
Fakultät für Mathematik
Universität Regensburg
93040 Regensburg
Deutschland<EMAIL_ADDRESS>https://homepages.uni-regensburg.de/~nad22969/
and  Maria Yakerson  Institute for Mathematical Research (FIM)
ETH Zürich
Rämistr. 101
8092 Zürich
Switzerland<EMAIL_ADDRESS>https://www.muramatik.com

###### Abstract.

In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint.
For a smooth variety $X$ over a field of characteristic zero and an Azumaya
algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic
spectral sequence, by computing the slices of the motivic twisted algebraic
$K$-theory spectrum as a twisted form of motivic cohomology. This generalizes
previous results due to Kahn-Levine where $\mathcal{A}$ is assumed to be
pulled back from a base field. Our methods use interaction between the slice
filtration and birational geometry. Along the way, we prove a representability
result, expressing the motivic space of twisted $K$-theory as an extension of
the twisted Grassmannian by the sheaf of “twisted integers”. This leads to a
proof of cdh descent and Milnor excision for twisted homotopy $K$-theory.

###### Contents

  1. 1 Introduction
  2. 2 Recollection on twisted K-theory
  3. 3 The motivic space of twisted K-theory
  4. 4 The motivic spectrum of twisted K-theory
  5. 5 Twisted motivic cohomology

##  1\. Introduction

Twists naturally occur in mathematics whenever locally-defined entities fail
to glue to a global object. We are particularly interested in twists by Brauer
classes in the context of algebraic geometry. We first recall some of the
cases when such twists appear, see also [Lie13] for more examples.

######  Example 1.1.

Suppose that $p\colon X\rightarrow Y$ is a projective morphism such that all
of its geometric fibers are isomorphic to a projective space of fixed rank.
Examples of such morphisms arise from taking the projectivization of a vector
bundle $\mathcal{E}$ on $Y$. However, there are many more examples: $X$ is a
_Severi-Brauer_ scheme if it is étale-locally on $Y$ isomorphic to
$\mathbf{P}^{n}\times Y$. The failure of $p$ to be a projective bundle is
measured by the associated Brauer class $\alpha\in\mathrm{Br}(Y)$, which is an
étale $2$-cocycle valued in $\mathbf{G}_{\mathrm{m}}$.

######  Example 1.2.

If $X$ is a smooth projective variety over a field, one can consider a moduli
space of sheaves (with stability or semistability conditions) $\mathcal{N}$.
Such a moduli space usually exists a coarse space but may not exist as a fine
moduli space. The problem is that the universal sheaf exists locally but may
not glue together to form a global sheaf. In certain cases, Căldăraru proves
that a Brauer class $\alpha$ on $\mathcal{N}$ obstructs the existence of a
universal sheaf and that a universal $\alpha$-_twisted_ sheaf exists instead.
He further proves a derived equivalence between the perfect complexes on $X$
and twisted perfect complexes on $\mathcal{N}$ [Cal00, Theorem 5.5.1].

The goal of this paper is to improve certain tools for studying twisted
version of algebraic K-theory. For a smooth $k$-scheme $X$ and $\mathcal{A}$
an Azumaya algebra over $X$, the _$\mathcal{A}$ -twisted K-theory_ space
$K^{\mathcal{A}}(X)$ is the K-theory space of the stable $\infty$-category of
$\mathcal{A}$-twisted perfect complexes on $X$. By construction, the
$\mathcal{A}$-twisted K-theory only depends on the equivalence class of
$\mathcal{A}$ in the Brauer group of $X$; in other words, it is _Morita
invariant_. The $\mathcal{A}$-twisted K-theory groups
$K^{\mathcal{A}}_{*}(X)$, defined as the homotopy groups of the space
$K^{\mathcal{A}}(X)$, first appeared in Quillen’s computation of the
$K$-theory of Severi-Brauer varieties [Qui73].

######  Remark 1.3.

The space $K^{\mathcal{A}}(X)$ can be defined more generally for any Brauer
class $\mathcal{A}$, not necessarily representable by an Azumaya algebra.
While in this paper we will concentrate for simplicity on the case when such a
representing object exists, all the proofs will work in the more general case
as well, as they always proceed by reduction to the case where $\mathcal{A}$
is pulled back from a field where all Brauer classes are representable.

When $\mathcal{A}$ is trivial in the Brauer group, one of the main tools for
accessing the $K$-theory groups $K_{*}(X)$ is the _motivic spectral sequence_

(1.4) $E_{2}^{p,q}=CH^{-q}(X;-p-q)\Rightarrow K_{-p-q}(X),$

where $CH^{*}(X;*)$ are Bloch’s _higher Chow groups_. The spectral sequence
(1.4) has many implications: for example, it is a key ingredient in
Voevodsky’s proof of the Quillen-Lichtenbaum conjectures [Voe11]. The spectral
sequence degenerates after rationalization and implies (a variant of) the
Grothendieck-Riemann-Roch theorem:

$K_{0}(X)_{\mathbf{Q}}\simeq\bigoplus^{\dim(X)}_{i=0}CH^{i}(X)_{\mathbf{Q}}.$

Bloch’s higher Chow groups provide a cohomology theory for schemes which
satisfies Beilinson’s desiderata for a theory of “motivic cohomology”; see for
example [FSV00] for the list of properties. Voevodsky’s construction of
motivic cohomology groups compares with Bloch’s higher Chow groups via an
isomorphism

$CH^{q}(X;2q-p)\simeq H^{p}(X;\mathbf{Z}(q))$

for smooth schemes over a perfect field [Voe02a]. In light of the last
isomorphism, the main result of this paper is an extension of (1.4) to the
twisted setting.

######  Theorem 1.5 (Corollary 5.18).

Let $k$ be a field of characteristic zero, $X$ a smooth $k$-scheme and
$\mathcal{A}$ an Azumaya algebra over $X$. Then there is a strongly convergent
spectral sequence

(1.6)
$E^{p,q}_{2}=H^{p-q}_{\mathrm{mot}}(X;\mathbf{Z}^{\mathcal{A}}(-q))\Rightarrow
K_{-p-q}^{\mathcal{A}}(X),$

where $E_{2}$-terms are the $\mathcal{A}$-twisted motivic cohomology groups.
For a field $k$ of characteristic $p$, the same result holds after inverting
$p$.

When $\mathcal{A}$ is pulled back from the base field $k$, the spectral
sequence (1.6) is one of the main results of the work of Kahn and Levine on
twisted K-theory [KL10, Theorem 1]. In this case, the $\mathcal{A}$-twisted
motivic cohomology groups have a cycle-theoretic interpretation. Namely, they
can be defined as $\mathcal{A}$-twisted higher Chow groups: the generators are
algebraic cycles labelled by “$\mathcal{A}$-twisted integers” (see Proposition
5.15). The sheaf of $\mathcal{A}$-twisted integers $\mathbf{Z}^{\mathcal{A}}$
is the sheafification of the presheaf $K_{0}^{\mathcal{A}}$, and the twist
encodes the index of $\mathcal{A}$. In the case of a general $\mathcal{A}$,
the cycle-theoretic description holds locally, and we expect that it applies
globally as well (see Remark 5.16) and that the spectral sequence (1.6)
coincides with the spectral sequence of [KL10, Corollary 6.1.4].

The spectral sequence (1.6) results from the computation of slices of the
motivic spectrum $\mathrm{KGL}^{\mathcal{A}}$, representing twisted algebraic
K-theory. Under the assumptions of Theorem 1.5, we compute the slices $s_{q}$
with respect to Voevodsky’s slice filtration as follows (see Theorem 5.13):

(1.7)
$s_{q}\mathrm{KGL}^{\mathcal{A}}\simeq\Sigma^{q}_{\mathbf{P}^{1}}H\mathbf{Z}^{\mathcal{A}}\qquad\forall
q\in\mathbb{Z}.$

Here $H\mathbf{Z}^{\mathcal{A}}$ is the $\mathcal{A}$-twisted motivic
cohomology spectrum, which we construct in Definition 5.1. When the Brauer
class of $\mathcal{A}$ is trivial, the spectrum $H\mathbf{Z}^{\mathcal{A}}$
coincides with Spitzweck’s motivic cohomology spectrum [Spi18], which, in
turn, represents Bloch’s higher Chow groups for smooth schemes over a field.

The computation of slices of twisted K-theory in [KL10, Theorem 1] goes
through the twisted version of Levine’s homotopy coniveau tower [Lev08]. We
simplify some of the crucial steps in the proof by means of the techniques of
Bachmann and the first author developed in [BE21], which make use of
birational geometry. We elaborate more on the differences between our approach
and [KL10] in the end of the Introduction.

An important ingredient in our proof is representability of twisted K-theory,
which has its own implications. One of the main results of the foundational
work of Morel and Voevodsky on $\mathbf{A}^{1}$-homotopy theory [MV99] was the
representability of algebraic K-theory motivic space by the ind-smooth scheme
$\mathbf{Z}\times\mathrm{Gr}_{\infty}$, where $\mathrm{Gr}_{\infty}$ is the
infinite Grassmannian. This result allows to deduce that the motivic K-theory
spectrum $\mathrm{KGL}$ is stable under pullback, which in turn implies that
homotopy $K$-theory satisfies cdh descent [Cis13] and Milnor excision
[EHIK21].

An immediate analogue of representability for twisted K-theory fails, but we
show that there is a fiber sequence of motivic spaces (see Theorem 3.15):

(1.8) $\mathrm{Gr}^{\mathcal{A}}_{\infty}\to
K^{\mathcal{A}}\rightarrow\mathbf{Z}^{\mathcal{A}}.$

This sequence splits Zariski locally, and it splits globally if $\mathcal{A}$
is trivial.

A key geometric ingredient in our computation (1.7) of slices of the motivic
$\mathcal{A}$-twisted K-theory spectrum $\mathrm{KGL}^{\mathcal{A}}$ is
rationality of the ind-scheme $\mathrm{Gr}_{\infty}^{\mathcal{A}}$. Moreover,
we show that the terms of the fiber sequence (1.8) are stable under pulback
and deduce the following result.

######  Theorem 1.9 (Theorem 4.8).

Let $X$ be a regular scheme of finite Krull dimension, and $\mathcal{A}$ an
Azumaya algebra over $X$. Then the $\mathcal{A}$-twisted homotopy K-theory
presheaf of spectra on the category of $X$-schemes satisfies Milnor excision
and cdh descent.

### Vista

Let $X$ be a smooth $k$-variety with the structure morphism $p\colon
X\rightarrow\operatorname{Spec}k$ and $\alpha\in{\mathrm{Br}}(X)$ be a Brauer
class. We can define the _$\alpha$ -twisted motive of $X$_ to be

$M(X,\alpha):=p_{\\#}H\mathbf{Z}^{\alpha}.$

This motivic spectrum is canonically an $H\mathbf{Z}$-module and thus defines
an object in the category of Voevodsky’s motives $\mathbf{DM}(k)$. When
$X=\operatorname{Spec}k$, this object has been studied in [KL10], but it is
new otherwise, and so we can ask for “twisted” analogs of some conjectures and
theorems on smooth varieties with many interesting Brauer classes.

One family of examples of such viarieties is given by $K3$ surfaces; see
[Huy16, Chapter 18] for a discussion on the Brauer-theoretic aspects of a $K3$
surface. A fascinating result, due to Huybrechts, is that twisted derived
equivalent $K3$ surfaces give rise to isomorphic _rational_ Chow motives
[Huy19, Huy18]. Since twists disappear after rationalization, one can then
wonder if there is an integral refinement of Huybrechts’ results.

######  Question 1.10.

If $(X,\alpha)$ and $(X^{\prime},\alpha^{\prime})$ are two twisted $K3$
surfaces with equivalent categories of twisted perfect complexes, are their
motives $M(X,\alpha)$ and $M(X^{\prime},\alpha^{\prime})$ equivalent?

Answering Question 1.10 requires a better understanding of the twisted
spectral sequence (1.4), since $(X,\alpha)$ and $(X^{\prime},\alpha^{\prime})$
have isomorphic twisted $K$-theory groups. It is subject to further
investigation by the authors.

### Comparison with other works

Let us elaborate on the comparison of Theorem 1.5 with [KL10, Theorem 1], when
the Azumaya algebra $\mathcal{A}$ is pulled back from the base field $k$. Then
the statement of [KL10, Theorem 1] differs in two ways. First, in [KL10] the
slices are computed in the category of $\mathbf{S}^{1}$-spectra
$\mathbf{SH}^{\mathbf{S}^{1}}(k)$, i.e., the category of
$\mathbf{A}^{1}$-invariant, Nisnevich sheaves of spectra, rather than the
category of motivic $\mathbf{P}^{1}$-spectra. Second, their result only
assumes that $k$ is a perfect field, without inverting the characteristic.

In Theorem 5.7 we give a different proof of the original result of Kahn-Levine
in the same generality as the original: there is no assumption on the
characteristic of $k$, as long as $\mathcal{A}$ is pulled back from the base
field. To get a statement in the form presented in [KL10], i.e. on the level
of $\mathbf{S}^{1}$-spectra, one can use [Voe02b, Conjecture 3], which asserts
a compatibility of slice towers under the canonical functor
$\omega^{\infty}\colon\mathbf{SH}(k)\rightarrow\mathbf{SH}^{\mathbf{S}^{1}}(k)$.
This conjecture was proved by Levine [Lev06, Theorem 9.0.3] and has a
different proof in [BE21, Theorem 17] in the flavour of the present paper.

One of the main points of the present work is to emphasize that the machinery
of motivic homotopy theory provides a rather simple candidate for
$\mathcal{A}$-twisted motivic cohomology, see Definition 5.1. The cohomology
groups appearing in Theorem 1.5 (and its proof) do not make reference to
Bloch’s cycle complexes, which present real technical challenges in the
theory. In particular, our proof avoids the key localization lemma [KL10,
Lemma 6.3.1]. On the other hand, when the twist $\mathcal{A}$ is trivial, our
definition coincides with all other definitions of motivic cohomology (this
latter statement, unfortunately, does not avoid the use of moving lemmas).

Another approach for twisting the motivic K-theory spectrum was introduced by
Spitzweck and Østvær in [SØ12]. Their approach closely follows the formalism
of twisted cohomology theories in topology. In particular, they constructed a
version of of the twisted motivic spectral sequence in [SØ12, Theorem 5.17].
However, their formalism only works for twists by classes in the group
$H_{\mathrm{mot}}^{3}(X;\mathbf{Z}(1))$. For a general motivic space $X$, this
group might not be trivial (for example, for the motivic sphere
$\Sigma\mathbf{P}^{1}$), but it is trivial whenever $X$ is a smooth scheme
over a perfect field. It would be interesting to compare and unify their
approach to twisted K-theory with ours.

### Summary

In Section 2 we recall the formation of twisted algebraic $K$-theory, as well
as the necessary prerequisites on Brauer groups and Azumaya algebras. In
Section 3, we study the motivic $K$-theory space and its representability via
the twisted Grassmannian. This leads us to Section 4, where the motivic
twisted K-theory spectrum is introduced, and the cdh descent and Milnor
excision for twisted homotopy K-theory are proved. Finally, in Section 5 we
compute the slices of twisted K-theory and obtain the twisted motivic spectral
sequence.

### Acknowledgments

This project has originally started together with Ben Antieau, and we have
benefitted a lot from his help. We also thank sincerely Tom Bachmann, Marc
Hoyois, Daniel Huybrechts, Marc Levine, and David Rydh for their useful
comments and suggestions. Yakerson was supported by the Hermann-Weyl-
Instructorship and is grateful to the Institute of Mathematical Research (FIM)
and to ETH Zürich for providing perfect working conditions. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No.
101001474).

### Notation

$S$ denotes the quasi-compact quasi-separated base scheme, $\mathrm{Sch}_{S}$
denotes the category of $S$-schemes, $\mathrm{Sm}_{S}$ that of smooth
$S$-schemes. We denote by $\mathrm{PSh}{}(\mathrm{Sm}_{S})$ the
$\infty$-category of presheaves of spaces on $\mathrm{Sm}_{S}$,
$\mathbf{H}(S)$ the $\infty$-category of motivic spaces over $S$, and
$\mathbf{SH}(S)$ that of motivic spectra, see [BH21, §2.2 and §4.1]. The Tate
sphere $\mathbf{T}\in\mathbf{H}(S)$ is defined as
$\mathbf{A}^{1}/(\mathbf{A}^{1}-0)\simeq\Sigma\mathbf{G}_{\mathrm{m}}\simeq(\mathbf{P}^{1},\infty)$.

We denote by $L_{\mathrm{Zar}}$ and $L_{\mathrm{Nis}}$ the Zariski and
Nisnevich sheafifications on $\mathrm{PSh}{}(\mathrm{Sm}_{S})$ respectively.
The $\mathbf{A}^{1}$-localization
$L_{\mathbf{A}^{1}}\colon\mathrm{PSh}{}(\mathrm{Sm}_{S})\to\mathrm{PSh}{}(\mathrm{Sm}_{S})$
(resp. motivic localization
$L_{\mathrm{mot}}\colon\mathrm{PSh}{}(\mathrm{Sm}_{S})\to\mathrm{PSh}{}(\mathrm{Sm}_{S})$)
is the localization functor onto the full subcategory of
$\mathbf{A}^{1}$-invariant presheaves of spaces (resp. of
$\mathbf{A}^{1}$-invariant Nisnevich sheaves). A morphism $f$ in
$\mathrm{PSh}{}(\mathrm{Sm}_{S})$ is called an $\mathbf{A}^{1}$-equivalence
(resp. a motivic equivalence) if $L_{\mathbf{A}^{1}}(f)$ (resp.
$L_{\mathrm{mot}}(f)$) is an equivalence.

##  2\. Recollection on twisted K-theory

In this section we recall the main definitions related to twisted algebraic
K-theory.

Let $R$ be a commutative ring. Then an associative algebra $A$ over $R$ has a
structure of left module over the algebra $A\otimes_{R}A^{\mathrm{op}}$ given
by $(x\otimes y)\cdot z=xzy$. In particular we have an associative algebra
homomorphism
$A\otimes_{R}A^{\mathrm{op}}\to\operatorname{\mathrm{End}}_{R}(A)$.

######  Proposition 2.1.

Let $R$ be a commutative ring and $A$ an associative algebra over $R$. Then
the following are equivalent:

  1. (1)

$A$ is a finitely generated projective module of positive rank and the map
$A\otimes_{R}A^{\mathrm{op}}\to\operatorname{\mathrm{End}}_{R}(A)$ is an
isomorphism;

  2. (2)

$A$ is a finitely generated projective module of positive rank, it is
projective as an $A\otimes A^{\mathrm{op}}$-module and its center is equal to
$R$;

  3. (3)

$A$ is a finitely generated projective $R$-module and for any map
$R\to\mathcal{O}$ where $\mathcal{O}$ is a strict henselian ring, there exists
a $d\geqslant 1$ and an isomorphism of $\mathcal{O}$-algebras
$A\otimes_{R}\mathcal{O}\simeq\mathrm{M}_{d}(\mathcal{O})$;

  4. (4)

$A$ is a finitely generated projective $R$-module and for every map $R\to k$
where $k$ is a separably closed field, there exists an integer $d\geqslant 1$
and an isomorphism of $k$-algebras $A\otimes_{R}k\simeq\mathrm{M}_{d}(k)$;

  5. (5)

$A$ is a finitely generated projective $R$-module and for every map $R\to k$
where $k$ is a field, the algebra $A\otimes_{R}k$ is a central simple algebra.

We say that an algebra $A$ is an Azumaya algebra if it satisfies any of the
above conditions.

###### Proof.

The equivalence of (1) and (2) is [DMI06, Theorem 3.4]. The equivalence of (2)
with (3), (4) and (5) is [Mil80, Proposition IV.1.2] together with the fact
that property (2) is étale-local, and the Artin-Wedderburn theorem. ∎

######  Definition 2.2.

Let $X$ be a scheme. An _Azumaya algebra_ over $X$ is a sheaf of quasicoherent
$\mathcal{O}_{X}$-algebras $\mathcal{A}$ such that for every open affine
$\operatorname{Spec}R\subseteq X$ the $R$-algebra $\mathcal{A}(R)$ is Azumaya.
Equivalently, $\mathcal{A}$ is Zariski-locally an étale twisted form of
$\mathrm{M}_{d}(\mathcal{O}_{X})$. The integer $d$ is called the _degree_ of
the Azumaya algebra (and it is constant on each connected component of $X$).

We denote by $\mathcal{A}\mathrm{z}(X)$ (resp. $\mathcal{A}\mathrm{z}_{d}(X)$)
the groupoid of Azumaya algebras over $X$ (resp. of degree $d$) with
$\mathcal{O}_{X}$-algebra isomorphisms and by $\mathcal{A}\mathrm{z}$ the
corresponding stack.

Recall that the group scheme $\mathrm{PGL}_{d}$ is defined as the quotient of
$\mathrm{GL}_{d}$ by its center $\mathbf{G}_{\mathrm{m}}$. By Skolem-Noether
theorem [Mil80, Proposition IV.1.4], the action by conjugation
$\mathrm{GL}_{d}\rightarrow\operatorname{\mathrm{Aut}}_{\mathrm{Alg}}(\mathrm{M}_{d}(\mathcal{O}))$
factors through $\mathrm{PGL}_{d}$ and induces an isomorphism
$\mathrm{PGL}_{d}\simeq\operatorname{\mathrm{Aut}}_{\mathrm{Alg}}(\mathrm{M}_{d}(\mathcal{O}))$
of group schemes. This fact implies the following presentation of the groupoid
of Azumaya algebras on $X$, aka étale twisted forms of
$\mathrm{M}_{d}(\mathcal{O}_{X})$.

######  Proposition 2.3.

The natural map

$\mathrm{Map}(X,\mathrm{B}_{\mathrm{\acute{e}t}}\mathrm{PGL}_{d})\rightarrow\mathcal{A}\mathrm{z}_{d}(X)$

is an equivalence of groupoids, where the left side denotes the mapping space
in étale sheaves of spaces.

In particular, we get that

$\mathbf{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathrm{PGL}_{d})\simeq\pi_{0}\mathcal{A}\mathrm{z}_{d}(X)$

is the set of isomorphism classes of Azumaya algebras and

$\mathrm{PGL}_{d}(X)=\mathbf{H}^{0}_{\mathrm{\acute{e}t}}(X,\mathrm{PGL}_{d})\simeq\pi_{1}(\mathcal{A}\mathrm{z}_{d}(X),\mathrm{M}_{d}(\mathcal{O}_{X}))$

is the group of automorphisms of the trivial Azumaya algebra
$\mathrm{M}_{d}(\mathcal{O}_{X})$.

######  Definition 2.4.

Two Azumaya algebras $\mathcal{A}$ and $\mathcal{B}$ on $X$ are called _Brauer
equivalent_ if there are vector bundles $\mathcal{E}$ and $\mathcal{F}$ on $X$
such that
$\mathcal{A}\otimes_{\mathcal{O}}\operatorname{\mathrm{End}}(\mathcal{E})\simeq\mathcal{B}\otimes_{\mathcal{O}}\operatorname{\mathrm{End}}(\mathcal{F})$
as $\mathcal{O}_{X}$-algebras. The _Brauer group_ ${\mathrm{Br}}(X)$ is the
quotient of the set of isomorphism classes of Azumaya algebras on $X$ by
Brauer equivalence. We write $[\mathcal{A}]$ for the class of $\mathcal{A}$ in
${\mathrm{Br}}(X)$.

######  Definition 2.5.

An _index_ of an Azumaya algebra $\mathcal{A}$ over $X$, denoted
$\mathrm{ind}_{X}(\mathcal{A})$, is the gcd of degrees of Azumaya algebras
over $X$ that are Brauer equivalent to $\mathcal{A}$ (for a non-connected
scheme, the index is defined componentwise).

The central extension

$1\rightarrow\mathbf{G}_{\mathrm{m}}\rightarrow\mathrm{GL}_{d}\rightarrow\mathrm{PGL}_{d}\rightarrow
1$

induces a sequence

$\mathbf{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathrm{GL}_{d})\rightarrow\mathbf{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathrm{PGL}_{d})\xrightarrow{\delta}\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}}).$

This sequence is exact in the sense that the fiber of the map
$\delta:\mathbf{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathrm{PGL}_{d})\rightarrow\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}})$
over $0\in\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}})$ is
the image of $\mathbf{H}^{1}_{\mathrm{\acute{e}t}}(X,\mathrm{GL}_{d})$.
Moreover, by [Ser94, Proposition 44] there is an injective map

${\mathrm{Br}}(X)\hookrightarrow\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}}).$

An Azumaya algebra of degree $d$ is $d$-torsion in the Brauer group. Hence if
$X$ is quasicompact, ${\mathrm{Br}}(X)$ is a torsion subgroup of
$\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}})$, and thus
have an inclusions:

${\mathrm{Br}}(X)\hookrightarrow{\mathrm{Br}}^{\prime}(X):=\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}})_{\mathrm{tors}}\hookrightarrow\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}}).$

Next, we define twisted versions of vector bundles, perfect complexes etc. By
abuse of notation, for $f\colon Y\to X$ and
$\mathcal{A}\in\mathcal{A}\mathrm{z}(X)$ we denote
$f^{*}(\mathcal{A})\in\mathcal{A}\mathrm{z}(Y)$ also by $\mathcal{A}$, when
$f$ is clear from the context.

######  Definition 2.6.

Let $\mathcal{A}$ be an Azumaya algebra over a scheme $X$. We define the
category of _$\mathcal{A}$ -twisted sheaves on $X$_ as

$\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}(X)=\mathrm{Mod}_{\mathcal{A}}(\mathcal{Q}\mathrm{Coh}(X))$

(by convention, $\mathrm{Mod}_{\mathcal{A}}$ stands for left modules).

By Grothendieck’s descent theorem, the map

$f\colon Y\to X\,\mapsto\,\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}(Y)$

is an étale sheaf of categories on $\mathrm{Sch}_{S}$, which gives the stack
of categories $\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}$.

######  Remark 2.7.

Let $\alpha\in\mathbf{H}^{2}_{\mathrm{\acute{e}t}}(X,\mathbf{G}_{\mathrm{m}})$
be a class, then we can associate to it a $\mathbf{G}_{\mathrm{m}}$-gerbe
$\mathcal{X}\rightarrow X$. Lieblich has defined an $\alpha$-twisted sheaf as
a quasicoherent sheaf on $\mathcal{X}$ such that the inertial action (on the
left) of $\mathbf{G}_{\mathrm{m}}$ agrees with the
$\mathcal{O}_{\mathcal{X}}$-module structure. The comparison between
Definition 2.6 and his definition can be parsed together from the comparison
results in Lieblich’s thesis [Lie04, Section 2.1.3] and in Căldăraru’s [Cal00,
Theorem 1.3.7].

Let $f\colon Y\to X$ be an étale cover of $X$ such that
$f^{*}\mathcal{A}\simeq M_{d}(\mathcal{O}_{Y})$. Then there is an equivalence

(2.8)
$\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}(Y)\simeq\mathcal{Q}\mathrm{Coh}(Y),$

given by sending an $M_{d}(\mathcal{O}_{Y})$-module $M$ to
$(\mathcal{O}_{Y}^{\oplus d})^{\vee}\otimes_{M_{d}(\mathcal{O}_{Y})}M$.

######  Definition 2.9.

The groupoid of _$\mathcal{A}$ -twisted vector bundles on $X$_, denoted
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}(X)$, is given by the
$\mathcal{A}$-twisted sheaves on $X$ that are étale-locally free, and their
isomorphisms. Precisely, $V\in\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}(X)$ is an
$\mathcal{A}$-twisted vector bundle if there is an étale cover $f\colon Y\to
X$ such that $f^{*}\mathcal{A}\simeq M_{d}(\mathcal{O}_{Y})$ and
$f^{*}V\in\mathcal{Q}\mathrm{Coh}(Y)$ is a vector bundle on $Y$, under the
identification (2.8).

Similarly, the category of _$\mathcal{A}$ -twisted perfect complexes on $X$_,
denoted $\mathcal{P}\mathrm{erf}^{\mathcal{A}}(X)$, is given by complexes of
$\mathcal{A}$-twisted sheaves on $X$ that are étale-locally perfect. These
data form the corresponding stacks $\mathcal{V}\mathrm{ect}^{\mathcal{A}}$ and
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}$.

######  Remark 2.10.

$\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}$ can be interpreted as an étale twisted
form of $\mathcal{Q}\mathrm{Coh}$. Indeed, since there is an isomorphism of
$\mathcal{Q}\mathrm{Coh}$-modules:

$\operatorname{\mathrm{Aut}}_{\mathcal{Q}\mathrm{Coh}}(\mathcal{Q}\mathrm{Coh})\simeq\mathrm{Pic},$

the étale twisted forms of $\mathcal{Q}\mathrm{Coh}$ are in bijection with
$H^{1}(S,\mathrm{Pic})\simeq H^{2}(S,\mathbf{G}_{\mathrm{m}})$, and
$\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}$ corresponds to $[\mathcal{A}]\in
H^{2}(S,\mathbf{G}_{\mathrm{m}})$. Similarly,
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}$ and
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}$ are étale twisted forms of
$\mathcal{V}\mathrm{ect}$ and $\mathcal{P}\mathrm{erf}$ respectively.

######  Remark 2.11.

If $\mathcal{A}$ and $\mathcal{B}$ are Brauer equivalent Azumaya algebras,
they are Morita equivalent, i.e.
$\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}\simeq\mathcal{Q}\mathrm{Coh}^{\mathcal{B}}$.
Hence the stacks $\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}$,
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}$,
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}$ only depend on the Brauer equivalence
class of $\mathcal{A}$, as well as the $\mathcal{A}$-twisted K-theory and the
related constructions. We twist with respect to a representative in the Brauer
equivalence class only for the sake of convenience.

The following lemma will be important for studying $\mathcal{A}$-twisted
K-theory.

######  Lemma 2.12.

Let $V$ be an $\mathcal{A}$-twisted sheaf. Then the following are equivalent:

  1. (1)

$V$ is an $\mathcal{A}$-twisted vector bundle;

  2. (2)

For every affine subscheme $i\colon U\hookrightarrow X$ the
$\mathcal{A}(U)$-module $V(U)$ is finitely generated and projective;

  3. (3)

$V$ is locally free of finite rank as an $\mathcal{O}_{X}$-module;

###### Proof.

The equivalence between (2) and (3) is [DMI06, Proposition II.2.3]. The
equivalence of (1) with (3) follows because the property of being locally free
of finite rank is fpqc-local by [Stacks, 05B2]. ∎

With these definitions at hand, we can define $\mathcal{A}$-twisted K-theory.

######  Definition 2.13.

Let $\mathcal{A}$ be an Azumaya algebra over a qcqs scheme $S$. The
_$\mathcal{A}$ -twisted K-theory_ is the presheaf of spaces on
$\mathrm{Sm}_{S}$ defined by

$K^{\mathcal{A}}(U)=K(\mathcal{P}\mathrm{erf}^{\mathcal{A}}(U)),$

where $\mathcal{P}\mathrm{erf}^{\mathcal{A}}(U)$ is considered as a Waldhausen
category.

Alternatively, $K^{\mathcal{A}}$ can be characterized as the Zariski sheaf of
spaces that on affine S-schemes is given by
$K^{\mathcal{A}}(U)=\mathcal{V}\mathrm{ect}^{\mathcal{A}}(U)^{\mathrm{gp}}$,
where $\mathrm{gp}$ stands for group completion (with respect to direct sum on
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}(U)$). Such a characterization is
possible because $\mathcal{V}\mathrm{ect}^{\mathcal{A}}$ is a Zariski sheaf by
Lemma 2.12.

We define the _$\mathcal{A}$ -twisted K-theory groups_ of a smooth $S$-scheme
$U$ as

$K^{\mathcal{A}}_{i}(U)=\pi_{i}K^{\mathcal{A}}(U).$

######  Remark 2.14.

$K^{\mathcal{A}}$ is naturally a module over the presheaf of
${\mathcal{E}_{\infty}}$-ring spaces $K$, since
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}$ is a $\mathcal{P}\mathrm{erf}$-module
via tensor product.

######  Definition 2.15.

We define the _$\mathcal{A}$ -twisted rank map_ as the map of presheaves of
spaces

$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon\mathcal{V}\mathrm{ect}^{\mathcal{A}}\to\mathbf{Z}\qquad
P\mapsto\sqrt{\operatorname{\mathrm{rk}}_{\mathcal{O}}\operatorname{\mathrm{End}}_{\mathcal{A}}(P)}$

that sends a locally free $\mathcal{A}$-module $P$ to the square root of the
rank of $\operatorname{\mathrm{End}}_{\mathcal{A}}(P)$ as
$\mathcal{O}$-module. This is well-defined because
$\mathcal{Q}\mathrm{Coh}^{\mathcal{A}}$ is a $\mathcal{Q}\mathrm{Coh}$-module
hence enriched in $\mathcal{Q}\mathrm{Coh}$, and when $\mathcal{A}\simeq
M_{n}(\mathcal{O})$ it coincides with the usual rank under the identification
(2.8) . In particular the map $\operatorname{\mathrm{rk}}^{\mathcal{A}}$ takes
values in the sheaf $\mathbf{Z}$ of integers, because étale-locally it gives
an integer. Note that
$\operatorname{\mathrm{rk}}^{\mathcal{A}}(\mathcal{A})=d$ for $\mathcal{A}$ an
Azumaya algebra of degree $d$.

The stack $\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{m}$ of _$\mathcal{A}$
-twisted vector bundles of rank $m$_ is defined as the fiber of
$\operatorname{\mathrm{rk}}^{\mathcal{A}}-m\colon\mathcal{V}\mathrm{ect}^{\mathcal{A}}\to\mathbf{Z}$.

Since $\mathbf{Z}$ is a sheaf of groups, the monoid map
$\operatorname{\mathrm{rk}}^{\mathcal{A}}$ factors through $K^{\mathcal{A}}$,
and we get

$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon K^{\mathcal{A}}\to\mathbf{Z},$

which in turn factors through $K_{0}^{\mathcal{A}}$.

######  Definition 2.16.

The sheaf of _$\mathcal{A}$ -twisted integers_ is the following subsheaf of
$\mathbf{Z}$:

$\mathbf{Z}^{\mathcal{A}}=L_{\mathrm{Zar}}\operatorname{\mathrm{Im}}(\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon
K^{\mathcal{A}}\to\mathbf{Z}).$

######  Lemma 2.17.

The induced map of presheaves

$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon
K_{0}^{\mathcal{A}}\to\mathbf{Z}^{\mathcal{A}}$

is a Zariski-local equivalence. Moreover, for $R$ a local ring, the inclusion
$K_{0}^{\mathcal{A}}(R)\simeq\mathbf{Z}^{\mathcal{A}}(R)\subset\mathbf{Z}$ is
given by

$\mathrm{ind}_{R}(\mathcal{A})\cdot\mathbf{Z}\subset\mathbf{Z}.$

###### Proof.

By Morita invariance of $K_{0}^{\mathcal{A}}$ and [DeM69, Corollary 1], the
map $\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon
K_{0}^{\mathcal{A}}\to\mathbf{Z}$ is an injective homomorphism on local rings,
hence it identifies $L_{\mathrm{Zar}}K_{0}^{\mathcal{A}}$ with
$\mathbf{Z}^{\mathcal{A}}$. Over a local ring $R$, the image is given by
$\mathrm{ind}_{R}(\mathcal{A})\cdot\mathbf{Z}\subset\mathbf{Z}$ again by
[DeM69, Corollary 1]. ∎

######  Corollary 2.18.

The sheaf inclusion $\mathbf{Z}^{\mathcal{A}}\subset\mathbf{Z}$ becomes an
isomorphism after tensoring with $\mathbf{Q}$.

By Lemma 2.17, our sheaf $\mathbf{Z}^{\mathcal{A}}$ is isomorphic to the sheaf
(with transfers) $\mathbf{Z}_{\mathcal{A}}$ of [KL10, Section 5.3], and our
subsheaf inclusion $\mathbf{Z}^{\mathcal{A}}\subset\mathbf{Z}$ is the reduced
norm map $\mathrm{Nrd}$ of [KL10, Lemma 5.2.1]. The following properties of
$\mathbf{Z}^{\mathcal{A}}$ are proved in [KL10].

######  Lemma 2.19.

Let $S$ be regular of finite type. Then $\mathbf{Z}^{\mathcal{A}}$ is an
$\mathbf{A}^{1}$-invariant Nisnevich sheaf on $\mathrm{Sm}_{S}$. Moreover,
$\mathbf{Z}^{\mathcal{A}}$ is a birational sheaf, i.e. locally-constant in the
Zariski topology.

###### Proof.

This is [KL10, Lemma 5.1.1]. ∎

######  Lemma 2.20.

Let $S$ be regular of finite type. Then the sheaf $\mathbf{Z}^{\mathcal{A}}$
has a canonical structure of Voevodsky’s transfers, and
$\mathbf{Z}^{\mathcal{A}}\subset\mathbf{Z}$ is an inclusion of sheaves with
transfers on $\mathrm{Sm}_{S}$.

###### Proof.

This is [KL10, Section 5.3]. ∎

##  3\. The motivic space of twisted K-theory

Let $S$ be a quasi-compact quasi-separated base scheme, $\mathcal{A}$ an
Azumaya algebra of degree $d$ over $S$. We want to understand the motivic
homotopy type of the twisted K-theory space $K^{\mathcal{A}}$, which is a
sheaf on $\mathrm{Sm}_{S}$, given on any affine $S$-scheme
$\operatorname{Spec}R$ by
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)^{\mathrm{gp}}$. First, we show that
it is indeed a motivic space when $S$ is regular.

######  Lemma 3.1.

Let $X$ be a quasi-compact quasi-separated scheme and $Z\subseteq X$ a closed
subset such that $U=X-Z$ is quasi-compact. Then there is a functorial fiber
sequence of spaces

$K(\mathcal{P}\mathrm{erf}^{\mathcal{A}}(X\textrm{ on }Z))\to
K^{\mathcal{A}}(X)\to K^{\mathcal{A}}(U)\,.$

###### Proof.

The claim follows by tensoring the $\mathcal{P}\mathrm{erf}(X)$-linear exact
sequence

$\mathcal{P}\mathrm{erf}(X\textrm{ on
}Z)\to\mathcal{P}\mathrm{erf}(X)\to\mathcal{P}\mathrm{erf}(U)$

by $\mathcal{P}\mathrm{erf}^{\mathcal{A}}(X)$ and using [Lur17, Theorem
4.8.5.16.(4)]. ∎

######  Proposition 3.2.

The presheaf of spaces $K^{\mathcal{A}}$ satisfies Nisnevich descent. If
moreover $S$ is regular and noetherian of finite Krull dimension, then
$K^{\mathcal{A}}$ is $\mathbf{A}^{1}$-invariant.

###### Proof.

The claim about Nisnevich descent follows from Lemma 3.1 as in [TT90,
Corollary 10.10] by the étale descent for $\mathcal{A}$-twisted perfect
complexes.

Let us prove the $\mathbf{A}^{1}$-invariance. Let $d$ be the degree of
$\mathcal{A}$ and $\mathrm{SB}(\mathcal{A})$ be the Severi-Brauer variety
associated to $\mathcal{A}$. By [Qui73, Theorem 4.1], for every smooth
$S$-scheme $X$ there is an isomorphism

$K(X\times_{S}\mathrm{SB}(\mathcal{A}))\simeq\prod_{i=0}^{d-1}K^{\mathcal{A}^{\otimes
i}}(X)\,.$

Since $\mathrm{SB}(\mathcal{A})$ is a smooth $S$-scheme, the pullback map

$p^{*}\colon K(X\times_{S}\mathrm{SB}(\mathcal{A}))\to
K(\mathbf{A}^{1}_{X}\times_{S}\mathrm{SB}(\mathcal{A}))$

is an equivalence. Since the decomposition is natural, it follows that the
pullback map induces an equivalence on each component, in particular on the
first component, which is what we wanted to prove. ∎

Let $\operatorname{Spec}R$ be a smooth affine $S$-scheme. By Lemma 2.12,
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)$ is the groupoid of finitely
generated projective $\mathcal{A}_{R}$-modules, i.e. summands of free
$\mathcal{A}_{R}$-modules (by the usual abuse of notation, we write further on
$\mathcal{A}$ for $\mathcal{A}_{R}$). Hence, we have

$\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)^{\mathrm{gp}}\simeq\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)[-\mathcal{A}].$

By the McDuff-Segal group completion theorem [MS76] (specifically, we use
[Nik17, Proposition 6]), we get an equivalence of
${\mathcal{E}_{\infty}}$-spaces:

$\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)^{\mathrm{gp}}\simeq\mathrm{sVect}^{\mathcal{A}}(R)^{+},$

where
$\mathrm{sVect}^{\mathcal{A}}(R)=\operatorname*{colim}(\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)\xrightarrow{+\mathcal{A}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}(R)\xrightarrow{+\mathcal{A}}\dots)$
is the groupoid of stable $\mathcal{A}$-twisted vector bundles on
$\operatorname{Spec}R$, and $+$ stands for Quillen’s +-construction.

Recall that +-construction, applied to $\mathrm{sVect}^{\mathcal{A}}(R)$, is
invisible up to $\mathbf{A}^{1}$-homotopy by Robalo’s criterion [Rob15,
Insight 1.1] (which originates from [Voe98, Theorem 4.3]). More precisely, we
have the following result.

######  Proposition 3.3.

On affine schemes, there is a canonical $\mathbf{A}^{1}$-equivalence of
${\mathcal{E}_{\infty}}$-spaces:

$\mathrm{sVect}^{\mathcal{A}}\simeq\mathcal{V}\mathrm{ect}^{\mathcal{A},\mathrm{gp}}.$

###### Proof.

By [BEH+21, Proposition 5.1], it’s enough to show that on any affine
$S$-scheme $\operatorname{Spec}R$ the cyclic permutation of $\mathcal{A}^{3}$
is $\mathbf{A}^{1}$-homotopic to identity. The action of the cyclic group
$C_{3}$ on $\mathcal{A}^{3}$ induces a map of stacks
$BC_{3}\to\mathcal{V}\mathrm{ect}^{\mathcal{A}}$. This map factors through the
stack $B\mathrm{SL}_{3}$, and the group scheme $\mathrm{SL}_{3}$ is
$\mathbf{A}^{1}$-connected, so the claim follows. ∎

######  Corollary 3.4.

There is a canonical $\mathbf{A}^{1}$-equivalence on affines:

$K^{\mathcal{A}}\simeq\mathrm{sVect}^{\mathcal{A}}.$

######  Remark 3.5.

Since twisted K-theory is Morita-invariant, it follows from Corollary 3.4 that
for Brauer equivalent Azumaya algebras $\mathcal{A}$ and $\mathcal{B}$ there
is a canonical $\mathbf{A}^{1}$-equivalence on affines between
$\mathrm{sVect}^{\mathcal{A}}$ and $\mathrm{sVect}^{\mathcal{B}}$.

######  Proposition 3.6.

For any affine scheme $\operatorname{Spec}R$, there is a _non-canonical_
equivalence of spaces

$\mathrm{sVect}^{\mathcal{A}}(R)\simeq K_{0}^{\mathcal{A}}(R)\times
BGL(\mathcal{A}).$

###### Proof.

This follows since $\pi_{0}\mathrm{sVect}^{\mathcal{A}}(R)\simeq
K_{0}^{\mathcal{A}}(R)$, and the automorphism group of any stable
$\mathcal{A}$-twisted vector bundle is isomorphic to $GL(\mathcal{A})$. The
latter is explained for non-twisted vector bundles in [EHK+20, Section 2.1],
and the argument in _loc.cit._ applies verbatim for twisted vector bundles, by
replacing $R$ with $\mathcal{A}_{R}$. ∎

The map of presheaves
$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon\mathcal{V}\mathrm{ect}^{\mathcal{A}}\to\mathbf{Z}^{\mathcal{A}}$
induces a map

$\overline{\operatorname{\mathrm{rk}}}^{\mathcal{A}}\colon\mathrm{sVect}^{\mathcal{A}}\to\mathbf{Z}^{\mathcal{A}}$

that sends $V\in\mathcal{V}\mathrm{ect}^{\mathcal{A}}$ in the $n$-th component
to $\operatorname{\mathrm{rk}}^{\mathcal{A}}(V)-nd$ (this makes sense because
the degree of an Azumaya algebra is stable under base change).

Consider the telescope

$\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}=\operatorname*{colim}(\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{0}\xrightarrow{+\mathcal{A}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{d}\xrightarrow{+\mathcal{A}}\dots).$

We get a fiber sequence of presheaves of spaces:

(3.7)
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}\to\mathrm{sVect}^{\mathcal{A}}\xrightarrow{\overline{\operatorname{\mathrm{rk}}}^{\mathcal{A}}}\mathbf{Z}^{\mathcal{A}}.$

######  Remark 3.8.

In the non-twisted case (i.e. when $\mathcal{A}$ is trivial), this fiber
sequence splits canonically, since there is a map of sheaves
$\mathbf{Z}\to\mathcal{V}\mathrm{ect}$ sending an integer to a trivial vector
bundle of the corresponding rank. However, this construction does not work in
the twisted case. Indeed, while the $\mathcal{A}$-twisted rank map
$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon
K_{0}^{\mathcal{A}}(R)\to\mathbf{Z}^{\mathcal{A}}(R)$ is surjective on all
affine schemes $\operatorname{Spec}R$, the authors do not know if it is
possible to choose the splitting in a natural way, and therefore whether the
map of sheaves $K_{0}^{\mathcal{A}}\to\mathbf{Z}^{\mathcal{A}}$ has a section.

A related phenomenon can be observed in [AW14], where the authors show that
the index is not necessarily the image of an $\mathcal{A}$-twisted vector
bundle (although it will be the image of an $\mathcal{A}$-twisted perfect
complex).

######  Lemma 3.9.

The fiber sequence (3.7) splits Zariski-locally.

###### Proof.

Let $R$ be a local ring. By [DeM69, Corollary 1], there is an Azumaya algebra
$\mathcal{B}$ of degree $\mathrm{ind}_{R}(\mathcal{A})$ in the Brauer
equivalence class of $\mathcal{A}$ on $\operatorname{Spec}R$. Under the Morita
equivalence
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}\simeq\mathcal{V}\mathrm{ect}^{\mathcal{B}}$,
the $\mathcal{B}$-module $\mathcal{B}$ corresponds to some
$\mathcal{A}$-twisted vector bundle $V$ such that
$\operatorname{\mathrm{rk}}^{\mathcal{A}}(V)=\mathrm{ind}_{R}(\mathcal{A})$.
By Lemma 2.17, the subgroup $\mathbf{Z}^{\mathcal{A}}(R)\subset\mathbf{Z}$ is
given by the inclusion
$\mathrm{ind}_{R}(\mathcal{A})\cdot\mathbf{Z}\subset\mathbf{Z}$. Hence we can
define the splitting of the surjection
$\mathrm{sVect}^{\mathcal{A}}(R)\to\mathbf{Z}^{\mathcal{A}}(R)$ by sending
$m\cdot\mathrm{ind}_{R}(\mathcal{A})$ to $V^{m}$. ∎

######  Proposition 3.10.

After applying motivic localization, the sequence (3.7) becomes a fiber
sequence of motivic spaces in $\mathbf{H}(S)$.

###### Proof.

It is enough to show the claim for the fiber sequence

(3.11)
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}\to\mathrm{sVect}^{\mathcal{A}}\xrightarrow{\overline{\operatorname{\mathrm{rk}}}^{\mathcal{A}}}\mathbf{Z}.$

For a local ring $R$, the sequence

$L_{\mathbf{A}^{1}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}(R)\to
L_{\mathbf{A}^{1}}\mathrm{sVect}^{\mathcal{A}}(R)\to
L_{\mathbf{A}^{1}}\mathbf{Z}(R)$

is a fiber sequence by [Rez14, Proposition 5.4], since the sheaf
$L_{\mathbf{A}^{1}}\mathbf{Z}$ is constant in the simplicial direction. Hence,
the sequence (3.11) remains a fiber sequence after applying
$L_{\mathrm{Zar}}L_{\mathbf{A}^{1}}$. Note that
$L_{\mathrm{Zar}}L_{\mathbf{A}^{1}}\mathrm{sVect}^{\mathcal{A}}\simeq
K^{\mathcal{A}}$ is a motivic space, and so is
$L_{\mathrm{Zar}}L_{\mathbf{A}^{1}}\mathbf{Z}\simeq\mathbf{Z}$. Hence, the
fiber
$L_{\mathrm{Zar}}L_{\mathbf{A}^{1}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}$
is also a motivic space, and the claim follows. ∎

Next, we want to replace $\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}$ with
a more geometric object.

######  Definition 3.12.

The _$\mathcal{A}$ -twisted Grassmannian_ is the presheaf
$\mathrm{Gr}^{\mathcal{A}}_{m,n}$ on affine $S$-schemes sending
$\operatorname{Spec}R$ to the set of $\mathcal{F}\subseteq\mathcal{A}^{\oplus
n}_{R}$ of rank $m$ left $\mathcal{A}_{R}$-submodules such that
$\mathcal{A}^{\oplus n}_{R}/\mathcal{F}$ is projective as a left
$\mathcal{A}_{R}$-module. This construction generalizes Severi-Brauer schemes
(in the case $m=1$), and has been studied in $K$-theoretic contexts by various
authors [LSW89, Pan89, Bae16].

######  Lemma 3.13.

The presheaf $\mathrm{Gr}^{\mathcal{A}}_{m,n}$ is represented by a smooth
proper $S$-scheme.

###### Proof.

The Azumaya algebra $\mathcal{A}$ has an underlying locally free sheaf
$\mathcal{E}$. The classical Grassmannian $\mathrm{Gr}_{m,n}(\mathcal{E})$ is
represented by a smooth proper scheme. We observe that
$\mathrm{Gr}^{\mathcal{A}}_{m,n}\subset\mathrm{Gr}_{m,n}(\mathcal{E})$ is a
closed subfunctor, hence it is represented by a proper subscheme. This scheme
is smooth, because smoothness is an étale-local condition on the target of the
forgetful map $\mathrm{Gr}^{\mathcal{A}}_{m,n}\to S$, and the classical
Grassmannian $\mathrm{Gr}_{m,n}$ is smooth. ∎

We define
$\mathrm{Gr}^{\mathcal{A}}_{m,\infty}=\operatorname*{colim}_{n}\mathrm{Gr}^{\mathcal{A}}_{m,n}$,
with maps in the colimit induced by $(\mathrm{id},0)\colon\mathcal{A}^{\oplus
n}\hookrightarrow\mathcal{A}^{\oplus(n+1)}$. We stabilize further and set

$\mathrm{Gr}^{\mathcal{A}}_{\infty}=\operatorname*{colim}(\mathrm{Gr}^{\mathcal{A}}_{0,\infty}\xrightarrow{+\mathcal{A}}\mathrm{Gr}^{\mathcal{A}}_{d,\infty}\xrightarrow{+\mathcal{A}}\dots).$

######  Proposition 3.14.

The forgetful maps
$\mathrm{Gr}^{\mathcal{A}}_{m,\infty}\to\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{m}$
and
$\mathrm{Gr}^{\mathcal{A}}_{\infty}\to\mathcal{V}\mathrm{ect}^{\mathcal{A}}_{\infty}$
are $\mathbf{A}^{1}$-equivalences on affines.

###### Proof.

When $\mathcal{A}$ is trivial, this is shown in [HJN+20, Proposition 4.7]. The
argument applies verbatim for general $\mathcal{A}$, by replacing $A$ with
$\mathcal{A}_{A}$ in _loc.cit._ ∎

Altogether, we get the following result, which gives a geometric
interpretation of the twisted K-theory as a motivic space.

######  Theorem 3.15.

There is a fiber sequence of motivic spaces

$\mathrm{Gr}^{\mathcal{A}}_{\infty}\to
K^{\mathcal{A}}\xrightarrow{\overline{\operatorname{\mathrm{rk}}}^{\mathcal{A}}}\mathbf{Z}^{\mathcal{A}}.$

###### Proof.

This is a combination of Proposition 3.10, Corollary 3.4 and Proposition 3.14.
∎

##  4\. The motivic spectrum of twisted K-theory

In this section, we assume that the base scheme $S$ is regular and of finite
Krull dimension. First, we show that the $\mathcal{A}$-twisted K-theory
satisfies Bott periodicity.

######  Proposition 4.1.

We have an isomorphism of $K(\mathbf{P}^{1}_{S})$-modules:
$K^{\mathcal{A}}(\mathbf{P}^{1}_{S})\simeq
K(\mathbf{P}^{1}_{S})\otimes_{K(S)}K^{\mathcal{A}}(S)$.

###### Proof.

The tensor product map

$\mathcal{P}\mathrm{erf}(\mathbf{P}^{1}_{S})\otimes_{\mathcal{P}\mathrm{erf}(S)}\mathcal{P}\mathrm{erf}^{\mathcal{A}}(S)\to\mathcal{P}\mathrm{erf}^{\mathcal{A}}(\mathbf{P}^{1}_{S})$

is an isomorphism étale-locally, hence an isomorphism of stacks. Consider the
Beilinson orthogonal decomposition of $\mathcal{P}\mathrm{erf}(S)$-modules:

$\mathcal{P}\mathrm{erf}(\mathbf{P}^{1}_{S})\simeq\langle\mathcal{P}\mathrm{erf}(S),\mathcal{P}\mathrm{erf}(S)\rangle.$

Upon tensoring the decomposition with
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}(S)$, we get

$\mathcal{P}\mathrm{erf}^{\mathcal{A}}(\mathbf{P}^{1}_{S})\simeq\langle\mathcal{P}\mathrm{erf}^{\mathcal{A}}(S),\mathcal{P}\mathrm{erf}^{\mathcal{A}}(S)\rangle.$

The proposition then follows by additivity of K-theory. ∎

######  Corollary 4.2.

We have $\Omega_{\mathbf{P}^{1}}K^{\mathcal{A}}\simeq K^{\mathcal{A}}$, i.e.
$K^{\mathcal{A}}$ satisfies motivic Bott periodicity.

Corollary 4.2 gives an infinite $\mathbf{P}^{1}$-delooping of
$K^{\mathcal{A}}$, and so we can define the motivic spectrum of twisted
K-theory.

######  Definition 4.3.

The _$\mathbf{P}^{1}$ -spectrum of $\mathcal{A}$-twisted K-theory_ is defined
as follows (with bonding maps specified by Corollary 4.2):

$\mathrm{KGL}^{\mathcal{A}}=(K^{\mathcal{A}},K^{\mathcal{A}},\dots)\in\mathbf{SH}(S).$

######  Remark 4.4.

Since $K^{\mathcal{A}}$ is a $K$-module by Remark 2.14, and the action is
compatible with Bott periodicity, the motivic spectrum
$\mathrm{KGL}^{\mathcal{A}}$ is a $\mathrm{KGL}$-module.

Similarly, $\mathrm{kgl}^{\mathcal{A}}$, the effective cover of
$\mathrm{KGL}^{\mathcal{A}}$, is naturally a $\mathrm{kgl}$-module. In [Bac20,
Corollary 4.3], it was shown that a $\mathrm{kgl}$-module structure is
equivalent to a structure of (coherent) pushforwards along finite locally free
maps of schemes. Under this equivalence, the structure of transfers on
$\mathrm{kgl}^{\mathcal{A}}$ comes from finite locally free covariance of
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}$.

Let us write $KH^{\mathcal{A}}(X)$ for the $\mathbf{A}^{1}$-localization of
the Bass K-theory spectrum $K^{B}$ of
$\mathcal{P}\mathrm{erf}^{\mathcal{A}}(X)$. That is

$KH^{\mathcal{A}}(X)=\operatorname*{colim}_{n\in\Delta^{\mathrm{op}}}K^{B}(\mathcal{P}\mathrm{erf}^{\mathcal{A}}(X\times\Delta^{n}))\,.$

When $\mathcal{A}$ is trivial $KH^{\mathcal{A}}$ is the cohomology theory
represented by $\mathrm{KGL}$. We want to show that the same is true for a
general $\mathcal{A}$.

######  Proposition 4.5.

Let $X$ be a quasi-compact quasi-separated scheme and $\mathcal{A}$ an Azumaya
algebra over $X$. Then there is an equivalence of spectra

$\operatorname{map}_{\mathbf{SH}(X)}(\Sigma^{\infty}X_{+},\mathrm{KGL}^{\mathcal{A}})\simeq
KH^{\mathcal{A}}(X)$

###### Proof.

The statement follows from [Cis13, Corollaire 2.11] as in the proof of [Cis13,
Théorème 2.20]. ∎

One of the first achievements of motivic homotopy theory was a complete proof
of cdh descent for homotopy $K$-theory [Cis13]. The key insight is that any
motivic spectrum which is stable under base change satisfies cdh descent. We
now prove a base change property for twisted K-theory spectrum to establish
its cdh descent. To do so, we will first show that the sheaf
$\mathbf{Z}^{\mathcal{A}}$ satisfies base change. We note that cdh descent for
twisted homotopy $K$-theory can also be established using methods of Land-
Tamme [LT19] as explored in the twisted setting by Stapleton [Sta20]. We will
later give a _motivic refinement_ of the cdh descent of twisted homotopy
$K$-theory in the sense that we will endow it with a motivic filtration whose
associated graded pieces do enjoy cdh descent.

######  Proposition 4.6.

Let $f\colon T\to S$ be a morphism. Then the canonical map
$f^{*}\mathbf{Z}^{\mathcal{A}}\to\mathbf{Z}^{f^{*}\mathcal{A}}$ is an
isomorphism of Zariski sheaves on $\mathrm{Sm}_{T}$.

###### Proof.

Since $f^{*}\mathbf{Z}^{\mathcal{A}}$ and $\mathbf{Z}^{f^{*}\mathcal{A}}$ are
0-truncated, it suffices to check that the map is an isomorphism on stalks.

The sheaf $f^{*}\mathbf{Z}^{\mathcal{A}}$ can be identified with the
sheafification of the presheaf on $\mathrm{Sm}_{T}$ sending

$[Z\to T]\mapsto\operatorname*{colim}_{Z\to W\to
S}\mathbf{Z}^{\mathcal{A}}(W)$

where the colimit is indexed by all factorizations $Z\to W\to S$ of $Z\to T\to
S$ where $W\to S$ is a smooth $S$-scheme. Let $Z\in\mathrm{Sm}_{T}$ and choose
a point $z\in Z$, then the stalk of $f^{*}\mathbf{Z}^{\mathcal{A}}$ at $z$ is
therefore

$(f^{*}\mathbf{Z}^{\mathcal{A}})_{z}\simeq\operatorname*{colim}_{U\ni
z}\operatorname*{colim}_{U\to W\to
S}\mathbf{Z}^{\mathcal{A}}(W)\simeq\operatorname*{colim}_{\operatorname{Spec}\mathcal{O}_{Z,z}\to
W\to S}\mathbf{Z}^{\mathcal{A}}(W)$

where the last colimit is indexed through all factorizations of
$\operatorname{Spec}\mathcal{O}_{Z,z}\to Z\to S$ through a smooth $S$-scheme
$W$. Note that $\mathbf{Z}^{\mathcal{A}}$ commutes with filtered colimits, as
it is the Zariski sheafification of $K_{0}^{\mathcal{A}}$ by Lemma 2.17. Hence
we can replace $W$ with $\mathcal{O}_{W,w}$ (where $w$ is the image of $z$)
and rewrite this colimit as

$(f^{*}\mathbf{Z}^{\mathcal{A}})_{z}\simeq\operatorname*{colim}_{\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Spec}R\to
S}\mathbf{Z}^{\mathcal{A}}(\operatorname{Spec}R)\simeq\operatorname*{colim}_{\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Spec}R\to
S}K^{\mathcal{A}}_{0}(R)$

where the colimit now ranges through all factorizations through an essentially
smooth local $S$-scheme $\operatorname{Spec}R$ such that the map
$R\to\mathcal{O}_{Z,z}$ is a local ring homomorphism. Moreover, the stalk of
$\mathbf{Z}^{f^{*}\mathcal{A}}$ at $z$ is precisely
$K_{0}^{\mathcal{A}}(\mathcal{O}_{Z,z})$, and the map we need to prove is an
isomorphism is the pullback map

$\operatorname*{colim}_{\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Spec}R\to
S}K^{\mathcal{A}}_{0}(R)\to K^{\mathcal{A}}_{0}(\mathcal{O}_{Z,z})\,.$

We will prove separately that it is injective and surjective.

For surjectivity it suffices to show that every $\mathcal{A}$-twisted vector
bundle $V$ over $\mathcal{O}_{Z,z}$ can be obtained by pullback from some
essentially smooth local ring $R$ over $S$. But, by choosing a set of
generators as an $\mathcal{A}$-module, we can obtain it as a pullback from
$\mathrm{Gr}^{\mathcal{A}}_{\mathcal{O}_{Z,z}}$ and therefore from
$\mathrm{Gr}^{\mathcal{A}}_{\mathcal{O}_{S,s}}$ where $s$ is the image of $z$
in $S$. Since the $\mathcal{A}$-twisted Grassmannian is smooth by Lemma 3.13,
this proves surjectivity.

Let us now prove injectivity. Suppose we have an essentially smooth local ring
$R$ and a class $[V]-[V^{\prime}]\in K^{\mathcal{A}}_{0}(R)$ that is sent to 0
by the pullback along
$f\colon\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Spec}R$. Up to
adding a suitable free $\mathcal{A}$-twisted vector bundle to both $V$ and
$V^{\prime}$, this means that there exists an isomorphism $\varphi\colon
f^{*}V\simeq f^{*}V^{\prime}$. Therefore the map
$\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Spec}R$ factorizes as

$\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Iso}_{\mathcal{A}}(V,V^{\prime})\to\operatorname{Spec}R$

where $\operatorname{Iso}_{\mathcal{A}}(V,V^{\prime})$ is the smooth affine
scheme of Lemma 4.7 below. Therefore, by taking
$\operatorname{Spec}R^{\prime}$ to be the localization of
$\operatorname{Iso}_{\mathcal{A}}(V,V^{\prime})$ at the image of $z$, there is
a factorization

$\operatorname{Spec}\mathcal{O}_{Z,z}\to\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R$

where $R^{\prime}$ is an essentially smooth $S$-scheme where $V$ and
$V^{\prime}$ become isomorphic. Hence the class $[V]-[V^{\prime}]$ becomes 0
in the colimit. ∎

######  Lemma 4.7.

Let $S$ be a qcqs scheme, $\mathcal{A}$ an Azumaya algebra over $S$ and $V$
and $W$ be two $\mathcal{A}$-twisted vector bundles. Consider the presheaf
$\operatorname{Iso}_{\mathcal{A}}(V,W)$ sending every $S$-scheme $p\colon T\to
S$ to the set $\\{\varphi\colon p^{*}V\to p^{*}W\\}$ of isomorphisms of
$\mathcal{A}$-twisted vector bundles over $T$. Then
$\operatorname{Iso}_{\mathcal{A}}(V,W)$ is represented by a smooth affine
$S$-scheme.

###### Proof.

Since the prestack of $\mathcal{A}$-modules is an fpqc stack by [Stacks,
023T], the presheaf $\operatorname{Iso}_{\mathcal{A}}(V,W)$ is an fpqc sheaf.
Moreover étale-locally we can find a trivialization of $\mathcal{A}$, so we
can assume $\mathcal{A}$ to be trivial. But then, étale-locally,
$\operatorname{Iso}_{\mathcal{A}}(V,W)$ is either empty (if
$\operatorname{\mathrm{rk}}V\neq\operatorname{\mathrm{rk}}W$) or a
$\mathrm{GL}_{\operatorname{\mathrm{rk}}V}$-torsor. In either case it is
represented by a smooth affine $S$-scheme. ∎

Using Proposition 4.6, we can deduce the base change property for
$\mathrm{KGL}^{\mathcal{A}}$.

######  Theorem 4.8.

Let $S$ be a regular scheme of finite Krull dimension, and $\mathcal{A}$ an
Azumaya algebra over $S$. The formation of the motivic twisted $K$-theory
spectrum

$p\colon X\rightarrow S\,\mapsto\,KGL^{p^{*}\mathcal{A}}$

defines a Cartesian section of the Cartesian fibration
$\int\mathbf{SH}\rightarrow\mathrm{Sch}_{S}$. In particular, the twisted
homotopy $K$-theory spectrum is a cdh sheaf and satisfies Milnor excision.

###### Proof.

The first statement follows once we know that for any morphism of $S$-schemes
$f\colon X\rightarrow Y$, with structure mas $p_{X}\colon X\rightarrow
S,p_{Y}\colon Y\rightarrow S$ the canonical comparison map in $\mathbf{SH}(X)$

$f^{*}KGL^{p_{X}^{*}\mathcal{A}}\rightarrow KGL^{p_{Y}^{*}\mathcal{A}}$

is an equivalence of motivic spectra. By construction of
$\mathrm{KGL}^{\mathcal{A}}$, it’s enought to check that the comparison map
$f^{*}K^{p_{X}^{*}\mathcal{A}}\rightarrow K^{p_{Y}^{*}\mathcal{A}}$ is an
equivalence of motivic spaces in $\mathbf{H}(X)$. By Theorem 3.15,
$K^{\mathcal{A}}$ as a motivic space is an extension of
$\mathbf{Z}^{\mathcal{A}}$ by $\mathrm{Gr}_{\infty}^{\mathcal{A}}$. Then the
claim follows from Proposition 4.6, because the twisted Grassmannians are
stable under pullbacks.

The base change property implies that $\mathrm{KGL}^{\mathcal{A}}$ is a cdh
sheaf by [Cis13, Proposition 3.7], and that it satisfies Milnor excision by
[EHIK21, Corollary 2]. ∎

##  5\. Twisted motivic cohomology

In this section, we compute the slices of $\mathcal{A}$-twisted K-theory,
which allows us to construct the $\mathcal{A}$-twisted motivic spectral
sequence. Previously, Kahn and Levine have computed the slices in the case
when the base scheme $S=\operatorname{Spec}k$ is the spectrum of a perfect
field $k$ [KL10, Theorem 6.5.5]. In this section, we will first provide a
different proof of this computation (Theorem 5.7) and then generalize it to
the case when the Azumaya algebra is not pulled back from the base field
(Proposition 5.13).

To define $\mathcal{A}$-twisted motivic cohomology, we employ the theory of
framed transfers, which was developed in [Voe01, GP18, EHK+21] and other
works. We summarize here the main structural results that we use.

  1. (1)

Every cohomology theory represented by a motivic spectrum has a unique
structure of framed transfers, i.e. a structure of a presheaf on the
$\infty$-category of framed correspondences
$\mathrm{Corr}^{\operatorname{fr}}(\mathrm{Sm}_{S})$ [Hoy21, Theorem 18]. More
precisely,

$\mathbf{SH}(S)\simeq\mathbf{SH}^{\operatorname{fr}}(S),$

where $\mathbf{SH}^{\operatorname{fr}}(S)$ is the $\infty$-category of framed
motivic spectra.

  2. (2)

The framed suspension spectrum
$\Sigma^{\infty}_{\operatorname{fr}}\colon\mathbf{H}^{\operatorname{fr}}(S)\to\mathbf{SH}^{\operatorname{fr}}(S)$,
going from framed motivic spaces to (framed) motivic spectra, is fully
faithful on grouplike objects, when $S$ is a perfect field [EHK+21, Theorem
3.5.14].

  3. (3)

There is a canonical functor
$\mathrm{Corr}^{\operatorname{fr}}(\mathrm{Sm}_{S})\to{\mathrm{Cor}}_{S}$ from
the $\infty$-category of framed correspondences to the category of Voevodsky’s
finite correspondences [EHK+21, Section 5.3]. In particular, every presheaf
with Voevodsky’s transfers has canonical framed transfers.

Recall that the presheaf of abelian groups $\mathbf{Z}^{\mathcal{A}}$ is a
motivic space with a canonical structure of Voevodsky’s finite transfers by
Lemmas 2.19 and 2.20, hence it represents a framed motivic space, which we
also denote $\mathbf{Z}^{\mathcal{A}}$.

######  Definition 5.1.

We define the _$\mathcal{A}$ -twisted motivic cohomology spectrum_ to be

$H\mathbf{Z}^{\mathcal{A}}:=\Sigma^{\infty}_{\operatorname{fr}}\mathbf{Z}^{\mathcal{A}}\in\mathbf{SH}^{\operatorname{fr}}(S)\simeq\mathbf{SH}(S).$

When $\mathcal{A}$ is trivial, the motivic spectrum
$H\mathbf{Z}^{\mathcal{A}}$ coincides with the Spitzweck motivic cohomology
spectrum, constructed in [Spi18], by [Hoy21, Theorem 21]. Moreover,
$H\mathbf{Z}^{\mathcal{A}}$ is stable under base change by Proposition 4.6,
and in particular, the corresponding cohomology theory satisfies cdh descent
by [Cis13, Proposition 3.7]. Therefore it makes sense to say that
$H\mathbf{Z}^{\mathcal{A}}$ represents $\mathcal{A}$-twisted Spitzweck motivic
cohomology.

We recall the definition of the slice filtration in motivic homotopy theory,
constructed in [Voe02b].

######  Definition 5.2.

Let $S$ be a scheme. Then the $\infty$-category
$\mathbf{SH}(S)^{\mathrm{eff}}$ of _effective motivic spectra_ (resp. the
$\infty$-category $\mathbf{SH}_{S^{1}}(S)^{\mathrm{eff}}$ of _effective
$S^{1}$-spectra_) is the localizing subcategory of $\mathbf{SH}(S)$ (resp.
$\mathbf{SH}_{S^{1}}(S)$) generated by the objects of the form
$\Sigma^{\infty}_{\mathbf{T}}X_{+}$ (resp. $\Sigma^{\infty}_{S^{1}}X_{+}$) for
$X\in\mathrm{Sm}_{S}$. For every $n\in\mathbf{Z}$ (resp. $n\in\mathbf{N}$) we
call $\Sigma^{n}_{\mathbf{T}}\mathbf{SH}(S)^{\mathrm{eff}}$ the
$\infty$-category of _$n$ -effective motivic spectra_ and we write $f_{n}$ for
the left adjoint to the inclusion
$\Sigma^{n}_{\mathbf{T}}\mathbf{SH}(S)^{\mathrm{eff}}\subseteq\mathbf{SH}(S)$.
Since
$\Sigma^{n+1}_{\mathbf{T}}\mathbf{SH}(S)^{\mathrm{eff}}\subseteq\Sigma^{n}_{\mathbf{T}}\mathbf{SH}(S)^{\mathrm{eff}}$,
for every $E\in\mathbf{SH}(S)$ we obtain a functorial $\mathbf{Z}$-indexed
tower

$\cdots\to f_{2}E\to f_{1}E\to\to\cdots\to E$

called the _slice tower_ of $E$. Each layer
$s_{i}E\coloneqq\mathrm{cofib}(f_{i+1}E\to f_{i}E)$ is called the _$i$ -th
slice_ of $E$.

We will use the following, more explicit description of 0-th slices. Consider
the functor
$L^{0}_{\mathrm{bir}}\colon\mathbf{H}^{\operatorname{fr}}(k)\to\mathbf{H}^{\operatorname{fr}}(k)$,
which is the localization generated by the birational open embeddings of
smooth varieties [BE21, Section 3]. The following result follows from the main
insight of [BE21].

######  Proposition 5.3.

Let $k$ be a perfect field and $X\in\mathbf{H}^{\operatorname{fr}}(k)$ be a
framed motivic space.

  1. (1)

The map $X\rightarrow L^{0}_{\mathrm{bir}}X$ is an equivalence after applying
$s_{0}\Sigma^{\infty}_{\operatorname{fr}}$.

  2. (2)

Assume that $L^{0}_{\mathrm{bir}}X$ is grouplike, then
$\Sigma^{\infty}_{\operatorname{fr}}L^{0}_{\mathrm{bir}}X$ is a 0-slice, i.e.,
we have an equivalence

$\Sigma^{\infty}_{\operatorname{fr}}L^{0}_{\mathrm{bir}}X\simeq
s_{0}\Sigma^{\infty}_{\operatorname{fr}}X.$

In particular, any $L^{0}_{\mathrm{bir}}$-local grouplike object in
$\mathbf{H}^{{\operatorname{fr}}}(k)$ becomes a 0-slice upon applying the
functor $\Sigma^{\infty}_{\operatorname{fr}}$.

###### Proof.

We will write $F:\mathbf{H}(k)_{\ast}\to\mathbf{H}^{{\operatorname{fr}}}(k)$
for the left adjoint to the functor
$\mathbf{H}^{{\operatorname{fr}}}(k)\to\mathbf{H}(k)_{\ast}$ that forgets the
framed transfers. To prove (1), consider the commutative diagram

$\leavevmode\hbox to105.2pt{\vbox
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}\hbox{${\mathbf{H}(k)_{\ast}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
16.10487pt\hfil&\hfil\hskip 40.65706pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbf{H}^{{\operatorname{fr}}}(k)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
16.65709pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
20.16098pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbf{SH}^{S^{1}}(k)}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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}\hbox{${\mathbf{SH}(k)^{\mathrm{eff}}}$} }}\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-32.4382pt}{2.97366pt}\pgfsys@lineto{-32.4382pt}{-14.40868pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\Sigma_{S^{1}}^{\infty}}$}
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{34.51373pt}{-7.95413pt}\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.07722pt}{-25.43053pt}\pgfsys@lineto{11.12279pt}{-25.43053pt}\pgfsys@stroke\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.80501pt}{-33.42773pt}\pgfsys@invoke{
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}\hbox{$\scriptstyle{\Sigma^{\infty}_{\mathbf{G}_{\mathrm{m}}}}$}
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$

According to [BE21, Lemma 13.(5)], the $L^{0}_{\mathrm{bir}}$-equivalences in
$\mathbf{H}(k)_{\ast}$ are sent to $s_{0}$-equivalences in $\mathbf{SH}(k)$
(since their cofibers are sent to $1$-effective spectra). Moreover the
$L^{0}_{\mathrm{bir}}$-equivalences in $\mathbf{H}^{{\operatorname{fr}}}(k)$
are, by definition, generated under colimits by the image under $F$ of the
$L^{0}_{\mathrm{bir}}$-equivalences of $\mathbf{H}(k)_{\ast}$. Therefore the
commutativity of the above diagram and the fact that all functors in it
preserve all colimits, imply that the $L^{0}_{\mathrm{bir}}$-equivalences in
$\mathbf{H}^{{\operatorname{fr}}}(k)$ are sent to $s_{0}$-equivalences in
$\mathbf{SH}(k)$. But this is the claim of (1).

We now prove (2). After (1), we need only to prove that
$\Sigma^{\infty}_{\operatorname{fr}}L^{0}_{\mathrm{bir}}X$ is a 0-slice,
assuming that $L^{0}_{\mathrm{bir}}X$ is grouplike. Since
$\Sigma^{\infty}_{\operatorname{fr}}L^{0}_{\mathrm{bir}}X$ is effective, it
suffices to show that its first effective cover is trivial, i.e., that the
homotopy sheaves
$\underline{\pi}_{\ast}(\Sigma^{\infty}_{{\operatorname{fr}}}L^{0}_{\mathrm{bir}}X)_{-1}$
are zero. For $\ast<0$, this follows since the image of
$\Sigma^{\infty}_{{\operatorname{fr}}}$ lands inside connective objects for
the homotopy t-structure in $\mathbf{SH}(k)$. For $\ast\geqslant 0$, we use
that $\Sigma^{\infty}_{{\operatorname{fr}}}$ is fully faithful on grouplike
objects by the cancellation theorem [AGP21]. Hence
$\underline{\pi}_{\ast}(\Sigma^{\infty}_{{\operatorname{fr}}}L^{0}_{\mathrm{bir}}X)_{-1}$
is the sheaf associated to the presheaf

$U\mapsto\pi_{\ast}\mathrm{Map}_{\mathbf{SH}(k)}(\Sigma^{\infty}_{{\operatorname{fr}}}F(\mathbf{G}_{m}\wedge
U_{+}),\Sigma^{\infty}_{{\operatorname{fr}}}L^{0}_{\mathrm{bir}}X)\simeq\pi_{\ast}\mathrm{Map}_{\mathbf{H}(k)^{{\operatorname{fr}}}}(F(\mathbf{G}_{m}\wedge
U_{+}),L^{0}_{\mathrm{bir}}X)\,.$

However, since $\mathbf{G}_{m}\times U\to\mathbf{A}^{1}\times U$ is a
birational map, we get

$\mathrm{Map}_{\mathbf{H}(k)^{{\operatorname{fr}}}}(F(\mathbf{G}_{m}\wedge
U_{+}),L^{0}_{\mathrm{bir}}X)\simeq\mathrm{Map}_{\mathbf{H}(k)^{{\operatorname{fr}}}}(F(\mathbf{A}^{1}\wedge
U_{+}),L^{0}_{\mathrm{bir}}X)\simeq\ast\,,$

which is what we wanted to show. Since an $L^{0}_{\mathrm{bir}}$-local object
$X$ satisfies $X\simeq L^{0}_{\mathrm{bir}}X$, the last claim of (2) follows.
∎

######  Remark 5.4.

It is plausible that
$\Sigma^{\infty}_{{\operatorname{fr}}}L^{0}_{\mathrm{bir}}X\simeq
s_{0}\Sigma^{\infty}_{{\operatorname{fr}}}X$ for
$X\in\mathbf{H}^{\operatorname{fr}}(k)$ such that $L^{0}_{\mathrm{bir}}X$ is
_not necessarily grouplike_ , which would then give a geometric formula for
the 0-th slice of $\Sigma^{\infty}_{{\operatorname{fr}}}X$. Since we do not
need this result, we leave it to the interested reader.

Our goal in this section is to prove that the $n$-th slice
$s_{n}\mathrm{KGL}^{\mathcal{A}}$ is equivalent to
$\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}}$. To do so, we will use the
techniques for slice computations developed in [BE21]. Recall that we write
$\mathrm{kgl}^{\mathcal{A}}$ for the effective cover of
$\mathrm{KGL}^{\mathcal{A}}$.

######  Lemma 5.5.

Let $X$ be a scheme essentially smooth over a field and $\mathcal{A}$ an
Azumaya algebra over $X$. Then the motivic spectrum
$\mathrm{kgl}^{\mathcal{A}}$ is very effective.

###### Proof.

By Theorem 4.8, [BH21, Lemma B.1 and Proposition B.3] it suffices to show the
thesis when $X$ is the spectrum of a field. The motivic spectrum
$\mathrm{kgl}^{\mathcal{A}}$ is effective by definition, so by [Hoy15, Theorem
2.3] it suffices to check that for $Y$ an essentially smooth henselian local
scheme, the spectrum

$\operatorname{map}_{\mathbf{SH}(X)}(\Sigma^{\infty}Y_{+},\mathrm{kgl}^{\mathcal{A}})\simeq\operatorname{map}_{\mathbf{SH}(X)}(\Sigma^{\infty}Y_{+},\mathrm{KGL}^{\mathcal{A}})\simeq
K^{B}(\mathcal{P}\mathrm{erf}^{\mathcal{A}}(Y))$

is connective. But by [Ber09, Theorem 4.1] this spectrum is a summand of the
Bass K-theory of $\mathrm{SB}(\mathcal{A})$, the Severi-Brauer scheme of
$\mathcal{A}$. Since $\mathrm{SB}(\mathcal{A})$ is smooth over $Y$, it is a
regular Noetherian scheme and so its Bass K-theory is connective. ∎

The following computation is the key geometric input for identifying the
slices of $\mathrm{KGL}^{\mathcal{A}}$.

######  Lemma 5.6.

The motivic space $\mathrm{Gr}_{\infty}^{\mathcal{A}}$ is
$L^{0}_{\mathrm{bir}}$-contractible.

###### Proof.

Let $d$ be the degree of $\mathcal{A}$. It suffices to show that
$\mathrm{Gr}^{\mathcal{A}}_{md,n}$ is $L^{0}_{\mathrm{bir}}$-contractible for
every $m$ and $n\geqslant md$, since
$\mathrm{Gr}_{\infty}^{\mathcal{A}}=\operatorname*{colim}_{n\geqslant
md}\mathrm{Gr}_{md,n}^{\mathcal{A}}$.

We want to show that $\mathrm{Gr}^{\mathcal{A}}_{md,n}$ contains an open dense
subscheme that is Zariski-locally $\mathbf{A}^{1}$-contractible. Let
$\operatorname{\mathrm{Hom}}_{\mathcal{A}}(\mathcal{A}^{m},\mathcal{A}^{n-m})$
be the sheaf of $\mathcal{A}$-linear maps
$\mathcal{A}^{m}\to\mathcal{A}^{n-m}$. Then there is a map

$\operatorname{\mathrm{Hom}}_{\mathcal{A}}(\mathcal{A}^{m},\mathcal{A}^{m-n})\to\mathrm{Gr}^{\mathcal{A}}_{md,n}$

sending a map $f:\mathcal{A}^{m}\to\mathcal{A}^{m-n}$ to its graph in
$\mathcal{A}^{m}\times\mathcal{A}^{n-m}\simeq\mathcal{A}^{n}$. This map is a
dense open embedding, since étale-locally $\mathcal{A}$ is trivial, and in
that case this map is one of the standard charts of the Grassmannian
$\mathrm{Gr}_{md,nd}$. We observe that

$\operatorname{\mathrm{Hom}}_{\mathcal{A}}(\mathcal{A}^{m},\mathcal{A}^{n-m})\simeq\mathcal{A}^{m(n-m)}.$

Therefore $\mathrm{Gr}^{\mathcal{A}}_{md,n}$ is birational to a vector bundle,
and thus it is $L^{0}_{\mathrm{bir}}$-trivial. ∎

Using the above lemma we can present an alternative proof of [KL10, Theorem
6.5.5].

######  Theorem 5.7.

Let $k$ be a perfect field, $\mathcal{A}$ be an Azumaya algebra over $k$ and
$f\colon S\to\operatorname{Spec}k$ be an essentially smooth $k$-scheme. Then
the map in $\mathbf{SH}(S)$

$\Sigma^{\infty}_{\operatorname{fr}}\mathcal{V}\mathrm{ect}^{f^{*}\mathcal{A}}\to\Sigma^{\infty}_{\operatorname{fr}}\mathbf{Z}^{f^{*}\mathcal{A}}=H\mathbf{Z}^{f^{*}\mathcal{A}}\,,$

induced by the $\mathcal{A}$-twisted rank map
$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon\mathcal{V}\mathrm{ect}^{\mathcal{A}}\to\mathbf{Z}^{\mathcal{A}}$,
can be identified with the canonical map

$\mathrm{kgl}^{f^{*}\mathcal{A}}\to s_{0}(\mathrm{kgl}^{f^{*}\mathcal{A}})\,.$

Therefore for every $n\in\mathbf{Z}$ there is an equivalence

$s_{n}\mathrm{KGL}^{f^{*}\mathcal{A}}\simeq
s_{n}\mathrm{KGL}\otimes_{H\mathbf{Z}}H\mathbf{Z}^{f^{*}\mathcal{A}}\simeq\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{f^{*}\mathcal{A}}\,.$

###### Proof.

By [Hoy15, Remark 4.20], [BH21, Theorem B.4], Proposition 4.6 and Theorem 4.8,
all terms in the statement of the theorem are stable under essentially smooth
base change. Therefore we can assume that $S=\operatorname{Spec}k$ is the
spectrum of a perfect field.

By the motivic recognition principle [EHK+21, Theorem 3.5.14] and Lemma 5.5 we
have

$\mathrm{kgl}^{\mathcal{A}}\simeq\Sigma^{\infty}_{\operatorname{fr}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}\simeq\Sigma^{\infty}_{\operatorname{fr}}K^{\mathcal{A}}.$

By Lemma 2.19, $\mathbf{Z}^{\mathcal{A}}$ is a birational sheaf. Hence the
motivic spectrum $H\mathbf{Z}^{\mathcal{A}}$ belongs to the image of the
localizing functor $s_{0}$ on effective motivic spectra by Lemma 5.3.(2).

Let now $\overline{\mathrm{kgl}}^{\mathcal{A}}$ be the fiber of the map
$\mathrm{kgl}^{\mathcal{A}}\to H\mathbf{Z}^{\mathcal{A}}$, induced by
$\operatorname{\mathrm{rk}}^{\mathcal{A}}$. It remains to be proven that
$s_{0}(\overline{\mathrm{kgl}}^{\mathcal{A}})=0$. Since for any smooth
$S$-scheme $U$, the map of sheaves of spectra

$K^{\mathcal{A}}(U)=\operatorname{map}(\Sigma^{\infty}U_{+},\mathrm{kgl}^{\mathcal{A}})\to\operatorname{map}(\Sigma^{\infty}U_{+},H\mathbf{Z}^{\mathcal{A}})=H(\mathbf{Z}^{\mathcal{A}}(U))$

is Zariski-locally 0-connected, it follows that $\mathrm{kgl}^{\mathcal{A}}$
and $\overline{\mathrm{kgl}}^{\mathcal{A}}$ are very effective. By Theorem
3.15 and [EHK+21, Theorem 3.5.14] we see that

(5.8)
$\overline{\mathrm{kgl}}^{\mathcal{A}}\simeq\Sigma^{\infty}_{\operatorname{fr}}\mathrm{Gr}_{\infty}^{\mathcal{A}},$

for some structure of framed transfers on
$\mathrm{Gr}_{\infty}^{\mathcal{A}}$.

But now we have equivalences:

$s_{0}(\overline{\mathrm{kgl}}^{\mathcal{A}})\simeq
s_{0}(\Sigma^{\infty}_{\operatorname{fr}}\mathrm{Gr}_{\infty}^{\mathcal{A}})\simeq
s_{0}(\Sigma^{\infty}_{{\operatorname{fr}}}L_{0}^{\mathrm{bir}}\mathrm{Gr}_{\infty}^{\mathcal{A}})=0.$

Indeed, the first equivalence comes from (5.8), the second equivalence is
Lemma 5.3.(1) and the third equivalence is Lemma 5.6 together with [BE21,
Lemma 13.(4)].

The final statement follows from Bott periodicity for
$\mathrm{KGL}^{\mathcal{A}}$, see Corollary 4.2. ∎

######  Corollary 5.9.

Let $S=\operatorname{Spec}k$ be the spectrum of a field of exponential
characteristic $e$ and $\mathcal{A}$ be an Azumaya algebra over $k$. Then the
map in $\mathbf{SH}(S)$

$\Sigma^{\infty}_{\operatorname{fr}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}[1/e]\to\Sigma^{\infty}_{\operatorname{fr}}\mathbf{Z}^{\mathcal{A}}[1/e]=H\mathbf{Z}^{\mathcal{A}}[1/e],$

induced by the $\mathcal{A}$-twisted rank map
$\operatorname{\mathrm{rk}}^{\mathcal{A}}\colon\mathcal{V}\mathrm{ect}^{\mathcal{A}}\to\mathbf{Z}^{\mathcal{A}}$,
can be identified with the canonical map

$\mathrm{kgl}^{\mathcal{A}}[1/e]\to s_{0}(\mathrm{kgl}^{\mathcal{A}}[1/e])\,.$

Moreover for every $n\in\mathbf{Z}$ there is an equivalence

$s_{n}\mathrm{KGL}^{\mathcal{A}}[1/e]\simeq
s_{n}\mathrm{KGL}\otimes_{H\mathbf{Z}}H\mathbf{Z}^{\mathcal{A}}[1/e]\simeq\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}}[1/e]\,.$

###### Proof.

Let $k^{p}$ be the perfection of $k$. Then by [EK20, Corollary 2.1.7] the
statement follows from Theorem 5.7 in $\mathbf{SH}(k^{p})[1/e]$. ∎

Our next goal is to generalize Corollary 5.9 to the case when the Azumaya
algebra $\mathcal{A}$ is not pulled back from the base field. To begin we will
show that $H\mathbf{Z}^{\mathcal{A}}$ is a 0-slice in this more general case.

######  Definition 5.10.

We say that a Noetherian scheme $X$ is _Grothendieck_ if for every point $x\in
X$ the local ring $\mathcal{O}_{X,x}$ is a G-ring [Stacks, Tag 07GH]. Every
scheme of finite type over a Grothendieck scheme, in particular every scheme
of finite type over a field, is again Grothendieck [Stacks, Tag 07PX].

######  Lemma 5.11.

Let $R$ be a Grothendieck ring, and let
$F\colon\mathrm{Sm}_{R}^{\mathrm{op}}\rightarrow\mathrm{Sp}$ be a Nisnevich
sheaf of spectra. Let
$\widetilde{F}\colon\mathrm{EssSm}_{R}^{\mathrm{op}}\rightarrow\mathrm{Sp}$ be
its finitary extension to essentially smooth $R$-schemes. Assume that for all
$x\in X$ where $X\in\mathrm{Sm}_{R}$, we have that
$\widetilde{F}(\widehat{\mathcal{O}_{X,x}})=0$, where
$\widehat{\mathcal{O}_{X,x}}$ is the completion of $\mathcal{O}_{X,x}$ at its
maximal ideal. Then $F=0$.

###### Proof.

Since $F$ is, in particular, a Zariski sheaf, it then suffices to prove that
for any smooth affine morphism $R\rightarrow S$, we have that $F(S)=0$. Now,
by [Stacks, Tag 07PX], $S$ is a $G$-ring since it is a finite type extension
of $R$. Therefore, for any prime ideal $\mathfrak{p}$ of $S$, the map
$S_{\mathfrak{p}}\rightarrow\widehat{S_{\mathfrak{p}}}$ is a regular
homomorphism. Popescu’s theorem [Stacks, Tag 07GC] then proves that this map
is ind-smooth and thus $\widetilde{F}(\widehat{\mathcal{O}_{X,x}})$ is well-
defined.

Now, since $F$ is a Nisnevich sheaf, it suffices to prove that
$\widetilde{F}(S^{h}_{\mathfrak{p}})=0$. The map
$S_{\mathfrak{p}}\rightarrow\widehat{S_{\mathfrak{p}}}$ factors through
$S^{h}_{\mathfrak{p}}\rightarrow\widehat{S_{\mathfrak{p}}}$ by [Stacks, Tag
06LJ]. By assumption, it then suffices to prove that
$\widetilde{F}(S^{h}_{\mathfrak{p}})\rightarrow\widetilde{F}(\widehat{S_{\mathfrak{p}}})$
is injective on homotopy groups. But now, since $S_{\mathfrak{p}}$ is by
definition a noetherian local $G$-ring, $S^{h}_{\mathfrak{p}}$ is a $G$-ring
by [Stacks, Tag 07QR]. Thus Popescu’s theorem applies again to prove that
$S^{h}_{\mathfrak{p}}\rightarrow\widehat{S_{\mathfrak{p}}}$ can be written as
a filtered colimit of smooth ring maps
$\\{f_{\alpha}:S^{h}_{\mathfrak{p}}\rightarrow B_{\alpha}\\}_{\alpha}$. By
construction, for each $B_{\alpha}$ we have a morphism
$B_{\alpha}\rightarrow\widehat{S_{\mathfrak{p}}}\rightarrow\kappa(\mathfrak{p})$
under $S^{h}_{\mathfrak{p}}$, i.e., a $\kappa(\mathfrak{p})$-section of the
map $S^{h}_{\mathfrak{p}}\rightarrow B_{\alpha}$. Therefore, by [Gru72,
Théorème I.8], we obtain a section of $S^{h}_{\mathfrak{p}}\rightarrow
B_{\alpha}$ extending the section over $\kappa(\mathfrak{p})$. This means that
the map
$\pi_{n}\widetilde{F}(S^{h}_{\mathfrak{p}})\rightarrow\pi_{n}\widetilde{F}(B_{\alpha})$
is injective for every $n\in\mathbf{Z}$. We conclude by observing that a
filtered colimit of injective morphisms of abelian groups is injective.

∎

######  Lemma 5.12.

Let $k$ be a field of exponential characteristic $e$, $X$ a regular
Grothendieck $k$-scheme, and $\mathcal{A}$ an Azumaya algebra over $X$. Then
the map

$\mathbf{Z}^{\mathcal{A}}[1/e]\to\Omega^{\infty}H\mathbf{Z}^{\mathcal{A}}[1/e]$

is an equivalence of Nisnevich sheaves. Furthermore,
$H\mathbf{Z}^{\mathcal{A}}[1/e]$ is a 0-slice.

###### Proof.

By Lemma 5.11 applied to the fiber and the hypercompleteness of the Nisnevich
topos of $X$, it suffices to prove the result after evaluating on
$\widehat{\mathcal{O}_{X,x}}$ where $x\in X$ is a point of $X$.

Let now
$i\colon\operatorname{Spec}\kappa(x)\hookrightarrow\operatorname{Spec}\widehat{\mathcal{O}_{X,x}}$
be the inclusion of the residue field. By the Cohen structure theorem [Stacks,
Tag 0C0S] there is an isomorphism
$\widehat{\mathcal{O}_{X,x}}\simeq\kappa(x)[\\![t_{1},\dots,t_{n}]\\!]$ with
$n=\dim_{x}X$. In particular, there is an ind-smooth map
$r\colon\widehat{\mathcal{O}_{X,x}}\to\operatorname{Spec}\kappa(x)$ such that
$r\circ i$ is the identity. Moreover, since $i^{*}$ is an equivalence on the
Brauer group by [Mil80, Corollary IV.2.13], the Azumaya algebras
$r^{*}i^{*}\mathcal{A}$ and $\mathcal{A}$ are Brauer equivalent and so
$\mathbf{Z}^{r^{*}i^{*}\mathcal{A}}\simeq\mathbf{Z}^{\mathcal{A}}$. Therefore,
by [Hoy15, Lemma A.7], the map we want to prove is an equivalence is the
pullback along $r$ of the map

$\mathbf{Z}^{i^{*}\mathcal{A}}[1/e]\to\Omega^{\infty}H\mathbf{Z}^{i^{*}\mathcal{A}}[1/e]$

in $\mathbf{SH}(\kappa(x))$. But this map is an equivalence by [EK20, Remark
3.2.8].

Let us now prove that $H\mathbf{Z}^{\mathcal{A}}[1/e]$ is a 0-slice in
$\mathbf{SH}(X)$. It suffices to show that for every smooth $X$-scheme $Y$ the
map

$\mathrm{Map}_{\mathbf{SH}(X)}(\Sigma^{\infty}(Y\times\mathbf{P}^{1})_{+},H\mathbf{Z}^{\mathcal{A}}[1/e])\to\mathrm{Map}_{\mathbf{SH}(X)}(\Sigma^{\infty}Y_{+},H\mathbf{Z}^{\mathcal{A}}[1/e])$

is an equivalence. As we already proved, this is equivalent to asking whether
the map

$\mathbf{Z}^{\mathcal{A}}(Y\times\mathbf{P}^{1})[1/e]\to\mathbf{Z}^{\mathcal{A}}(Y)[1/e]$

is an isomorphism, which is true by Lemma 2.19. ∎

Under the assumptions of Lemma 5.12 we can compute the slices of
$\mathrm{KGL}^{\mathcal{A}}$. This generalizes [KL10, Theorem 1] as explained
in the introduction.

######  Theorem 5.13.

Let $k$ be a field of exponential characteristic $e$, $X$ a regular
Grothendieck $k$-scheme, and $\mathcal{A}$ an Azumaya algebra over $X$. Then
for every $n\in\mathbf{Z}$ there is an equivalence

$s_{n}\mathrm{KGL}^{\mathcal{A}}[1/e]\simeq
s_{n}\mathrm{KGL}\otimes_{H\mathbf{Z}}H\mathbf{Z}^{\mathcal{A}}[1/e]\simeq\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}}[1/e]\,.$

###### Proof.

Consider the canonical map of framed spaces
$\mathcal{V}\mathrm{ect}^{\mathcal{A}}\to K^{\mathcal{A}}$. By adjunction, it
induces a map of motivic spectra
$\Sigma^{\infty}_{\operatorname{fr}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}{}[1/e]\to\mathrm{kgl}^{\mathcal{A}}{}[1/e]$,
which can be checked to be an equivalence Nisnevich-locally. Over the complete
Noetherian local schemes $\operatorname{Spec}\widehat{\mathcal{O}_{X,x}}$ we
argue as in the proof of Lemma 5.12. Indeed, our map is a pullback along an
ind-smooth retraction
$r\colon\operatorname{Spec}\widehat{\mathcal{O}_{X,x}}\to\operatorname{Spec}\kappa(x)$
of the map

$\Sigma^{\infty}_{\operatorname{fr}}\mathcal{V}\mathrm{ect}^{i^{*}\mathcal{A}}[1/e]\to\mathrm{kgl}^{i^{*}\mathcal{A}}[1/e]$

in $\mathbf{SH}(\kappa(x))$, which is an equivalence by the motivic
recognition principle [EHK+21, Theorem 3.5.14], since
$\mathrm{kgl}^{i^{*}\mathcal{A}}$ is very effective by Lemma 5.5.

We argue that the induced map

$\mathrm{kgl}^{\mathcal{A}}[1/e]\simeq\Sigma^{\infty}_{\operatorname{fr}}\mathcal{V}\mathrm{ect}^{\mathcal{A}}[1/e]\xrightarrow{\Sigma^{\infty}_{\operatorname{fr}}\operatorname{\mathrm{rk}}^{\mathcal{A}}}H\mathbf{Z}^{\mathcal{A}}[1/e]$

is equivalent to taking the 0-th slice. By Lemma 5.12 we know that
$H\mathbf{Z}^{\mathcal{A}}[1/e]$ is a 0-slice. It remains to show that the
fiber of
$\Sigma^{\infty}_{\operatorname{fr}}\operatorname{\mathrm{rk}}^{\mathcal{A}}$
is 1-effective after inverting $e$. Arguing as before we can reduce to the
case of fields (by checking on the complete local rings of $X$, where it is
pulled back from a field along an ind-smooth map). There it holds by Corollary
5.9. ∎

Let $k$ be a field and $X$ be a smooth $k$-scheme. Recall that the _$n$ -th
Bloch cycle complex_ is the simplicial abelian group $z^{n}(X)$ which in
degree $d$ is given by

$\bigoplus_{Z\subseteq X\times\Delta^{d}}\mathbf{Z}\cdot{}[Z]$

where the sum is over all the irreducible closed subsets of codimension $d$ of
$X\times\Delta^{d}$ that intersect all faces of $X\times\Delta^{d}$ properly.
The face maps in the simplicial abelian group are given by intersections with
the faces of $X\times\Delta^{d}$. By [Lev08, Theorem 6.4.2], there is a
homotopy equivalence of the mapping spectrum
$\operatorname{map}_{\mathbf{SH}(k)}(\Sigma^{\infty}X_{+},\Sigma^{n}_{\mathbf{T}}H\mathbf{Z})$
with (the $H\mathbf{Z}$-module spectrum represented by) $z^{n}(X)$.

######  Definition 5.14.

Let $\mathcal{A}$ be an Azumaya algebra over $X$. The _$\mathcal{A}$ -twisted
Bloch cycle complex_ is the subcomplex $z^{n}_{\mathcal{A}}(X)\subseteq
z^{n}(X)$ of those cycles $\sum_{Z}n_{Z}\cdot[Z]$ such that
$n_{Z}\in\mathbf{Z}^{\mathcal{A}}(Z)\subseteq\mathbf{Z}$.

######  Proposition 5.15.

Let $k$ be a perfect field and $\mathcal{A}$ an Azumaya algebra over $k$. Then
for every smooth $k$-scheme $X$ the map of mapping spectra

$\operatorname{map}_{\mathbf{SH}(k)}(\Sigma^{\infty}_{\mathbf{T}}X_{+},\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}})\to\operatorname{map}_{\mathbf{SH}(k)}(\Sigma^{\infty}X_{+},\Sigma^{n}_{\mathbf{T}}H\mathbf{Z})$

identifies the left hand side with $z^{n}_{\mathcal{A}}(X)\subseteq z^{n}(X)$.

###### Proof.

By Theorem 5.7 it follows that
$s_{n}(\Sigma^{n}H\mathbf{Z}^{\mathcal{A}})\simeq\Sigma^{n}H\mathbf{Z}^{\mathcal{A}}$.
Therefore, from [Lev08, Corollary 5.3.2 and Theorem 9.0.3] applied with $p=n$
and $E=\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}}$, it follows that the
map we want to study is induced by a map of simplicial spectra

$\bigoplus_{Z\subseteq
X\times\Delta^{\bullet}}(s_{0}H\mathbf{Z}^{\mathcal{A}})(Z)\cdot[Z]\to\bigoplus_{Z\subseteq
X\times\Delta^{\bullet}}(s_{0}H\mathbf{Z})(Z)\cdot[Z]$

where the sums are indexed by the closed subspaces of
$X\times\Delta^{\bullet}$ of codimension $n$ intersection all faces properly.
But since $s_{0}H\mathbf{Z}^{\mathcal{A}}\simeq H\mathbf{Z}^{\mathcal{A}}$ and
the right hand side is canonically identified with Bloch cycle complex, this
proves the thesis. ∎

######  Remark 5.16.

The authors know how to prove Proposition 5.15 only when $\mathcal{A}$ is
pulled back from a perfect field. However the argument in the proof of Lemma
5.12 shows that, up to inverting the exponential characteristic, this is true
Nisnevich locally whenever $X$ is smooth over a field. Therefore the
identification of
$\Omega^{\infty}\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}}$ with
$z^{n}_{\mathcal{A}}(X)$ holds Nisnevich locally in this generality. It seems
therefore reasonable to conjecture that this equivalence is true globally.

######  Definition 5.17.

The $(i,n)$-th _$\mathcal{A}$ -twisted motivic cohomology group_
$H^{i}_{\mathrm{mot}}(X;\mathbf{Z}^{\mathcal{A}}(n))$ is defined to be
$\pi_{2n-i}\operatorname{map}_{\mathbf{SH}(X)}(\Sigma^{\infty}_{\mathbf{T}}X_{+},\Sigma^{n}_{\mathbf{T}}H\mathbf{Z}^{\mathcal{A}})$.
When $\mathcal{A}$ is an Azumaya algebra over a perfect field $k$, the group
$H^{i}_{\mathrm{mot}}(X;\mathbf{Z}^{\mathcal{A}}(n))$ is isomorphic to the
$(i-2n)$-th cohomology group of the $\mathcal{A}$-twisted Bloch cycle complex
$z^{n}_{\mathcal{A}}(X)$ by Proposition 5.15, and therefore to the twisted
Chow groups of [KL10, Definition 5.6.3].

By considering the slice spectral sequence for $\mathrm{KGL}^{\mathcal{A}}$ we
obtain the following corollary, which recovers the spectral sequence of [KL10,
Corollary 6.1.4] when $\mathcal{A}$ is pulled back from the base field.

######  Corollary 5.18.

Let $k$ be a field of exponential characteristic $e$, $X$ a regular
Grothendieck $k$-scheme, and $\mathcal{A}$ an Azumaya algebra over $X$. Then
there is a strongly convergent spectral sequence

$E^{p,q}_{2}=H^{p-q}_{\mathrm{mot}}(X;\mathbf{Z}^{\mathcal{A}}[1/e](-q))\Rightarrow
K_{-p-q}^{\mathcal{A}}(X)[1/e]\,.$

Moreover if the field $k$ is perfect and $\mathcal{A}$ is pulled back from
$k$, then one can avoid inverting $e$, so that the spectral sequence has the
shape

$E^{p,q}_{2}=H^{p-q}_{\mathrm{mot}}(X;\mathbf{Z}^{\mathcal{A}}(-q))\Rightarrow
K_{-p-q}^{\mathcal{A}}(X)\,.$

###### Proof.

The slice spectral sequence for $\mathrm{KGL}^{\mathcal{A}}$ converges to
$K^{\mathcal{A}}(X)$ by Proposition 4.5 and Proposition 3.2. Moreover its
$E_{2}$-page can be identified with motivic cohomology by Theorem 5.7 when
$\mathcal{A}$ is pulled back from a perfect field and by Proposition 5.13
after inverting $e$.

Finally, the strong convergence of the spectral sequence follows as in [KL10,
Remark 6.1.2] from the fact that $\mathrm{kgl}^{\mathcal{A}}$ is very
effective, which was proved in Lemma 5.5. ∎

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