# Generic motives and motivic cohomology of fields

Frédéric Déglise

► To cite this version:

Frédéric Déglise. Generic motives and motivic cohomology of fields. 2024. hal-04825566

HAL Id: hal-04825566

https://hal.science/hal-04825566v1

Preprint submitted on 8 Dec 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# GENERIC MOTIVES AND MOTIVIC COHOMOLOGY OF FIELDS

F. DÉGLISE

ABSTRACT. These notes are devoted to the study of generic motives and their application to the structure of the motivic cohomology of fields. Related to Voevodsky's homotopy t-structure on motivic complexes, generic motives are also central to the conjectural motivic t-structure, as was shown by Beilinson. Morphisms of generic motives are closely related to the functoriality of Rost's cycle modules. We obtain some new computations of generic motives, and deduce computations of morphisms of generic motives, which draw a conjectural picture.

The structure of generic motives reflects into that of motivic cohomology of fields. After giving a new argument for determining weights in the K-theory of number fields using Borel's isomorphism, we deduce that many motivic cohomology groups of fields have infinite rank, usually of the same cardinality as the field itself. As an example motivic cohomology groups of the real or complex numbers are shown to be uncountable in many degrees and twists. Despite these results, we formulate a hope on their behavior with respect to transcendental elements, providing a counterpart to the Beilinson-Soulé vanishing conjecture.

This paper is dedicated to the memory of Jacob Murre, whose enthusiasm for, and commitment to, Grothendieck's theory of motives was a driving force for me.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Guide for reading</td><td>5</td></tr><tr><td>1.2. Conventions</td><td>6</td></tr><tr><td>1.3. Acknowledgement</td><td>7</td></tr><tr><td>2. Preliminary recall</td><td>7</td></tr><tr><td>2.1. Notation from motivic homotopy</td><td>7</td></tr><tr><td>2.2. Motivic cohomology</td><td>10</td></tr><tr><td>2.3. Pro-object</td><td>12</td></tr><tr><td>3. The category of Generic motives</td><td>12</td></tr><tr><td>3.1. Generic motives and points for the homotopy t-structures</td><td>12</td></tr><tr><td>3.2. Comparison with Beilinson's approach</td><td>16</td></tr><tr><td>3.3. Morphisms of generic motives and cycle modules</td><td>18</td></tr><tr><td>4. Computations of generic motives</td><td>23</td></tr></table>

Date: July 2024.
<table><tr><td>4.1. Rational curves</td><td>23</td></tr><tr><td>4.2. Positive genus curves</td><td>25</td></tr><tr><td>4.3. Higher dimension</td><td>28</td></tr><tr><td>5. Motivic cohomology of fields</td><td>31</td></tr><tr><td>5.1. Borel's computations and λ-structures</td><td>31</td></tr><tr><td>5.2. The effect of transcendental elements</td><td>39</td></tr><tr><td>5.3. Motivic cohomology with coefficients in abelian varieties</td><td>41</td></tr><tr><td>Appendix</td><td>43</td></tr><tr><td>6. Categorical presentation of cycle modules</td><td>43</td></tr><tr><td>7. Recall on Bousfield localizations</td><td>46</td></tr><tr><td>References</td><td>46</td></tr></table>

# 1. INTRODUCTION

One of the acts of birth of motivic cohomology was, retrospectively, Borel's 1974 computation of the K-theory of number fields as a corollary of that of the stable real cohomology of arithmetic groups [Bor74]. Higher K-theory had barely been introduced by Quillen in his celebrated paper [Qui73b], and Lichtenbaum had just embarked in a lifetime quest to develop higher regulators and analogs of Dirichlet's class number formula. Both works were published in the same lecture notes in 1973. The next step seemed to have been crystallized by two works in the year 1979: the first one is a paper due to Deligne [Del79] on special values of L-functions attached to motives, and the second one consists of lectures of Spencer Bloch, on algebraic cycles. They both lead to the second age of motives, led by the famous conjectures due to Beilinson, stated in [Bi84] in 1984. Not long after the statement of these conjecture, Voevodsky's PhD proposed a new approach via the famous h-topology, and the category of h-motivic complexes. This opened the way to a totally new era, which encompassed mixed motivic complexes into a homotopy theory for algebraic variety, now known as motivic homotopy theory. It led to many success, and most notably the solution to the Milnor and Bloch-Kato conjectures by Voevodsky ([Voe11]), with important contributions by Rost ([Ros96]).

At the time being, Beilinson's conjectures on motivic complexes have been positively answered, with the construction of several suitable categories of mixed motives, equipped with six functors and an $\ell$-adic realization functor compatible with the 6 operations (see [Voe92, VSF00, Ayo07, CD19, Ayo14, CD16, CD15, Par21, Spi18]).<fnref n="1" /> One main problem remains open, the question of the motivic t-structure. Though its definition is clear, its properties are bond to conjectures, such as the Beilinson-Soulé conjecture, which can be reduced to the question of the motivic

<footnote n="1">¹The publication of [CD19], available on arXiv in its almost final form since 2009, has been delayed more than reasonable by unfortunate events independent of the will of the authors.</footnote>
cohomology of fields. This is one of the reason to consider the so-called generic motives which is one of the central notion adressed in these notes.

Recall also that Voevodsky's theory of motivic complexes DM($k$) over a perfect field $k$ comes with a natural $t$-structure, called the homotopy $t$-structure. Generic motives were introduced in [Dég02] in order to study the objects of its heart, the homotopy modules with transfers.<fnref n="2" /> One of the main results of Voevodsky is that the corresponding sheaves (homotopy invariant sheaves with transfers) admits a Gersten resolution. Generic motives allow to enhance this result, showing that Voevodsky's homotopy modules with transfers are equivalent to Rost's cycle modules (see Theorem 3.3.7 for recall). Indeed, one of the main part of this result is to be able to recover the rich functoriality of cycle modules out of that of homotopy modules. One of the result of [Dég02] is that this functoriality is completely encoded into morphisms of generic motives (see recall in Section 3.3). The first goal of these notes is to improve the theory of generic motives, in several direction.

It is time to recall that the generic motive of a function field<fnref n="3" /> $F$ over $k$ is the pro-motive associated with all possible smooth schematic models $X$ over $k$:

$$ \hat{M}(F) := \text{``lim''} M(X), $$

where $M(X)$ is the homological motive over $k$ associated with $X$ — see also Definition 3.1.4. Such a generic motive can be twisted $\hat{M}(F)\{n\}$ with respect to an arbitray integer $n \in \mathbb{Z}$, with respect to $\mathbb{G}_m$-twists: $1\{1\} := 1(1)[1] \subset M(\mathbb{G}_m)$.

Fundamentally, generic motives form a conservative family of motivic points, with respect to the homotopy $t$-structure on DM($k$). In other words, $\hat{M}(F)\{i\}$ is derived pro-$t$-projective with respect to the $t$-category DM($k$): see Proposition 3.2.10. This echoes Beilinson's notion of correspondences at the generic point, which were introduced in [Bei02]. In his work, Beilinson considers the dual of generic motives, which are ind-cohomological motives; see Proposition 3.2.4 for details. In fact, he also proves that the latter allows to conjecturally build a conservative of derived $t$-injective objects with respect to the motivic $t$-structure on the triangulated category of $n$-motives, for an arbitrary large integer $n \ge 0$; see [Bei02, Prop. 4.5] for details.

Therefore, the category of generic motives is fundamental to understand motivic complexes. In particular, its morphisms can be considered as specialization morphisms, and admits a (partial) description in terms of the functoriality of cycle modules. We improve the computations of morphisms by explicitly computing generic motives associated with low dimensional varieties: curves and surfaces. This illustrates Beilinson's theory as it appears that the basic bricks of motives, given by jacobians for curves, and by the so-called transcendental motives of Kahn,

<footnote n="2">²Recall on Voevodsky's theory, effective and stable, and the homotopy $t$-structure are given in Section 2.1.</footnote>

<footnote n="3">³finitely generated field extension</footnote>
Pedrini and Murre, appears as constituent of the corresponding generic motives of their function fields:

**Proposition A.** Let $X$ be a smooth proper scheme over $k$.

(1) If $X = C$ is a curve with Albanese scheme $A$ (dual of the Jacobian), there exists a homotopy exact sequence of pro-motives over $k$:

$$ \text{``} \prod_{x \in C_{(0)}} \hat{M}(\kappa_x) \{1\} \xrightarrow{\prod_x \partial_x} \hat{M}(K) \xrightarrow{(\pi_0^K, \pi_1^K)} 1 \oplus \underline{A} $$

which is split if and only if $C$ is rational. We refer to Proposition 4.2.2 for more details and notation.

(2) If $X = S$ is a surface, with Albanese scheme $A$, first Betti number $b_1$, and homological transcendental motive $M_2^{tr}(X)$, one gets two homotopy exact sequences of pro-motives:

$$ \begin{tikzcd} \text{``} \prod_{s \in S^{(2)}} \hat{M}(\kappa_s) \{2\} \arrow[r] & \\ \text{``} \prod_{x \in S^{(1)}} \hat{M}(\kappa_x) \{1\} \arrow[r, "\prod_x \partial_x"] & \hat{M}(K) \arrow[r] & \hat{M}(S^{\le 1}) \arrow[d] \\ 1 \oplus \underline{A} \oplus b_1 \cdot 1(1)[2] \oplus M_2^{tr}(S) \oplus \underline{A}(1)[2] \oplus 1(2)[4] \arrow[d] & & \end{tikzcd} $$

The pro-motive $M(S^{\le 1})$ has a precise interpretation in term of the coniveau filtration: see Proposition 4.3.7 for details.

Generic motives are therefore extensions (non-split in general) of the fundamental bricks of (mixed) motives. Both results shed new light on morphisms of generic motives. They confirm the fact that morphisms of generic motives should be topologically generated by Rost's four basic maps (see Section 4.1).

Generic motives reflect the properties of motivic cohomology of fields. As stated previously, determining the latter is the next fundamental questions one faces in Beilinson's approach to mixed motives. At the time being, the only results at our disposal are: Quillen's finiteness theorem for the K-theory of number rings [Qui73a] and computation of algebraic K-theory of finite field [Qui72], Borel's computation of the rank of algebraic K-theory of number rings [Bor74], and Harder's analogous computation for function fields of curves over a finite field [Har77].

It is known how to compute the rank of motivic cohomology of number fields and rings (see the precise formulation in Theorem 5.1.1), out of the computation of Borel. It consists in comparing Borel and Beilinson's regulators: see [BG02]. This is rather indirect. By studying the compatibility of Borel's computation with $\lambda$-filtrations at each step, we give a direct proof of this computation in Section 5.1.
The main result of these notes, deduced from the computation of generic motives and the preceding result of Borel is the fact motivic cohomology $H_M^{**}(K)$ of a field $K$ of characteristic 0 is infinite in a wide range. More precisely, we prove:

**Theorem B.** Let $K$ be a field of characteristic 0 of transcendence degree $d > 0$.

Then for any $n \in [2, d+1]$ and any $i \ge n$, the abelian group $H_M^{n,i}(K)$ has infinite rank equal to $\mathrm{card}(K)$.

In particular, motivic cohomology of the real or complex numbers is uncountable in the indicated range. We refer the reader respectively to Proposition 5.2.2 and Proposition 5.2.4 in the text for proofs. In the range $n = i$, where motivic cohomology agrees with Milnor K-theory, the result was already known from Bass and Tate [BT73, Prop. 5.10]. For further information, in particular in positive characteristic, we refer the reader to Section 5.2.

Though this theorem shows that motivic cohomology of fields is usually enormous, the behavior with respect to purely transcendental extensions suggests the following plausible conjecture:

**Conjecture C.** Let $K$ be a field finitely generated over its prime field $F$, with Krull dimension:

$$ \delta(K) = \begin{cases} 1 + \mathrm{degtr}(K/\mathbb{Q}) & F = \mathbb{Q} \\ \mathrm{degtr}(K/\mathbb{F}_p) & F = \mathbb{F}_p \end{cases} $$

Then, one has the following vanishing of the rational motivic cohomology of $K$

$$ H_{\mathbb{B}}^{n,i}(K) = 0 $$

unless $(n, i) = (0, 0)$ or $n \in [1, \delta(K)]$.

The conjecture is known in any case for $\delta(K) \le 1$. We refer the reader to the end of Section 5.2 for further discussion.

**1.1. Guide for reading.** The conventions and main notation of this paper are gathered in the next Section 1.2. A more detailed summary of the theories used in these notes are given in the first section. This concerns Voevodsky's motives over a field, the known computation of motivic cohomology, and the homotopical theory of pro-object using $\infty$-categories. This section might serve as a reference section for the reader in the remaining of this text. Note we have also provided the reader with details of useful notions for the reading of the paper, in the Appendix: Section 6 on cycle modules is mainly due to Rost, and Section 7 recalls the basics of Bousfield localization.

Section 3, is concerned with the theory of generic motives, described previously. The definition follows that given in the PhD of the author [Dég02], but with the advantages of the $\infty$-categorical language. The importance of generic motives come from their close relation with Voevodsky's homotopy t-structure. The crucial property is Proposition 3.2.10. Secondly, generic motives can be compared with Beilinson's notion of cohomological motives at the generic point, which were shown
to be crucial for the motivic *t*-structure in [Bei02]. At the price<fnref n="4" /> of inverting the
characteristic exponent *e* of the base field *k* we get an anti-equivalence of categories
in Proposition 3.2.2. Section 5.3 finally recalls the rich functoriality of generic
motives, and the link with cycle modules (Theorem 3.3.7).

In Section 4, we give several computations of generic motives, and of morphisms
of generic motives: respectively, the case of the function field of rational curves
(§4.1), arbitrary curves (§4.2), surfaces, and arbitrary higher dimension (§4.3). The
attentive reader will have noticed a tentative conjecture to describe morphisms of
generic motives at the end of Section 4.1, which allows to make more precise a
question asked in the PhD of the author (see Remark 4.1.6).

In the last part of these notes, Section 5, we focus on motivic cohomology of
fields. We first consider the case of number fields: as explained previously, we show
how to deduce from Borel’s computation of the rank of their K-groups the one of
rational motivic cohomology, by a detailed study of the involved isomomorphisms
with respect to appropriate λ-structures. We then deduce from the computation of
particular generic motives, obtained in Section 4, some facts about motivic
cohomology of fields, and the conjecture mentioned above. The section closes with
some interesting cohomology theory that could be of interests.

1.2. **Conventions.** *Category theory.* We will work with $\infty$-categories, as defined in [Lur09, Lur17, Cis19]. By convention, all our $\infty$-categories will be stable presentable monoidal $\infty$-categories. We will often say functor for $\infty$-functors, when it does not lead to confusion. Given a stable $\infty$-category $\mathcal{C}$, we will denote by $\mathrm{Map}_\mathcal{C}$ the mapping space functor, seen in the $\infty$-category of spectra (see e.g. [Gal23, Constr. 5.50]). We let $\mathrm{Ho}\mathcal{C}$ be the associated homotopy category, with its canonical triangulated structure ([Lur17, Th. 1.1.2.13]). We simply denote by $\mathrm{Hom}_\mathcal{C}$ the abelian groups of morphism in $\mathrm{Ho}\mathcal{C}$, so that $\mathrm{Hom}_\mathcal{C}(X, Y) = \pi_0 \mathrm{Map}_\mathcal{C}(X, Y)$, where $\pi_0$ is the 0-th stable homotopy group of the corresponding spectrum.

The use of $\infty$-category is justified by the consideration of pro-objects in homotopy categories (such as Voevodsky motives) which is central in this paper; see Section 2.3.

We say monoidal for symmetric monoidal. We say *rigid* or *strongly dualizable* for an object in such a monoidal category which admits a strong dual.

Given an abelian group $A$, we put $A_Q = A \otimes_Z Q$ and $A_e = A[1/e]$ for $e > 0$.
If $C$ is a stable $\infty$-category, we let $C_Q$ (resp. $C_e$) be the $Q$-linearisation (resp.
$e$-localisation) of $C$.

*Algebraic geometry.* We will fix a base field $k$ of characteristic exponent $e: e = 1$ (resp. $p$) if $k$ is a $Q$-algebra (resp $\mathbb{F}_p$-algebra for a prime $p$). When using $\mathbb{Z}$-coefficients, the field $k$ is assumed to be perfect.<fnref n="5" /> In order to use de Jong’s

<footnote n="4">⁴theoretically for duality reasons, practically for using resolution of singularities</footnote>

<footnote n="5">⁵This assumption is used in the theory of motivic complexes as recalled in Section Section 2.1.</footnote>
resolution of singularities, we sometime use $\mathbb{Z}[1/e]$-coefficients. Then $k$ can be arbitrary as the categories becomes invariants under purely inseparable extensions.

We say smooth for smooth of finite type.

Given a scheme $X$, we let $X^{(n)}$ (resp. $X_{(n)}$) be the set of points of $X$ of codimension (resp. whose reduced closure has Krull dimension) $n$.

A function field over $k$ will be a separable finitely generated extension $F$ of $k$. Without precision, a valuation $v$ on $E/k$ will be assumed to be discrete and with ring of integers $O_v$ a $k$-algebra essentially of finite type. When one wants to be precise, one says that $v$ is geometric. We also simply says that $(E, v)$ is a valued function field.

*Motives.* The theory and notation of motivic complexes is recalled in Section 2.1. We denote byt $1_k$ the unit for the $\otimes$-structure on motivic complexes, effective and stable, and use the special notation for twists:

$$1\{1\} = 1(1)[1].$$

1.3. **Acknowledgement.** Among those who greatly inspired me in the theory of motives, Jacob Murre shares the place with Sasha Beilinson and Vladimir Voevodsky. In a more practical way, I thank Simon Pepin Lehalleur, Jan Nagel, Marc Levine, Bruno Kahn, Raphaël Ruimy and Swann Tubach for discussions that helped me greatly while conceiving the mathematics behind this paper. Special thanks go to Olivier Taïbi without whom Borel's regulator would still be inaccessible to me. He helped me figure out its crucial compatibility with $\lambda$-operations. I also greatly thank all the participants of the covid 2021 workgroup on Borel's paper housed in the ENS Lyon *matrix*.

The author is supported by the ANR project "HQDIAG", ANR-21-CE40-0015.

## 2. PRELIMINARY RECALL

2.1. **Notation from motivic homotopy.** We will use Voevodsky's theory of motivic complexes. This theory has been presented in many places: [VSF00, Chap. 5], [MVW06], [Dég07, Dég11]. We will use the notation of [Dég11, CD19] that we will now gather for the comfort of the reader.

**2.1.1. The effective category.** ([Dég11, §4.1, 5.1], [CD19, §11.1]) The category of effective motivic complexes $\mathrm{DM}^{\mathrm{eff}}(k)$ is the $\mathbb{A}^1$-localization of the derived category of $\mathbb{Z}$-linear sheaves with transfers $\mathrm{Sh}^{\mathrm{tr}}(k)$. As such, it can be identified with the full-subcategory of $\mathrm{D}(\mathrm{Sh}^{\mathrm{tr}}(k))$ made of $\mathbb{A}^1$-local complexes $K$. According to the fundamental theorem of Voevodsky, $K$ is $\mathbb{A}^1$-local if and only if its homotopy sheaves $\mathcal{H}_i(K)$ are $\mathbb{A}^1$-invariant. One deduces that $\mathrm{DM}^{\mathrm{eff}}(k)$ admits a $t$-structure, called the *homotopy t-structure*, whose heart $\mathrm{HI}(k)$ is made of $\mathbb{A}^1$-invariant sheaves with transfers. We use homological conventions for the homotopy $t$-structure.

The $\mathbb{A}^1$-localization functor of a complex $K$ in $\mathrm{D}(\mathrm{Sh}^{\mathrm{tr}}(k))$ is given by Suslin's singular complex $\mathrm{C}_*^{\mathrm{sus}}(K)$. The *motive* (aka *motivic complex*) of a smooth $k$-scheme $X$ of finite type is the complex $M(X) = \mathrm{C}_*^{\mathrm{sus}}(\mathbb{Z}^{\mathrm{tr}}(X))$ where $\mathbb{Z}^{\mathrm{tr}}(X)$ is
the sheaf with transfers represented by $X$.<fnref n="6" /> The tensor product on $\mathrm{DM}^{\mathrm{eff}}(k)$ is characterized by the property $M(X) \otimes M(Y) = M(X \times Y)$ for $X, Y$ smooth. We denote by $\mathbb{1}_k = M(k)$ the unit object for this tensor structure.

The category of *effective geometric motives* $\mathrm{DM}_{\mathrm{gm}}^{\mathrm{eff}}(k)$ is the subcategory of $\mathrm{DM}^{\mathrm{eff}}(k)$ made by finite sums, extensions, shifts and direct summands of motives of the form $M(X)$ for $X$ smooth. It coincides with the subcategory of compact objects of $\mathrm{DM}^{\mathrm{eff}}(k)$.

2.1.2. *The stable category.* ([Dég11, §4.2, 5.2], [CD19, §11.1]) One defines the *Tate twist* by the following formula:

$$ \mathbb{1}(1) = M(\mathbb{P}_k^1)/M(\{\infty\})[-2] = M(\mathbb{G}_m)/M(\{\infty\})[-1]. $$

Recall that the motivic complex $\mathbb{1}(1)$ is concentrated in degree 1 equal to the sheaf with transfers $\mathbb{G}_m$. It is convenient, and justified from the point of view of the homotopy t-structure, to introduce the following redundant notation for twists:

$$ \mathbb{1}\{1\} = \mathbb{1}(1)[1] = \mathbb{G}_m. $$

There exists a universal (monoidal stable presentable) $\infty$-category $\mathrm{DM}(k)$ equipped with a functor $\Sigma^\infty : \mathrm{DM}^{\mathrm{eff}}(k) \to \mathrm{DM}(k)$, called the infinite (Tate) suspension functor, such that $\Sigma^\infty \mathbb{1}(1)$ is $\otimes$-invertible.<fnref n="7" /> The functor $\Sigma^\infty$ admits a right adjoint $\Omega^\infty$, the infinite (Tate) loop space functor. We will still simply denote by $\mathbb{1}_k = \Sigma^\infty M(k)$ the unit for the tensor structure on $\mathrm{DM}(k)$.

There exists a unique $t$-structure on $\mathrm{DM}(k)$ such that $\Omega^\infty$ and the functor tensoring with $\mathbb{1}\{1\}$ are $t$-exact. Its heart is the abelian category $\mathrm{HM}(k)$ of homotopy modules with transfers. Recall that a homotopy module with transfers is a $\mathbb{Z}$-graded sheaf with transfers $F_* = (F_n)_{n \in \mathbb{Z}}$ equipped with a sequence of isomorphisms $(F_{n+1})_{-1} \simeq F_n$ for $n \in \mathbb{Z}$, where $?_{-1}$ is Voevodsky's $(-1)$-construction.<fnref n="8" />

Voevodsky's cancellation theorem ([Voe10]) asserts that $\Sigma^\infty$ is fully faithful. Therefore, we will use the same notation for $M(X)$ and $\Sigma^\infty M(X)$. One also deduces that $\Sigma^\infty$ induces a canonical fully faithful functor:

$$ \sigma^\infty : \mathrm{HI}(k) \to \mathrm{HM}(k). $$

As in the effective case, the category of *geometric motives* $\mathrm{DM}_{\mathrm{gm}}(k)$ is the subcategory of $\mathrm{DM}(k)$ made by finite sums, extensions, shifts and direct summands of motives of the form $M(X)(n)$ for $X$ smooth, $n \in \mathbb{Z}$. It coincides with the subcategory of compact objects of $\mathrm{DM}(k)$.

<footnote n="6">⁶The definition of $M(X)$ makes sense for a not necessarily smooth $k$-scheme $X$. In the non smooth case, $M(X)$ has good properties only after inverting $e$. In these notes, we will only use $M(X)$ for $X/k$ smooth.</footnote>

<footnote n="7">⁷See [Rob15, Def. 2.6].</footnote>

<footnote n="8">⁸This notion was introduced in the PhD of the author; see [Dég11, §2] for more details.</footnote>
**2.1.3. Duality and Chow motives.** Let $X$ be a smooth $k$-scheme. We let $h(X) := M(X)^{\vee} = \operatorname{Hom}(M(X), \mathbb{1}_k)$ be its canonical dual, the cohomological motive associated with $X$.<fnref n="9" />

If $X/k$ is smooth projective of dimension $d$, it follows from [Dég08a, Th. 5.23] that $M(X)$ is rigid in $\operatorname{DM}(k)$ with dual $h(X)$. Moreover, there exists a canonical isomorphism $h(X) \simeq M(X)(-d)[-2d]$. For a general smooth $k$-scheme $X$ of dimension $d$ (which can even be assumed to be non-smooth but of finite type), up to inverting the characteristic exponent $e$ of $k$, one deduces that $M(X)$ is rigid in $\operatorname{DM}(k)_e$ with dual $h(X)$. Moreover, one gets an isomorphism $h(X) \simeq M^c(X)(-d)[-2d]$, where $M^c(X)$ is the associated motive with compact support in $\operatorname{DM}(k)_e$ (see [VSF00, Chap. 5, 4.3.7] if $e = 1$, and [CD15, Prop. 8.10, Rem. 7.4] in general).<fnref n="10" />

Our reference for Chow motives will be [MNP13]. One gets a contravariant monoidal fully faithful embedding:

$$ \operatorname{CHM}(k)^{\mathrm{op}} \to \operatorname{HoDM}_{\mathrm{gm}}(k) $$

with source the pseudo-abelian category of Chow motives over $k$, under which the Chow motive $h(X) = (X, \Delta_X, 0)$ (op. cit. §2.2.1) of a smooth projective $k$-scheme $X$ is mapped to the (homological) motivic complex $M(X)$.<fnref n="11" />

**Remark 2.1.4.** The construction of the category $\operatorname{DM}(k)$ via sheaves with transfers has been extended to arbitrary noetherian finite dimensional bases in [CD19]. One should be careful that over a positive dimensional base, the $\mathbb{A}^1$-local complexes (which form the effective category, as above) do not admit a simple description as above. They are given by complexes whose (hyper)cohomology is $\mathbb{A}^1$-invariant. One can define a stable category, using essentially the same procedure as above. It admits all of the six functors but, up to now, the localization property for closed immersion has only been proved in some restricted cases (see [CD19, 11.4] for details).

It was shown in [CD15] that, up to replacing the Nisnevich topology with the cdh-topology, working over $k$-schemes $S$ for a fixed base field $k$, and inverting the characteristic exponent of $k$, then localization holds and therefore, all of the six functors formalism (including duality) is available for these “cdh-motives”. Besides, they coincide with “modules over motivic cohomology”, that is more precisely, motives over the motivic Eilenberg-MacLane spectrum as defined by Voevodsky.

The strategy of defining motives as modules over an appropriate spectrum was then developed by Markus Spitzweck in [Spi18], where he defines an appropriate Eilenberg-MacLane motivic spectrum over a Dedekind scheme (say $\mathbb{Z}$) using Bloch's higher Chow groups. Taking pullbacks in the stable motivic homotopy

<footnote n="9">⁹In term of six functors, one has $h(X) = p_*(\mathbb{1}_k)$, $p: X \to \operatorname{Spec}(k)$ the structural map.</footnote>

<footnote n="10">¹⁰In terms of six operations, one has $M^c(X) = f_*f^!(\mathbb{1}_k)$, loc. cit. 8.10.</footnote>

<footnote n="11">¹¹This is coherent with the fact $M(X) = h(X)^{\vee}$ as recalled above.</footnote>
category then provides the correct $E_\infty$-ring spectrum. Finally, modules over this $E_\infty$-spectrum inherits all of the properties of the 6 functors formalism from motivic stable homotopy — this last step was already highlighted in the abstract case and in the rational coefficients case in [CD19, §7 and Th. 14.2.9] so that we know that Spitzweck’s construction and the one highlighted above for the Nisnevich topology agrees with rational coefficients over geometrically unibranch schemes. For arbitrary singular schemes and rational coefficients, this also holds provided one uses the cdh-topology.

## 2.2. Motivic cohomology.

**2.2.1. Definition via motivic complexes.** Let $k$ be a field. Let $X$ be a smooth $k$-scheme, and $(n, i) \in \mathbb{Z}^2$ be a pair of integers. Following Voevodsky ([VSF00, chap. 5]), one can define the motivic cohomology of $X$ in degree $n$ and twists $i$ as morphisms in the category DM$(k)$:

$$H_M^{n,i}(X) = \operatorname{Hom}_{\operatorname{DM}(k)}(\operatorname{M}(X), \mathbb{1}(i)[n]).$$

It follows from the so-called continuity property (see [CD19, 11.1.24, 11.1.25]) that these cohomology groups commute with projective limits of schemes whenever the transition morphisms of an essentially affine pro-scheme $(X_\lambda)_{\lambda \in \Lambda}$ are dominant (see loc. cit. for recall). So when $X = \lim_{\lambda \in \Lambda} X_\lambda$ we get under the preceding assumptions:

$$(2.2.1.a) \qquad H_M^{n,i}(X) \simeq \operatorname{colim}_{\lambda \in \Lambda} H_M^{n,i}(X_\lambda).$$

These motivic cohomology groups can be computed with the Nisnevich cohomology of the $i$-th motivic complex $\mathbb{1}(i) = \mathbb{Z}(i) = C_*^{\text{sus}}(\mathbb{Z}^{\text{tr}}(\mathbb{G}_m^{\wedge,i}))^{12}$:

$$H_M^{n,i}(X) \simeq H_{\text{Nis}}^n(X, \mathbb{Z}(i)).$$

This isomorphism follows from Voevodsky’s theory (recalled above) when $k$ is perfect. In the general case, $k$ is essentially smooth over its prime field $F$, so that $X$ is also essentially smooth over $F$, and one reduces to the case of $F$ using the continuity property. Along with the (quasi-)isomorphisms of motivic complexes

$$\mathbb{1}(i) \simeq \begin{cases} \mathbb{Z} & i = 0, \\ \mathbb{G}_m[-1] & i = 1, \end{cases}$$

$$\bigoplus_{i=1}^{n} \mathbb{Z}^{\text{tr}}(\mathbb{G}_m^{i-1}) \to \mathbb{Z}^{\text{tr}}(\mathbb{G}_m^n)$$

obtained by considering all the possible inclusions.

<footnote n="12">¹²Here $\mathbb{G}_m^{\wedge,i}$ means the smash product of the pointed scheme $(\mathbb{G}_m, 1)$, and $\mathbb{Z}^{\text{tr}}(\mathbb{G}_m^{\wedge,i})$ denotes the associated sheaf with transfers. Equivalently, $\mathbb{Z}^{\text{tr}}(\mathbb{G}_m^{\wedge,i})$ is the cokernel of the map:</footnote>
this implies the following computations of these motivic cohomology groups:

$$ (2.2.1.b) \qquad H_{\mathbb{M}}^{n,i}(X) = \begin{cases} \mathbb{Z}^{\pi_0(X)} & i = n = 0, \\ \mathcal{O}_X(X)^{\times} & i = n = 1, \\ \mathrm{Pic}(X) & (n, i) = (2, 1), \\ 0 & i < 0 \text{ or } n > 2i \text{ or } n - i > \dim(X). \end{cases} $$

One further gets for any integer $n \ge 0$ the following isomorphisms:

$$ (2.2.1.c) \qquad H_{\mathbb{M}}^{2n,n}(X) \simeq \mathrm{CH}^n(X) $$

$$ (2.2.1.d) \qquad H_{\mathbb{M}}^{n,n}(E) \simeq K_n^M(E) $$

with respectively is the classical Chow groups of codimension $n$ algebraic cycles of $X$ modulo rational equivalence, and the $n$-th Milnor K-theory of any function field $E/k$.

Recall finally that it is known due to a theorem of Voevodsky that motivic cohomology coincides with Bloch's higher Chow groups,<fnref n="13" /> according to the following formula:

$$ (2.2.1.e) \qquad H_{\mathbb{M}}^{n,i}(X) \simeq \mathrm{CH}^i(X, 2n - i). $$

**2.2.2. Rational coefficients.** Recall that the rational part of the above motivic cohomology groups can be compared with Quillen higher algebraic K-theory according to the following fundamental formula:

$$ (2.2.2.a) \qquad H_{\mathbb{M}}^{n,i}(X)_{\mathbb{Q}} \simeq K_{2i-n}^{(i)}(X) $$

where we have denoted by $K_{2i-n}^{(i)}(X)$ the $i$-th graded parts for the $\gamma$-filtration on rational K-theory $K_{2i-n}(X)_{\mathbb{Q}}$. In [Sou85], the right hand-side was actually taken as a definition of motivic cohomology, following the lead of Beilinson. For this reason, we also use the notation (due to Riou):

$$ H_{\mathbb{D}}^{n,i}(X) = H_{\mathbb{M}}^{n,i}(X)_{\mathbb{Q}}. $$

In the case of a smooth $k$-scheme, the above formula is a consequence Voevodsky's isomorphism (2.2.1.e), and the corresponding isomorphism for rational higher Chow groups due to Bloch ([Blo86]). A more general proof, valid for an arbitrary regular scheme $X$ (without a base field), is given in [CD19, 14.2.14, 16.1.4].<fnref n="14" />

<footnote n="13">¹³Bloch's definition, motivated by earlier results of Kratzer and Soulé, and by the ongoing excitement that emerged from the picture laid down by Lichtenbaum and Beilinson, appeared around 10 years before Voevodsky's definition. For the record, it was the will of repairing a mistake in the proof of the localization property for higher Chow groups that motivated Suslin to introduce Suslin homology in a talk in the CIRM at Luminy in 1987. This seminal definition was a major contribution that shaped Voevodsky's theory of motivic complexes a few years later. Meanwhile, the mistake was fixed by Bloch [Blo94] and Levine [Lev94].</footnote>

<footnote n="14">¹⁴For singular schemes $X$, several works have attempted to propose a suitable definition for an integral version of the left hand-side of (2.2.2.a), whose rational part coincides with Soulé's definition. This is still an active area of research.</footnote>
2.3. **Pro-object.** Our references for pro-objects is [AM69, Appendix] for 1-categories, and [Lur09, §5.3], [BHH17] for $\infty$-categories.

Let $\mathcal{C}$ be a presentable $\infty$-category. The $\infty$-category pro-$\mathcal{C}$ admits cofiltered (homotopy) projective limits. In particular, given an $\infty$-functor $X : \mathcal{J} \to \mathcal{C}, i \mapsto X_i$, one can consider the associated projective limit in pro-$\mathcal{C}$ that we will denote by "$\lim_{i \in \mathcal{J}}$ $X_i$ following a convention of Deligne.

According to [BHH17, Prop. 5.1.1, 5.1.2], given two $\infty$-functors $X : \mathcal{J} \to \mathcal{C}$ and $Y : \mathcal{J} \to \mathcal{C}$, there exists a canonical equivalence:

$$ \mathrm{Map}(\text{"lim}_{i \in \mathcal{J}} X_i, \text{"lim}_{j \in \mathcal{J}} Y_j) \simeq \lim_{j \in \mathcal{J}} \mathrm{colim}_{i \in \mathcal{J}} \mathrm{Map}(X_i, Y_j). $$

One should be careful that in general: $\mathrm{Ho}(\mathrm{pro-}\mathcal{C}) \neq \mathrm{pro-Ho}\mathcal{C}$ (but see Lemma 3.1.11).

### 3. THE CATEGORY OF GENERIC MOTIVES

3.1. **Generic motives and points for the homotopy t-structures.** We will use the following handy definition.

**Definition 3.1.1.** A $k$-scheme $\mathcal{X}$ is pro-smooth (resp. essentially smooth) if it is the projective limit of a pro-scheme $(X_i)_{i \in I}$ such that $X_i$ is a $k$-scheme of finite type, and the transition maps are étale (resp. open immersions) and affine.

According to [EGA4, 8.13.2], the category of pro-smooth (resp. essentially smooth) $k$-schemes can be viewed as a full sub-category of pro-Sm. In fact, the functor which to a pro-scheme $(X_i)_{i \in I}$ as above associate the projective limit $\mathcal{X} = \lim_{i \in I} X_i$ seen as an object of $\overline{\mathrm{Sm}}$ is an equivalence of categories. We will say that $(X_i)_{i \in I}$ is a presentation of $\mathcal{X}$, keeping in mind the choice of a particular presentation is irrelevant. In particular, we will consider the embedding$^{15}$:

$$ (3.1.1.a) \qquad \nu : \overline{\mathrm{Sm}} \to \mathrm{pro-Sm}_k. $$

**Example 3.1.2.** (1) Given any smooth scheme $X$, and any point $x \in X$, the localization $X_{(x)} = \mathrm{Spec}(\mathcal{O}_{X,x})$ and henselization $X_{(x)}^h = \mathrm{Spec}(\mathcal{O}_{X,x}^h)$ of $X$ at $x$ are respectively essentially smooth and pro-smooth over $k$.

(2) Our main example will come from function fields $E$ over $k$: $\mathrm{Spec}(E)$ is essentially smooth over $k$.

As in [Dég08b], one can use a canonical presentation of $\mathrm{Spec}(E)$, by considering the set $\mathcal{M}(E)$ of smooth sub-$k$-algebras $A \subset E$ with fraction field equal to $E$. Ordered by inclusion, this set is filtered and one has:

$$ \mathrm{Spec}(E) = \lim_{A \in \mathcal{M}(E)^{op}} \mathrm{Spec}(A). $$

<footnote>$^{15}$Of course, it formally corresponds to the choice of a reciprocal equivalence to the projective limit functor</footnote>
One gets another presentation by considering a smooth model<fnref n="16" /> of E: that is a connected smooth k-scheme X with a chosen k-isomorphism $\phi: E \simeq \kappa(X)$. Then one gets a canonical isomorphism:

$$ \mathrm{Spec}(E) = \lim_{U \subset X} U $$

where U runs over the non-empty open subset of X.

**3.1.3. The composite functor:**

$$ \mathrm{Sm}_k \xrightarrow{M} \mathrm{DM}_{\mathrm{gm}}(k) \to \mathrm{pro-DM}_{\mathrm{gm}}(k), X \mapsto \mathrm{M}(X) $$

admits a right Kan extension $\hat{M}$ along the inclusion $\mathrm{Sm}_k \to \overline{\mathrm{Sm}}_k$, in the $\infty$-categorical sense ([Cis19, §6.4]). This formally follows from the fact $\mathrm{pro-DM}_{\mathrm{gm}}(k)$ is complete<fnref n="17" /> as an $\infty$-category, but one gets a direct construction by using the following commutative diagram:

<figure><img src="image_2.png" /></figure>

In other words,

$$ \hat{M}(\mathcal{X}) = \text{``lim''}_{i \in I} \mathrm{M}(X_i) $$

for any presentation $(X_i)_{i \in I}$ of the essentially smooth k-scheme $\mathcal{X}$. As in [Dég08b, Def. 3.3.1], one highlights the following pro-motives:

**Definition 3.1.4.** Using the above notation, one defines the *generic motive* associated with a function field $E/k$ as the pro-motive $\hat{M}(E) := \hat{M}(\mathrm{Spec}(E))$.

The $\infty$-category $\mathrm{DM}^{(0)}(k)$ of generic motives (over $k$) is the full sub-$\infty$-category of $\mathrm{pro-DM}_{\mathrm{gm}}(k)$ whose objects are of the form $\hat{M}(E)\{n\}$ for any function field $E/k$ and any integer $n \in \mathbb{Z}$.

*Remark 3.1.5.* One should be careful that the above definition of generic motives is sensibly different to that of *loc. cit.* as it includes the higher homotopy information, contained in the mapping spaces of the $\infty$-category $\mathrm{DM}^{(0)}(k)$. However, we will see in Remark 3.1.12 that the homotopy category of the later coincides with the category defined in *loc. cit.*

**3.1.6.** The reason to focus on these particular pro-objects is that they form a conservative family of points for the t-category $\mathrm{DM}(k)$ equipped with its homotopy t-structure. Before giving a precise meaning to this sentence in the following corollary, we introduce another notation. Given an $R$-linear presheaf $F$ on $\mathrm{Sm}_k$,

<footnote n="16">¹⁶Resolution of singularities leads to the existence of a smooth projective model. Therefore, such a model exists in characteristic 0 or when the transcendence degree of $E/k$ is less than 3.</footnote>

<footnote n="17">¹⁷i.e. admits all small limits; which follows by construction</footnote>
we denote by $\hat{F}$ its left Kan extension along the inclusion $\mathrm{Sm}_k \to \overline{\mathrm{Sm}}_k$. Concretely, for any essentially smooth $k$-scheme $\mathcal{X}$, we have:

$$ \hat{F}(\mathcal{X}) = \colim_{i \in I} F(X_i) $$

for any presentation $(X_i)_{i \in I}$ of $\mathcal{X}$.

**Proposition 3.1.7.** Let $\mathcal{O}$ be an essentially smooth henselian semi-local $k$-algebra, $(n, i) \in \mathbb{Z}^2$ a pair of integers, and $\mathbf{E}$ be a stable motivic complex over $k$. Then there exists a canonical isomorphism:

$$ \mathrm{Hom}_{\mathrm{pro-DM}(k)}(\hat{\mathrm{M}}(\mathcal{O})\{i\}[n], \mathbf{E}) = \hat{\mathrm{H}}_{n,-i}(\mathbf{E})(\mathcal{O}) $$

where $\hat{\mathrm{H}}_{n,*}(\mathbf{E})$ is the $n$-th homology object associated with $\mathbf{E}$, seen as a homotopy module.<fnref n="18" />

*Proof.* First recall that $\hat{\mathrm{H}}_{n,*}(\mathbf{E}\{-i\}) = \hat{\mathrm{H}}_{n,*-i}(\mathbf{E})$. This reduces to the case $i = 0$. Then, using the cancellation theorem (recalled in 2.1.2), one can compute the relevant group morphism in $\mathrm{DM}^{\mathrm{eff}}(k)$. Then one uses the hypercohomology spectral sequence for the homotopy $t$-structure and one concludes as $\mathcal{O}$ is acyclic for Nisnevich cohomology with coefficients in an object of $\mathrm{HI}(k)$ (see e.g. [Ros96, Th. 6.1] taking into account [Dég11, 3.10, 3.11]). $\square$

**Example 3.1.8.** Let $X$ be a smooth projective $k$-scheme of dimension $d$. Using duality (see 2.1.3) and the continuity property (2.2.1.a), one immediately deduces the following computation:

$$ \hat{\mathrm{H}}_{n,-i}(\mathrm{M}(X))(E) = \mathrm{Hom}_{\mathrm{pro-DM}(k)}(\hat{\mathrm{M}}(E)\{i\}[n], \mathrm{M}(X)) = \mathrm{H}_{\mathrm{M}}^{2d-n-i,d-i}(X_E) $$

where $X_E = X \times_k E$.

One deduces the announced interpretation of generic motives as a conservative family of points with respect to the homotopy $t$-structure.<fnref n="19" />

**Corollary 3.1.9.** *The family of exact functors:*

$$ \mathrm{DM}(k) \to \mathrm{Mod}_R, \mathbf{E} \mapsto \mathrm{Hom}_{\mathrm{pro-DM}(k)}(\hat{\mathrm{M}}(E)\{i\}[n], \mathrm{M}(X)) $$

indexed by a function field $E/k$ and a pair of integers $(n, i) \in \mathbb{Z}^2$, is conservative.

In other words, $(\hat{\mathrm{M}}(E)\{i\})_{E/k, i \in \mathbb{Z}}$ is a conservative family of points for the homotopy $t$-structure, and this implies that the family of functors:

$$ \mathrm{HM}(k) \to \mathrm{Mod}_R, F_* \mapsto \mathrm{Hom}_{\mathrm{pro-HM}(k)}(\hat{\mathrm{H}}_0(\hat{\mathrm{M}}(E))\{i\}, F_*) = \hat{F}_{-i}(E) $$

is conservative.

The corollary follows from the fact that the homotopy $t$-structure is non-degenerate and from the corresponding fact for homotopy sheaves: [Dég07, Prop. 4.7].

<footnote n="18">¹⁸So the symbol $*$ refers to the natural graduation of homotopy modules, which actually coincides with the $\mathbb{G}_m$-graduation on $\mathbf{E}$.</footnote>

<footnote n="19">¹⁹In fact, a much stronger property holds: see Theorem 3.3.7 and Remark 3.3.8.</footnote>
Remark 3.1.10. Of course, one also obtains that $(\hat{M}(E))_{E/k}$ is a conservative family of points for the homotopy t-structure on $DM^{\text{eff}}(k)$ (resp. for homotopy sheaves).

The preceding proposition is the first step to compute morphisms in the homotopy category of generic motives. More precisely:

**Lemma 3.1.11.** Let $E, F$ be function fields over $k$, and $n, m \in \mathbb{Z}$. Let $X, Y$ be respective smooth $k$-models of $E$ and $F$. Then there exists a canonical isomorphism, where $V$ and $U$ runs over non-empty open subsets:

$$ \pi_0 \left( \operatorname{holim}_{V \subset Y} \operatorname{hocolim}_{U \subset X} \operatorname{Map}_{\mathrm{DM}(k)}(\mathrm{M}(U), \mathrm{M}(V)) \right) \\ \simeq \lim_{V \subset Y} \operatorname{colim}_{U \subset X} (\pi_0 \operatorname{Map}_{\mathrm{DM}(k)}(\mathrm{M}(U), \mathrm{M}(V))). $$

*Proof.* As filtered colimits are exact, one first deduces, for any $V \subset Y$ an isomorphism:

$$ \pi_0 \left( \operatorname{hocolim}_{U \subset X} \operatorname{Map}_{\mathrm{DM}(k)}(\mathrm{M}(U), \mathrm{M}(V)) \right) \simeq \operatorname{colim}_{U \subset X} (\pi_0 \operatorname{Map}_{\mathrm{DM}(k)}(\mathrm{M}(U), \mathrm{M}(V))). $$

According to the preceding proposition, one further gets:

$$ \operatorname{colim}_{U \subset X} (\pi_0 \operatorname{Map}_{\mathrm{DM}(k)}(\mathrm{M}(U), \mathrm{M}(V))) \simeq \hat{\mathrm{H}}_0(V)(E). $$

But if $j: W \subset V$ is a non-empty open subset, it follows from [Dég07, Cor 3.22] that the morphism of homotopy invariant sheaves with transfers $\hat{\mathrm{H}}_0(W) \to \hat{\mathrm{H}}_0(V)$ is an epimorphism. This implies that the inductive sytem: $(\hat{\mathrm{H}}_0(V)(E))_{V \subset Y}$ satisfies the Mittag-Leffler condition. In particular, one deduces (see e.g. [Goe96, Th. B]):

$$ \pi_0 \left( \operatorname{holim}_{V \subset Y} \hat{\mathrm{H}}_0(V)(E) \right) \simeq \lim_{V \subset Y} \hat{\mathrm{H}}_0(V)(E) $$

and this concludes.

□

Remark 3.1.12. Note in particular that one deduces a fully faithful functor:

$$ \mathrm{HoDM}^{(0)}(k) \to \mathrm{pro-HoDM}_{\mathrm{gm}}(k) $$

whose image is the category of generic motives of [Dég08b, Def. 3.3.1].

Combining the above proposition and the preceding lemma, one deduces an embedding of the homotopy category of generic motives into an abelian category.

**Theorem 3.1.13.** *The functor*

$$ \mathrm{HoDM}^{(0)}(k) \to \mathrm{pro-HM}(k), \hat{M}(E)\{n\} \mapsto \underline{\mathrm{H}}_{0,*,n}(\hat{M}(E)) = \underline{\mathrm{H}}_{0,*}(\hat{M}(E))\{n\} $$

is fully faithful.
## 3.2. Comparison with Beilinson’s approach.

**3.2.1.** Recall that the stable $\infty$-category $DM(k)$ is generated by its compact objects, which are precisely the geometric motives (as recalled in 2.1.3). This readily implies an interpretation of the “big” category of motives $DM(k)$ as ind-geometric objects:

$$ (3.2.1.a) \qquad DM(k) \simeq \mathrm{ind-DM}_{\mathrm{gm}}(k). $$

Thus using that, after inverting the characteristic exponent $e$ of $k$, $DM_{\mathrm{gm}}(k)_e$ is made of rigid objects (see loc. cit.), one further deduces the following interpretation of pro-geometric motives.

**Proposition 3.2.2.** *The canonical functor:*

$$ (\mathrm{pro-DM}_{\mathrm{gm}}(k)_e)^{\mathrm{op}} \to DM(k)_e, \quad (\mathbf{E} \simeq \underset{i \in I}{\mathrm{\text{lim}}} \mathbf{E}_i) \mapsto \mathrm{hocolim}_{i \in I} \mathbf{E}_i^{\vee} $$

*is an equivalence of categories.*

**3.2.3.** The preceding functor maps the motive $M(X)$ of a $k$-scheme of finite type $X$ to its dual $h(X) := \mathrm{Hom}(M(X), 1) = f_*(1_X)$, the cohomological motive of $X$ (see 2.1.3).

The preceding proposition allows to translate the results of 3.1.3 to cohomological motives (without necessarily inverting $e$). There exists a right Kan extension $\hat{h}$ of the functor $h: \mathrm{Sm}_k \to DM(k)$ along the inclusion $\nu: \mathrm{Sm}_k \to \overline{\mathrm{Sm}}_k$:

$$ \begin{array}{ccc} \mathrm{Sm}_k^{\mathrm{op}} & \xrightarrow{h} & DM(k) \\ \nu \downarrow & & \hat{h} \\ \overline{\mathrm{Sm}}_k^{\mathrm{op}} & \xrightarrow{\hat{h}} & \overline{\mathrm{Sm}}_k \end{array} $$

Moreover, this extension can be computed as

$$ \hat{h}(\mathcal{X}) = \mathrm{hocolim}_{i \in I} h(X_i) $$

for an essentially smooth scheme $\mathcal{X}$ with a pro-presentation $\mathcal{X} = \lim_{i \in I} X_i$.

In particular, one can define the cohomological motive attached to function fields $E/k$:

$$ \hat{h}(E) = \underset{U \subset X}{\mathrm{hocolim}} h(U) $$

for any smooth model $X$ of $E/k$. This is exactly Beilinson’s “motive of the generic point” of $X$ from [Bei02, 3.1] — after identifying Voevodsky’s motivic spectra $DM(k)$ with ind-geometric motives according to (3.2.1.a). The preceding proposition readily gives the following comparison with the preceding notion of generic motives.

**Proposition 3.2.4.** *The canonical functor:*

$$ \mathrm{DM}^{(0)}(k)_e^{\mathrm{op}} \to \mathrm{DM}(k)_e, \hat{\mathrm{M}}(E)\{i\} \mapsto \hat{h}(E)\{-i\} $$
is fully faithful and induces an anti-equivalence of categories between the category of generic motives and the sub-category of motivic spectra made by $\mathbb{G}_m$-twists of Beilinson's cohomological motives of generic points.

Beilinson has already explained the conjectural injective nature of the ind-motives $\hat{h}(E)$ with respect to the motivic $t$-structure (see [Bei02, Cor. 3.3]). In fact, if one uses the homotopy $t$-structure, one can establish analogous but non-conjectural facts for generic motives. Let us first recall some definitions (see [Bei02, before Cor. 3.3], [GLVdB21, §6.1]).

**Definition 3.2.5.** Let $\mathcal{T}$ be a $t$-category.<fnref n="20" /> An object $M$ of the heart $\mathcal{T}^{\heartsuit}$ is called $t$-projective if $\operatorname{Hom}_{\mathcal{T}}(M, \mathcal{T}_{>0}) = 0$.

An object $K$ of $\mathcal{T}$ is called derived $t$-projective if the following two conditions hold:

(i) For all $L$ in $\mathcal{T}$, $\operatorname{Hom}_{\mathcal{T}}(K, L) \simeq \operatorname{Hom}_{\mathcal{T}^{\heartsuit}}(H_0(K), H_0(L))$.

(ii) The object $H_0(K)$ is projective in the abelian category $\mathcal{T}^{\heartsuit}$.

One easily derives that $t$-projective for an object of the heart of $\mathcal{T}$ implies derived $t$-projective. Also, a derived $t$-projective object is necessarily (homologically) non-negative.

*Remark 3.2.6.* The preceding definition can be dualized in an obvious way to define the notion of $t$-injective and derived $t$-injective objects.

To deal with the case of generic motives, one needs to extend slightly the previous definition.

**Definition 3.2.7.** Let $\mathcal{T}$ be a $t$-category. Let $\mathcal{M}$ be a pro-object of $\mathcal{T}^{\heartsuit}$. One says that $\mathcal{M}$ is pro-$t$-projective if $\operatorname{Hom}_{\mathcal{M}, \mathcal{T}}(\mathcal{M}, \mathcal{T}_{>0}) = 0$.<fnref n="21" />

Let $\mathcal{K}$ be a pro-object of $\mathcal{T}$. One says that $\mathcal{K}$ is derived pro-$t$-projective if the following conditions hold:

(i) For all $L$ in $\mathcal{T}$, $\operatorname{Hom}_{\mathcal{M}, \mathcal{T}}(\mathcal{K}, L) \simeq \operatorname{Hom}_{\mathcal{M}, \mathcal{T}^{\heartsuit}}(H_0(\mathcal{K}), H_0(L))$.

(ii) The functor $\mathcal{T}^{\heartsuit} \to \mathcal{A}b, N \mapsto \operatorname{Hom}_{\mathcal{M}, \mathcal{T}^{\heartsuit}}(H_0(\mathcal{K}), N)$ is exact.

*Remark 3.2.8.* The second condition sometimes appears in the literature under the name “pro-projective”.

**3.2.9.** Given a stable $\infty$-category $\mathcal{C}$ equipped with a $t$-structure, one can always define a unique $t$-structure on pro-$\mathcal{C}$ such that a pro-object $\mathcal{K}$ is positive (resp. negative) if there exists a presentation $\simeq$“lim”$K_i$ such that for all $i \in \mathcal{I}$, $K_i$ is positive (resp. negative).

With this definition, $\mathcal{K}$ is derived pro-$t$-projective with respect to $\mathcal{C}$ if and only if the following conditions hold:

<footnote n="20">²⁰A triangulated category equipped with a $t$-structure.</footnote>

<footnote n="21">²¹We see $\mathcal{T}_{>0}$ as a subcategory of pro-$\mathcal{T}$ by the constant pro-object functor.</footnote>
(i) The pro-object $\mathcal{K}$ is homologically non-$t$-negative in pro-$\mathcal{C}$.

(ii) For all $L$ in $\mathcal{T}$, $\mathrm{Hom}_{\mathrm{pro-}\mathcal{T}}(\mathcal{K}, L) \simeq \mathrm{Hom}_{\mathrm{pro-}\mathcal{T}\diamond}(\mathcal{K}, H_0(L))$.

(iii) the pro-object $H_0(\mathcal{K})$ of $\mathcal{T}\diamond$ is projective.

One should be careful however that derived pro-$t$-projective with respect to the $t$-category $\mathrm{Ho}\mathcal{C}$ does not imply derived $t$-projective with respect to the $t$-category $\mathrm{Ho}\mathrm{pro-}\mathcal{C}$.

Taking into account the previous definition and remark, one gets the following corollary of Proposition 3.1.7.

**Proposition 3.2.10.** Let $E/k$ be a function field and $i \in \mathbb{Z}$ an integer. Then the generic motive $\hat{M}(E)\{i\}$ is derived pro-$t$-projective with respect to the homotopy $t$-structure on $\mathrm{DM}(k)$.

*Remark 3.2.11.* The dual statement for the cohomological motives $\hat{h}(E)\{i\}$ in $\mathrm{DM}(k)$ is a bit trickier to state. One has to restrict to geometric motives $M$ for once and use the isomorphism:

$$ \mathrm{Hom}_{\mathrm{pro-DM}_{\mathrm{gm}}(k)}(M, \hat{h}(E)\{i\}) \simeq \mathrm{Hom}_{\mathrm{pro-DM}_{\mathrm{gm}}(k)}(\hat{M}(E)\{-i\}, M^{\vee}) $$

where $M^{\vee}$ denotes the strong dual of $M$. We leave the precise formulation to the reader.

### 3.3. Morphisms of generic motives and cycle modules.

**3.3.1.** Generic motives enjoy a rich functorial structure, which parallels that of Milnor K-theory as theorized by Rost in the structure of a cycle module [Ros96, Def 1.1]. In fact, this functoriality can be described by a category $\tilde{\mathcal{E}}_k$ whose objects are pair $(E, i)$ made by a function field $E$ and an integer $i \in \mathbb{Z}$, and morphisms are described by generators and relations: see the appendix, Definition 6.0.1. Then the aforementioned functoriality can be stated as the existence of a contravariant functor (see *op. cit.* Th. 5.1.1):

$$ (3.3.1.a) \qquad \rho : (\tilde{\mathcal{E}}_k)^{\mathrm{op}} \to \mathrm{HoDM}^{(0)}(k), (E, i) \mapsto \hat{M}(E)\{-i\}. $$

Below we will briefly recall the definition of this functor on the generators of morphisms and refer to *loc. cit.* for the proof of the needed relations. We use notation (D1*), (R1a*) and so on to indicate the data and relations on morphisms of generic motives that correspond to the functor $\rho$, according to the notation of Definition 6.0.1.

*Remark 3.3.2.* Using cohomological motives and given Proposition 3.2.4, the above construction amounts to giving a covariant functor:

$$ \rho^{\vee} : \tilde{\mathcal{E}}_k \to \mathrm{HoDM}(k), (E, i) \mapsto \hat{h}(E)\{i\}. $$

The reader might prefer to use this convention (at the price of inverting $e$).

**3.3.3.** We start with the first two types of functoriality satisfied by generic motives:
(D1*) **Covariant functoriality in function fields.**— This functoriality simply comes from the fact generic motives are obtained by restriction and $\mathbb{G}_m$-twists of the $\infty$-functor $\hat{M}: \overline{\mathbb{S}m}_k \to \mathrm{DM}_{\mathrm{gm}}(k)$. In particular, an arbitrary morphism $\varphi: E \to L$ of function fields over $k$ induces a canonical map:

$$ \varphi^*: \hat{M}(L)\{i\} \to \hat{M}(E)\{i\} $$

which actually corresponds to an $\infty$-functor from the category function fields over $k$ to $\mathrm{DM}^{(0)}(k)$.

(D2*) **Transfers: contravariance in function fields.**— Voevodsky's homological motives are by construction functorial in the category of finite correspondences $\overline{\mathrm{Sm}}_k^{\mathrm{cor}}$. This really corresponds to an $\infty$-functor:

$$ M: \overline{\mathrm{Sm}}_k^{\mathrm{cor}} \to \mathrm{pro-DM}_{\mathrm{gm}}(k). $$

One can extend the definition of finite correspondences to essentially smooth $k$-schemes $\mathcal{X}$ and $\mathcal{Y}$. The corresponding abelian group $c(\mathcal{X}, \mathcal{Y})$ of finite correspondences from $\mathcal{X}$ to $\mathcal{Y}$ is made by algebraic cycles in $\mathcal{X} \times_k \mathcal{Y}$ whose support is finite and equidimensional over $\mathcal{X}$. Composition can be defined as in [Dég07, Def. 1.16] and enjoys the same properties. In particular, one can define a category $\overline{\mathrm{Sm}}_k^{\mathrm{cor}}$ of essentially smooth $k$-schemes with these finite correspondences as morphisms.

Using the continuity property of [Dég07, Prop. 1.24], one deduces that the preceding functor admits a left Kan extension:

$$ (3.3.3.a) \qquad \hat{M}: \overline{\mathbb{S}m}_k^{\mathrm{cor}} \to \mathrm{pro-DM}_{\mathrm{gm}}(k). $$

This readily gives the announced functoriality. A finite extension of function field $\varphi: E \to L$ corresponds to a finite surjective (thus equidimensional) morphism $f: \mathcal{Y} = \mathrm{Spec}(L) \to \mathrm{Spec}(E) = \mathcal{X}$. Taking the transpose of its graph defines a finite correspondence: $f \in c(\mathcal{X}, \mathcal{Y})$ and therefore a transfer (or Gysin) map:

$$ \varphi_!: \hat{M}(E)\{i\} \to \hat{M}(L)\{i\}. $$

Note that this actually gives an $\infty$-functor, from function fields over $k$ equipped with finite morphisms to $\mathrm{DM}^{(0)}(k)$.

*Remark 3.3.4.* Note that the functorialities (D1*) and (D2*), together with relations (R1a*), (R1b*), (R1c*) they satisfy, can be gathered using the category $\overline{\mathrm{Sm}}_k^{\mathrm{cor}}$. In fact, one can consider the full sub-category $\overline{\mathrm{Sm}}_k^{\mathrm{cor},(0)}$ of $\overline{\mathrm{Sm}}_k^{\mathrm{cor}}$ whose object are the spectrum of function fields over $k$. Then, by restriction of (3.3.3.a), one obtains an $\infty$-functor

$$ \mathrm{Sm}_k^{\mathrm{cor},(0)} \to \mathrm{DM}^{(0)}(k), \mathrm{Spec}(E) \mapsto \hat{M}(E) $$

which gives a highly structured version of transfers on generic motives. Note moreover that his functors, exactly covers datas (D1*) and (D2*), as well as relations (R1*), of the functor $\rho$.
**3.3.5.** Let us now describe the two other kind of functoriality of generic motives: (D3*) **Action of Milnor K-theory.**– Let $E/k$ be a function field, and $n \ge 0$ be an integer. One deduces from (2.2.1.d) the following isomorphism:

$$ (3.3.5.a) \qquad \operatorname{Hom}_{\operatorname{DM}^{(0)}(k)}(\hat{M}(E), 1\{n\}) \simeq \operatorname{H}_{M}^{n,n}(E) \simeq \operatorname{K}_{n}^{M}(E). $$

Thus, one can associated to any symbol $\sigma = \{u_1, \dots, u_n\} \in \operatorname{K}_n^M(E)$ a canonical map $\mu(\sigma) : \hat{M}(E) \to 1\{n\}$.

Using the diagonal map $\delta : \operatorname{Spec}(E) \to \operatorname{Spec}(E) \times_k \operatorname{Spec}(E)$, or equivalently the multiplication map $E \otimes_k E \to E$, one deduces the required morphism of generic motives:

$$ \gamma_{\sigma} : \hat{M}(E) \xrightarrow{\delta_{*}} \hat{M}(E) \otimes \hat{M}(E) \xrightarrow{\mathrm{Id} \otimes \mu(\sigma)} \hat{M}(E)\{n\}. $$

(D4*) **Residues.**– Let $(E, v)$ be a valued function field over $k$. Let $\mathcal{O}_v$ and $\kappa(v)$ be the associated ring of integers and residue (class) field respectively. One deduces a closed immersion between essentially smooth $k$-schemes $i : \mathcal{Z} = \operatorname{Spec}(\kappa(v)) \to \operatorname{Spec}(\mathcal{O}_v) = \mathcal{X}$, with open complement $j : \mathcal{U} = \operatorname{Spec}(E) \to \operatorname{Spec}(\mathcal{O}_v)$. The Gysin triangle extends to this generality and yields a (split) homotopy exact sequence in the stable $\infty$-category pro-$\operatorname{DM}_{\operatorname{gm}}^{\mathrm{eff}}(k)$:

$$ \hat{M}(\kappa_v)\{1\} \xrightarrow{\partial_v} \hat{M}(E) \xrightarrow{j_*} \hat{M}(\mathcal{O}_v) \xrightarrow{i^!} \hat{M}(\kappa_v)(1)[2] $$

We call $\partial_v$ the residue map associated with $(E, v)$.

This conclude the description of the functor $\rho$ of (3.3.1.a) on morphisms. We refer the reader to [Dég08b] for the proof of the relations.

**3.3.6.** Let $\mathbf{E}$ be a motivic complex in $\operatorname{DM}(k)$. Seen as a constant pro-object, it represents a functor $\hat{\mathbf{E}} : \operatorname{Hom}_{\operatorname{pro-DM}(k)}(-, \mathbf{E})$, that can be restricted to the category of generic motives. Composing further with the functor $\rho$, one obtains a functor:

$$ \hat{\mathbf{E}} : \tilde{\mathcal{E}}_k \xrightarrow{\rho^{\mathrm{op}}} \operatorname{DM}^{(0)}(k)^{\mathrm{op}} \xrightarrow{\hat{\mathbf{h}}_{\mathbf{E}}} \mathcal{A}b $$

which is by definition a cycle premodule. We call it the *Rost transform* of $\mathbf{E}$. One can check it is in fact a cycle module. Further, according to [Dég11, Th. 3.7], one gets:

**Theorem 3.3.7.** For any perfect field $k$, the functor

$$ \mathcal{R} : \operatorname{HM}(k) = \operatorname{DM}(k)^{\heartsuit} \to \mathcal{M}\operatorname{Cycl}_k, \mathbf{E} \mapsto \hat{\mathbf{E}} $$

is an equivalence of categories with quasi-inverse the functor which associates to a cycle module $M$ its 0-th Chow group functor $\underline{\operatorname{HM}} := \mathcal{A}^0(-, M)$.

Moreover, if $k$ is non-perfect of characteristic $p > 0$, then the pair of functors $(\mathcal{R}[1/p], \underline{\operatorname{HM}}[1/p])$ obtained after inverting $p$, is an equivalence of categories.

Note the last assertion was not proved in *loc. cit.* but it easily follows as, after inverting $p$, both categories $\operatorname{HM}(k)[1/p]$ and $\mathcal{M}\operatorname{Cycl}_k[1/p]$ become invariant under purely inseparable extensions of the base field $k$.
Remark 3.3.8. According to Proposition 3.1.7, for any motivic complexe E, one gets:

$$ \hat{\mathbf{E}}(K, n) = \mathrm{Hom}_{\mathrm{pro-DM}(k)}(\hat{\mathrm{M}}(K)\{n\}, \mathbf{E}) \simeq \mathbf{H}_{0, -n}(\mathbf{E})(K). $$

In particular, the functor $\mathbf{E} \mapsto \hat{\mathbf{E}}$ factorizes through the homological functor $\mathbf{H}_{0,*}$ associated with the homotopy t-structure on DM$(k)$.

Moreover, the abelian group $\hat{\mathbf{E}}(K, n)$ can be interpreted as the fiber of the Nisnevich sheaf $\mathbf{H}_{0, -n}(\mathbf{E})$ at the point of the Nisnevich site $\mathrm{Sm}_k$ corresponding to the function field $K$. We have interpreted generic motives as points for the t-category DM$(k)$, equipped with its homotopy t-structure: Corollary 3.1.9, Proposition 3.2.10. The Rost transform functor $\mathcal{R}$ is therefore the restriction to this category of points. Remarkably, the restriction a homotopy module $M$ to the category made by these points, enriched via the functoriality defined by the functor $\rho$, allows to fully reconstruct $M$ (as well as morphisms of homotopy modules).

Rost morphisms $\mathbb{R}^*$ can be interpreted as specialization maps between these various points, but one should be careful they are not isomorphisms. The category DM$^{(0)}$(k) is not a groupoid. Nevertheless, from this point of view, cycle modules appear as a generalized version of local systems. Homotopy modules are the sheaf counter-part of these local systems.

**3.3.9.** In the PhD of the author ([Dég02, §9.3.2]), the question was raised as to whether morphisms of type (D*) generate all morphisms between generic motives. More precisely, is the above functor $\rho$ full, and/or faithful ?

This hope will be contradicted later (Remark 4.1.6), but one can still observe interesting cases where the functor $\rho$ does induce an isomorphism on morphisms.<fnref n="22" />

**Proposition 3.3.10.** Let $(E, n)$ and $(F, m)$ be objects in $\tilde{\mathcal{E}}_k$ such that $F/k$ is finite. Then the following map:

$$ \bigoplus_{x \in \mathrm{Spec}(E \otimes_k F)} K_{m-n}^{M}(\kappa(x)) \to \mathrm{Hom}_{\mathrm{DM}^{(0)}(k)}(\hat{\mathrm{M}}(E)\{n\}, \hat{\mathrm{M}}(F)\{m\}) \\ \sigma_x \mapsto \psi_x^* \circ \gamma_{\sigma_x} \circ \varphi_x! $$

where $\varphi_x : E \to \kappa(x)$ and $\psi_x : F \to \kappa(x)$ are the field extensions the morphism induced by the prime ideal $x \subset E \otimes_k F$.

In particular, the map:

$$ \mathrm{Hom}_{\tilde{\mathcal{E}}_k}((E, n), (F, m)) \to \mathrm{Hom}_{\mathrm{DM}^{(0)}(k)}(\hat{\mathrm{M}}(E)\{n\}, \hat{\mathrm{M}}(F)\{m\}) $$

is an isomorphism.

In particular, only (D1), norm maps (D2) and multiplication by symbols (D3) are sufficient to describe all morphisms of generic motives in the preceding case.

<footnote n="22">²²The next two results was both observed in the PhD of the author [Dég02, §9.3].</footnote>
Proof. The proposition follows from the following computation:

$$
\begin{align*}
\operatorname{Hom}_{\operatorname{DM}^{(0)}(k)} (\hat{M}(E)\{n\}, \hat{M}(F)\{m\}) &\stackrel{(1)}{\simeq} \operatorname{Hom}_{\operatorname{pro-DM}(k)} (\hat{M}(E) \otimes \hat{M}(F), 1\{m-n\}) \\
&\simeq \operatorname{H}_{M}^{m-n, m-n}(E \otimes_k F) \\
&\simeq \bigoplus_{x \in \operatorname{Spec}(E \otimes_k F)} K_{m-n}^{M}(\kappa(x))
\end{align*}
$$

where the first two isomorphisms follow as in Example 3.1.8, using in particular
that $\hat{M}(F) = M(F)$ is rigid, with strong dual $M(F)^* = M(F)$, and the last one
follows from (2.2.1.d). The assertion about the description of the isomorphism
follow from the construction of data (D1), (D2), (D3) recalled above. $\square$

The presentation of the category $\tilde{\mathcal{E}}_k$ given in Proposition 6.0.4 implies a vanishing
of morphisms in $\tilde{\mathcal{E}}_k$ in certain degree. It is notable that this vanishing does hold in
the category of generic motives, at least after inverting the characteristic exponent
$e$ of $k$.

**Proposition 3.3.11.** For any function field $E$ and $F$ over $k$, and any couple of integers $(n, m) \in \mathbb{Z}^2$ such that $n - m > \operatorname{degtr}_k(F)$, one has:

$$
\operatorname{Hom}_{\operatorname{DM}^{(0)}(k)}(\hat{M}(E)\{n\}, \hat{M}(F)\{m\})_e = 0.
$$

Proof. One can assume that $n = 0$ to simplify the notation. Put $d = \operatorname{degtr}_k(F)$.
Let $X$ be a smooth model of $F$ over $k$. According to Example 3.1.2(2) and
Lemma 3.1.11, one gets:

$$
\operatorname{Hom}_{\operatorname{DM}^{(0)}(k)}(\hat{M}(E), \hat{M}(F)\{m\}) = \text{``lim''}_{U\subset X} \operatorname{Hom}_{\operatorname{pro-DM}_{\operatorname{gm}}(k)}(\hat{M}(E), \operatorname{M}(U)\{m\})
$$

where the projective limit runs over the non-empty open subschemes of $X$. Ac-
cording to duality, which holds under our assumptions (see 2.1.3), one gets:

$$
\operatorname{Hom}_{\operatorname{pro-DM}_{\operatorname{gm}}(k)}(\hat{M}(E), \operatorname{M}(U)\{m\})_e \\
\simeq \operatorname{Hom}_{\operatorname{pro-DM}_{\operatorname{gm}}(k)}(\hat{M}(E) \otimes \operatorname{M}^c(U), 1(m+d)[m+2d])_e
$$

Moreover, the motive with compact support $M^c(U)$ is effective.<fnref n="23" /> Therefore, the
cancellation theorem implies that the left hand-side vanishes whenever $m + d < 0$
as required. $\square$

$$
M^c(X - Z) \to M^c(X) \to M^c(Z)
$$

which holds in DM($k$) thanks to the six functors formalism and [CD15].

<footnote n="23"><sup>23</sup>In characteristic 0, this is [VSF00, Chap. 5, Cor. 4.1.6]. In general, one uses resolution of singularities by alteration and the homotopy exact sequence:</footnote>
Then for any extension of function fields $\varphi : E \to L$, the map $\varphi^* : \hat{M}(L) \to \hat{M}(E)$ is a split epimorphism after tensoring with $R$.

*Proof.* The case where $L/E$ is finite follows from the degree formula<fnref n="24" />:

$$ \varphi^*\varphi! = d.\mathrm{Id} $$

where $d$ is the degree of $L/E$, invertible in the coefficient ring $R$ by our assumptions. In the general case, $L$ is a finite extension of a purely transcendental extension $E(t_1, ..., t_n)$. We are reduced to the purely transcendental case which holds according to the paragraph preceding the statement. □

The following proposition contradicts the hope of 3.3.9: the functor $\rho$ of (3.3.1.a) cannot be full in general.

**Proposition 4.1.5.** Let $F/k$ be a function field and $n$ be an integer.

Then there exists an isomorphism of abelian groups:

$$ \operatorname{Hom}_{\mathrm{DM}^{(0)}(k)} (\hat{M}(F), \hat{M}(k(t))\{n\}) \xleftarrow{\sim} \operatorname{K}_n^M(F) \oplus \prod_{\substack{x \in \mathbb{A}_{k,(0)}^1 \\ y \in \operatorname{Spec}(F \otimes_k \kappa_x)}} \operatorname{K}_{n+1}^M(\kappa_y) $$

which to a symbol $\sigma \in \operatorname{K}_n^M(F)$ associates the morphism

$$ \hat{M}(F) \xrightarrow{\gamma_{\sigma}} \hat{M}(F)\{n\} \xrightarrow{\varphi_F^*} \hat{M}(k)\{n\} \xrightarrow{s_t} \hat{M}(k(t))\{n\} $$

and to the data of a point $x \in \mathbb{A}_{k,(0)}^1$, of a composite extension field $\kappa_y$ of $F$ and $\kappa_x$ over $k$

$$ F \xrightarrow{\psi_y} \kappa_y \xleftarrow{\varphi_y} \kappa_x $$

and a symbol $\tau \in \operatorname{K}_{n+1}^M(\kappa_y)$ associates the morphism:

$$ \hat{M}(F) \xrightarrow{\psi_{y!}} \hat{M}(\kappa_y) \xrightarrow{\gamma_{\tau}} \hat{M}(\kappa_y)\{n+1\} \xrightarrow{\varphi_y^*} \hat{M}(\kappa_x)\{n+1\} \xrightarrow{\partial_x} \hat{M}(k(t))\{n\} $$

*Proof.* This follows from the preceding proposition and duality:

$$ \operatorname{Hom}_{\mathrm{DM}^{(0)}(k)}(\hat{M}(F), \hat{M}(k(t))\{n\}) \\ \simeq \operatorname{Hom}_{\mathrm{DM}^{(0)}(k)}(\hat{M}(F), \mathbb{1}\{n\}) \oplus \prod_{x \in \mathbb{A}_{k,(0)}^1} \underbrace{\operatorname{Hom}_{\mathrm{DM}^{(0)}(k)}(\hat{M}(F), \hat{M}(\kappa_x)\{n\})}_{=\operatorname{Hom}_{\mathrm{DM}^{(0)}(k)}(\hat{M}(F)\otimes\hat{M}(\kappa_x), \mathbb{1}\{n\})} $$

One finally concludes using the isomorphism (3.3.5.a) and of course, as $\kappa_x/k$ is finite separable:

$$ \operatorname{Spec}(F) \times_k \operatorname{Spec}(\kappa_x) = \operatorname{Spec}(F \otimes_k \kappa_x) = \bigsqcup_{y \in \operatorname{Spec}(F \otimes_k \kappa_x)} \operatorname{Spec}(\kappa_y). $$

□

<footnote n="24">²⁴This formula is for example a consequence of (R2c) and the fact that the norm map associated to $\varphi$ is multiplication by $d$ on the degree 0 part of Milnor K-theory</footnote>
Remark 4.1.6. The preceding proposition shows that the functor $\rho$ is not full, negatively answering a question in [Dég02, 9.3.2]. This comes from the fact generic motives are pro-objects. In particular, a correct formulation of the conjecture describing morphisms of generic motives could be formulated as follows.

**Conjecture.** There exists a natural topology on morphisms of generic motives such that morphisms of type (Dn*), for $1 \le n \le 4$, are topological generators. Relations among these topological generators are described by relations (Rm*) as spelled out in Definition 6.0.1.

## 4.2. Positive genus curves.

**4.2.1.** Let $K$ be a function field over $k$ of transcendence degree 1. It corresponds to a smooth projective curve $\bar{C}/k$. We fix a closed point $x_0 \in \bar{C}_{(0)}$, and consider the corresponding affine curve $C = \bar{C} - \{x_0\}$. This is a smooth affine model of $K$.

Let $A$ be the albanese scheme of $\bar{C}$ — isomorphic to the dual abelian variety of the Jacobian variety $J(\bar{C}, x_0)$. There is a canonical map of $k$-schemes.

$$C \to \bar{C} \to A.$$

We let $\underline{A}$ be the sheaf of abelian groups represented by $A$ on $\mathrm{Sm}_k$. It is $\mathbb{A}^1$-invariant and admits transfers (see e.g. [Org04, SS03]), and therefore an object of $\mathrm{HI}(k)$. The preceding map induces a morphism of sheaves with transfers

$$\underline{Z}^{\mathrm{tr}}(C) \to \underline{A}$$

which therefore induces a map $\pi_1 : \mathrm{M}(C) \to \underline{A}$ in $\mathrm{DM}^{\mathrm{eff}}(k)$. We let $\pi_0 = p_* : \mathrm{M}(C) \to 1$ be the map induced by the projection.

According to a theorem of Suslin and Voevodsky (see [VSF00, Chap. 5, 3.4.2]), the following map of $\mathrm{DM}^{\mathrm{eff}}(k)$

$$(4.2.1.a) \qquad \mathrm{M}(C) \xrightarrow{(\pi_0, \pi_1)} 1 \oplus \underline{A}$$

is an isomorphism. One deduces the following computation of generic motives.

**Proposition 4.2.2.** Consider the above notation, and let $\pi_i^K$ be the composition of $\pi_i$ with the canonical map $\hat{\mathrm{M}}(K) \to \mathrm{M}(C)$.

Then we have a (homotopy) exact sequence in pro-$\mathrm{DM}_{\mathrm{gm}}(k)$:

$$\text{``$\prod$'' } \hat{\mathrm{M}}(\kappa_x)\{1\} \xrightarrow{\prod_x \partial_x} \hat{\mathrm{M}}(K) \xrightarrow{(\pi_0^K, \pi_1^K)} 1 \oplus \underline{A},$$

where we have denoted by $\partial_x$ the residue map $(3.3.5(D_4^*))$ associated with the valuation on $K$ corresponding to the closed point $x \in C$.

The boundary map $\partial : 1 \oplus \underline{A} \to \text{``$\prod$'' } \hat{\mathrm{M}}(\kappa_x)(1)[2]$ of this exact sequence is trivial on the first factor. As an element of the abelian group:

$$\prod_{x \in C_{(0)}} \mathrm{Hom}(\underline{A}, \hat{\mathrm{M}}(\kappa_x)(1)[2]) \simeq \prod_{x \in C_{(0)}} \mathrm{Pic}^0(\bar{C}_{\kappa_x})$$
it corresponds to the element $(x - x_0)_{x \in C(0)}$ where $(x - x_0)$ is seen as a 0-cycle of degree 0 on $\bar{C}_{\kappa_x}$.

Therefore, as soon as the field $K$ is not rational over $k$, the boundary $\partial$ is non trival, and the above homotopy exact sequence is non-split. Therefore, the motive $\tilde{M}(K)$ is in some sense “maximally mixed”.

*Proof.* Let $Z \subset C$ be a closed subscheme. As $k$ is perfect, $Z$ is smooth and one deduces a homotopy exact sequence in $\mathrm{DM}_{\mathrm{gm}}^{\mathrm{eff}}(k)$:

$$M(Z)\{1\} \xrightarrow{d^Z} M(C-Z) \xrightarrow{j_*^Z} M(C) \xrightarrow{i_Z^1} M(Z)(1)[2]$$

Note that by additivity $M(Z) = \prod_{x \in Z} \hat{M}(\kappa_x)$. Thus, taking the projective limit of the homotopy exact sequences over the open complement $U = C - Z$ yields a homotopy exact sequence in pro-$\mathrm{DM}_{\mathrm{gm}}^{\mathrm{eff}}(k)$ which has the required form up to using the mentioned additivity and the isomorphism (4.2.1.a). The assertion about the boundary map $\partial$, which is nothing else than the projective limit of the Gysin morphisms $i'_Z$, thus follows. $\square$

**4.2.3.** In [Dég11], we associate to any homotopy sheaf $F$ in $\mathrm{HI}(k)$ a homotopy module $F_* = \sigma^\infty(F)$ (see also 2.1.2) with the formula for $n \ge 0$:

$$F_n = F \otimes \mathbb{G}_m^{\otimes n} = \mathbb{H}_0(F\{n\}) \\ F_{-n} = \underline{\mathrm{Hom}}(\mathbb{G}_m^{\otimes n}, F)$$

where the tensor product and internal Hom are taken with respect to the canonical monoidal structure on $\mathrm{HI}(k)$.<fnref n="25" />

This construction can be described more precisely when $F = \underline{\mathcal{A}}$ is the homotopy sheaf associated with an abelian variety, as recalled above. Let us recall this description due to Kahn and Yamazaki.

**Proposition 4.2.4.** Let $A$ be an abelian variety over $k$, and $\underline{\mathcal{A}}$ be the associated homotopy sheaf as defined above.

Then for any $n > 0$ and any function field $F$, one has:

$$\underline{\mathcal{A}}_{-n}(F) = 0 \\ \underline{\mathcal{A}}_n(F) = \mathcal{K}_n^{\mathrm{SM}}(F; A)$$

where $\mathcal{K}_n^{\mathrm{SM}}(F; A) = K(F, A, \mathbb{G}_m, \dots, \mathbb{G}_m)$ is Somekawa K-theory associated with the semi-abelian varieties given by one copy of $A$ and $n$ copies of $\mathbb{G}_m$ (see [Som90], or rather [KY13, Def. 5.1] for the correct sign).

The first computation implies $\underline{\mathcal{A}}_{-1} = 0$, which is equivalent to say that the sheaf $\underline{\mathcal{A}}$ is a birational invariant: we refer the interested reader to [KS17] for more on birational (homotopy) sheaves.

<footnote n="25">²⁵Recall that $F_{-n}$ is also obtained by iterating Voevodsky’s (–1)-construction; see loc. cit. 1.13.</footnote>
Proof. The first computation simply follows as $A(\mathbb{G}_{m,F}) = A(F)$. The second computation follows from the main theorem of [KY13], Th. 1.12. □

We can now state the following corollary of Proposition 4.2.2.

**Corollary 4.2.5.** Consider the notation of the aforementioned proposition. Then for any integer $n \in \mathbb{Z}$, there exists a short exact sequence of abelian groups:

$$ \bigoplus_{\substack{x \in C_{(0)} \\ y \in \text{Spec}(F \otimes_k \kappa_x)}} K_{n+1}^{M}(\kappa_y) \to \text{Hom}_{\text{DM}^{(0)}(k)}(\hat{M}(F), \hat{M}(K)\{n\}) \to K_n^M(F) \oplus K_n^{SM}(F; A) \to 0 $$

where the first map sends a symbol $\tau \in K_{n+1}^M(\kappa_y)$ to the morphism

$$ \hat{M}(F) \xrightarrow{\psi_{y!}} \hat{M}(\kappa_y) \xrightarrow{\gamma_{\tau}} \hat{M}(\kappa_y)\{n+1\} \xrightarrow{\varphi_y^*} \hat{M}(\kappa_x)\{n+1\} \xrightarrow{\partial_x} \hat{M}(K)\{n\} $$

given the canonical morphisms $F \xrightarrow{\psi_y} \kappa_y \xleftarrow{\varphi_y} \kappa_x$, and the residue map $\partial_x$ associated with the valuation on $K$ corresponding to the point $x$ as defined in 3.3.5(D4*).

In particular, when $n = -1$, one gets an isomorphism:

$$ \text{Hom}_{\text{DM}^{(0)}(k)}(\hat{M}(F)\{1\}, \hat{M}(K)) \simeq \prod_{x \in C_{F,(0)}} \mathbb{Z}. $$

Proof. One applies the functor $\text{Hom}_{\text{pro-DM}_{\text{gm}}(k)}(\hat{M}(F), -)$ to the homotopy exact sequence of Proposition 4.2.2, and proceed as in the proof of Proposition 4.1.5 for the first term of the above exact sequence, use the preceding proposition to compute the third one, and then use the vanishing of motivic cohomology $H_M^{n+1,n}(L)$ for any field $L$ for the surjectivity assertion. □

We end-up this section by a remark about duality.

**Lemma 4.2.6.** Consider the notation of 4.2.1. Then the motive $\underline{A}$ is rigid in $\text{DM}(k)$ and there exists a canonical isomorphism:

$$ \underline{A}^{\vee} \simeq \underline{A}(-1)[-2]. $$

Proof. According to 2.1.3, the motive $M(C)$ is rigid, with dual $M^c(C)(-1)[-2]$. The homotopy exact sequence of the motive with compact support

$$ M^c(\{x_0\}) \to M^c(\bar{C}) \to M^c(C) $$

splits and gives a canonical isomorphism $M^c(C) = \text{coKer}(1 \to M(C)) = \underline{A} \oplus 1(1)[-2]$. Therefore, according to (4.2.1.a), one gets an isomorphism

$$ 1 \oplus \underline{A}^{\vee} \simeq \underline{A}(-1)[-2] \oplus 1. $$

One deduces the desired isomorphism as the factor $1$ is cut-off by the projection $p: C \to \text{Spec}(k)$. □
**Remark 4.2.7.** Recall that $\mathbb{1}(1)[1] = \mathbb{G}_m$. Therefore, the preceding formula reflects the duality of abelian variety. Note in particular that, as $A$ is the Albanese variety of a pointed smooth projective curve, it admits a polarization which explains the preceding formula. For a general abelian variety $A$, one would expect $A^\vee = (A^*)(-1)[-2]$ where $A^*$ is the dual abelian variety. This formula is proved for étale motives (and therefore for $\mathrm{DM}(k, \mathbb{Q})$) by Kahn and Barbieri-Viale in [BVK16, 4.5.3].

**4.3. Higher dimension.** Given the preliminary sections, the reader will not be surprised by the fact generic motives are gigantic in general. We can at least illustrate the first assertion by the following fact which partially extends Proposition 4.1.5.

**Proposition 4.3.1.** Let $\varphi : K \to L$ be an extension of function fields over $k$ of transcendence degree $n > 0$. To facilitate the notation, we consider the pro-motive $T = \text{“}\prod_{x \in A_{K,(0)}^1} \hat{M}(\kappa_x)\{1\}$.

Consider a transcendental basis $(t_1, ..., t_n)$ of $L/K$, and let $K_i = K(t_1, ..., t_i) \subset L$. We will consider the finite morphism $\psi : K_n \to L$ and for any $i \le j$, we let $\varphi_{i,j} : K_i \to K_j$ be the obvious inclusion.

Then the following map is a split epimorphism:

$$ \hat{M}(L) \xrightarrow{(\varphi^*, \pi_i^n \circ \psi^*)} \hat{M}(K) \oplus \bigoplus_{i=1}^n T\{i\} $$

where one defines the projector $\pi_i^n : \hat{M}(K_n) \to T\{i\}$ inductively using the formulas:

$$ \pi_1^n = \phi \circ \varphi_{1,n}^* \\ i > 1 : \pi_i^n = \pi_{i-1}^{n-1} \circ \partial_n $$

where $\phi : \hat{M}(K_1) \to T\{1\}$ is the projection map from Proposition 4.1.2, and $\partial_n : \hat{M}(K_n) \to \hat{M}(K_{n-1})\{1\}$ is the residue map coming from the valuation on $K_n$ associated with the variable $t_n$.

**Proof.** Using Corollary 4.1.4 (or the degree formula directly), $\psi^*$ is a split epimorphism, so that on reduces to the purely transcendental case $L = K_n$. Then it follows by induction on $n > 0$: the case $n = 1$ is precisely Proposition 4.1.2. Then we get a composite epimorphism:

$$ \begin{align*} \hat{M}(K_n) & \to \hat{M}(K_{n-1}) \oplus \hat{M}(K_{n-1})\{1\} & \text{Proposition 4.1.2 to } K/K_{n-1} \\ & \to \hat{M}(K_1) \oplus \hat{M}(K_{n-1})\{1\} & \text{Corollary 4.1.4} \\ & \to \hat{M}(K) \oplus T\{1\} \oplus \hat{M}(K_{n-1})\{1\} & \text{Proposition 4.1.2} \end{align*} $$

and we conclude by induction using the $(n-1)$-case (up to discarding the first factor). The formulas for the projectors follow from this induction. $\square$
Remark 4.3.2. The preceding proposition only gives a rough “lower bound” on the different pro-motives that a given generic contains. Giving the analysis on the preceding section, formula Proposition 4.1.2 does not allow to compute the motive of even a purely transcendental extension of degree more than 1. On the other hand, it shows that generic motives of transcendental extensions of degree $n$ are usually non-trivial extensions of all motives of dimension less than $n$.

**4.3.3. Surfaces.** We will illustrate the preceding remark in transcendental degree 2. Let $S$ be a geometrically connected smooth projective surface over a field $k$ of characteristic 0. We let $K$ be its function field.

Recall that Murre has shown in [Mur90] the existence of a Chow-Künneth decomposition of the cohomological motive of $S$:

$$h(S) = 1 \oplus h^1(S) \oplus h^2(S) \oplus h^3(S) \oplus 1(-2)[-4],$$

which we will view in the homotopy category $\text{HoDM}(k, \mathbb{Q})$. Moreover, according to [KMP07], this decomposition can be refined: the component $h^2(S)$ can be split into:

$$h^2(S) = h_{alg}^2(S) \oplus h_{tr}^2(S),$$

respectively the algebraic and transcendental part of $h^2$. Note that by definition, the algebraic part is of the form: $h_{alg}^2 = b_1 \cdot 1(-1)[-2]$ where $b_1$ is the first Betti number of the surface $S$. Beware that in loc. cit., the motive $h_{tr}^2(S)$ is denoted by $t_2(S)$. Recall that it is shown in Corollary 14.8.11 of loc. cit. that the functor $S \mapsto h_{tr}^2(S)$ is a birational invariant of $S$.<fnref n="26" /> In particular, it depends only on the function field $K$ of $S$.

Dually, we get a decomposition of the homological motive, in $\text{DM}(k, \mathbb{Q})$:

$$M(S) = 1 \oplus M_1(S) \oplus M_2^{alg}(S) \oplus M_2^{tr}(S) \oplus M_3(S) \oplus 1(2)[-4].$$

Letting $A$ be the Albanese scheme associated with $S$, we get an identification $M_1(S) = A$, and by duality: $M_3(S) = A^\vee(2)[-4] \simeq A(1)[2]$ (apply Lemma 4.2.6). Let us sumarize the above discussion.

**Lemma 4.3.4.** Consider the above notation. Then there exists a direct factor $M_2^{tr}(S) \subset M(S)$ in $\text{DM}_{\text{gm}}(k, \mathbb{Q})$ called the transcendental part of the homological motive, and a (refined) Chow-Künneth decomposition in $\text{DM}_{\text{gm}}(k, \mathbb{Q})$:

$$M(S) = 1 \oplus A \oplus b_1 \cdot 1(1)[2] \oplus M_2^{tr}(S) \oplus A(1)[2] \oplus 1(2)[-4].$$

Remark 4.3.5. We can view the above decomposition in the effective category $\text{DM}^{\text{eff}}(k, \mathbb{Q})$ of motivic complexes. Then it is remarkable that, apart the mysterious part $M_2^{tr}(S)$ for which we do not know, all the components of the decomposition lives in homotopical degree $[0, \dim(S)]$.

<footnote n="26">²⁶As recalled to me by Jan Nagel, this property can also be shown directly by using the blow-up formula for Chow motives combined with the fact any birational morphism of surfaces in characteristic 0 can be written as a sequence of blow-ups of closed points.</footnote>
4.3.6. To go further, we will need to use the coniveau filtration viewed internally inside pro-motives, as in [Dég12] or [DP24].<sup>27</sup> We still consider a geometrically smooth projective surface S over k, with function field K.

Recall that a flag in S is a decreasing sequence $(Z^p)_{p \ge 0}$ of closed subschemes of S such that $\mathrm{codim}_S(Z^p) \ge p$. The set $\mathcal{F}lag(S)$ of flags of S, ordered by term-wise inclusion, is filtered. We define an increasing filtration of S by pro-schemes, the *coniveau filtration*, by setting:

$$S^{\le p} := \varinjlim_{Z^* \in \mathcal{F}lag(S)} S - Z^{p+1}.$$

By applying the functor M term-wise, we then get a canonical homotopy exact sequence in pro-DM<sub>gm</sub>(k):

$$(4.3.6.a) \qquad M(S^{\le p-1}) \to M(S^{\le p}) \to M(S^{=p})$$

where the last term stands for the homotopy cofiber. Moreover, it follows from [Dég12, Lem. 1.14] that there exists a canonical isomorphism:

$$M(S^{=p}) \simeq \varinjlim_{x \in S^{(p)}} \hat{M}(\kappa_x)(p)[2p].$$

Then one can extend the computation of Proposition 4.2.2 in the transcendental 2 case as follows.

**Proposition 4.3.7.** Consider the above notation. Then there exists homotopy exact sequences in pro-DM<sub>gm</sub>(k, Q):

$$\varinjlim_{x \in S^{(1)}} \hat{M}(\kappa_x)\{1\} \xrightarrow{\Pi_x \partial_x} \hat{M}(K) \to \hat{M}(S^{\le 1})$$

$$\varinjlim_{s \in S^{(2)}} \hat{M}(\kappa_s)\{2\} \to \hat{M}(S^{\le 1}) \to 1 \oplus \underline{A} \oplus b_1.1(1)[2] \oplus M_2^{tr}(S) \oplus \underline{A}(1)[2] \oplus 1(2)[4]$$

where the $\partial_x$ is the residue map (see 3.3.5(D4*)) associated with the valuation on K corresponding to the divisor with generic point $x \in S^{(1)}$.

In particular, the motive of the transcendental degree 2 function field $K/k$ can be computed in terms of a bi-extension of generic motives associated with fields of lower transcendental degrees on the one hand, and of pure motives, including the transcendental motive of S, on the other hand.

*Proof.* This follows immediately from the exact sequences (4.3.6.a) and the preceding discussion, once one notices that $M(S^{\le 2}) = M(S)$ and $M(S^{\le 0}) = M(S^{=0}) = \hat{M}(K)$. $\square$

<footnote n="27"><sup>27</sup>We follow the notation of the latter which seems better suited.</footnote>
## 5. MOTIVIC COHOMOLOGY OF FIELDS

**5.0.1.** The structure of the generic motive of a function field $F/k$ is reflected in motivic cohomology, as follows from the formula:

$$ (5.0.1.a) \qquad H_M^{n,i}(K) = \operatorname{Hom}_{\operatorname{pro-DM}(k)}(\hat{M}(K), \mathbb{1}(i)[n]). $$

(Apply the continuity property (2.2.1.a).) We will illustrate this fact in this section.

The latter group as several explicit presentation. For the non trivial case $i > 0$, we consider the non positively graded homological complex made of free abelian groups

$$ Z_0(\mathbb{G}_m^{\wedge, i}) = c_K(\Delta^0, \mathbb{G}_m^{\wedge, i}) \xrightarrow{d_0} \dots \xrightarrow{d_{n-1}} c_K(\Delta^n, \mathbb{G}_m^{\wedge, i}) \xrightarrow{d_n} c_K(\Delta^{n+1}, \mathbb{G}_m^{\wedge, i}) \to \dots $$

where all schemes and products are taken implicitly over the field $K$, $\mathbb{G}_m^{\wedge, i}$ is the $i$-th smash product of the pointed scheme $(\mathbb{G}_m, 1)$ (see also 2.2.1 for this notation). Thus the group $c_K(\Delta^n, \mathbb{G}_m^{\wedge, i})$ is the free abelian group generated by integral closed subschemes $Z \subset \Delta^n \times \mathbb{G}_m^{\wedge, i}$ which are finite dominant over $\Delta_K^n$ and which does not belong to a closed subscheme of the form $\Delta^n \times (\mathbb{G}_m^{\wedge, j} \times \{1\} \times \mathbb{G}_m^{\wedge, i-j-1})$. The differentials are given by the alternate sum of the operation of taking intersection along faces $\Delta_i^{n-1} \times \mathbb{G}_m^{\wedge, n}$, operations which are always well-defined and can be computed using Serre's Tor formula.

Then one gets the following formula for motivic cohomology of $K$

$$ (5.0.1.b) \qquad H_M^{n,i} = \operatorname{Ker}(d_{i-n}) / \operatorname{Im}(d_{i-n-1}). $$

As both abelian groups $\operatorname{Ker}(d_{i-n})$ and $\operatorname{Im}(d_{i-n-1})$ are free, this is therefore an explicit presentation of motivic cohomology. Of course, it is useless for practical computations, but it gives at least a bound for the size of motivic cohomology of fields (see Proposition 5.2.4 for more).$^{28}$

**5.1. Borel's computations and $\lambda$-structures.** In this subsection, we recall the following fundamental computation of motivic cohomology, mainly due to Borel if one neglects weights, which were determined by Beilinson using his not less fundamental theory of (higher) regulators. The main argument of Beilinson was explained (and corrected!) with all details by Burgos in [BG02].

**Theorem 5.1.1.** Let $K$ be a number field. Let $r_1$ and $r_2$ be respectively the number of real and complex embeddings of $K$. Then for any couple of integers $(n, i)$, one has:

$$ H_{\mathbb{D}}^{n,i}(K) = \begin{cases} \mathbb{Q} & n = i = 0, \\ K^* \otimes_{\mathbb{Z}} \mathbb{Q} & n = i = 1, \\ \mathbb{Q}^{r_2} & n = 1, i > 1, i \text{ even}, \\ \mathbb{Q}^{r_1+r_2} & n = 1, i > 1, i \text{ odd}, \\ 0 & \text{otherwise.} \end{cases} $$

<footnote>$^{28}$One gets a similar presentation by considering Bloch's higher Chow groups.</footnote>
If $\mathcal{O}_K$ is the ring of integers of $K$, one gets:

$$ H_{\mathbb{D}}^{n,i}(\mathcal{O}_K) = \begin{cases} \mathbb{Q} & n = i = 0, \\ \mathcal{O}_K^* \otimes_{\mathbb{Z}} \mathbb{Q} & n = i = 1, \\ \mathbb{Q}^{r_2} & n = 1, i > 1, i \text{ even}, \\ \mathbb{Q}^{r_1+r_2} & n = 1, i > 1, i \text{ odd}, \\ \mathrm{Pic}(\mathcal{O}_K)_{\mathbb{Q}} & n = 2, i = 1, \\ 0 & \text{otherwise.} \end{cases} $$

We will give a proof of this theorem, or rather a reduction to Borel's computation of the stable real cohomology of arithmetic groups ([Bor74]). We first recall the arguments in low degree for the comfort of the reader (the latter are now well established facts).

The computations for $i \le 1$ follow from (2.2.1.b). The same holds for the vanishing $H_{\mathbb{D}}^{n,i}(K)$ (resp. $H_{\mathbb{D}}^{n,i}(\mathcal{O}_K)$) for $n - i > 0$ (resp. $n - i > 1$). We can consider the coniveau spectral sequences for the 1-dimensional regular scheme $X = \mathrm{Spec}(\mathcal{O}_K)$ with coefficients in $n$-th twisted rational motivic cohomology and rational K-theory:<fnref n="29" />

$$ E_{1,\mathbb{D}}^{p,q} = \bigoplus_{x \in X^{(p)}} H_{\mathbb{D}}^{q-p,n-p}(\kappa_x) \Rightarrow H_{\mathbb{D}}^{p+q,n}(\mathcal{O}_K), $$

$$ E_{1,K}^{p,q} = \bigoplus_{x \in X^{(p)}} K_{-p-q}(\kappa_x)_{\mathbb{Q}} \Rightarrow K_{-p-q}(\mathcal{O}_K). $$

According to Quillen's computation of the K-theory of a finite field $\mathbb{F}_q$ (see [Qui72]), one gets:

$$ H_{\mathbb{D}}^{n,i}(\mathbb{F}_q) = 0 \quad \text{iff } (n, i) \neq (0, 0), $$

$$ K_n(\mathbb{F}_q)_{\mathbb{Q}} = 0 \quad \text{iff } n \neq 0. $$

Therefore, the above spectral sequence implies:

$$ H_{\mathbb{D}}^{n,i}(\mathcal{O}_K) \xrightarrow{\sim} H_{\mathbb{D}}^{n,i}(K) \text{ if } i \ge 2, $$

$$ K_n(\mathcal{O}_K)_{\mathbb{Q}} \to K_n(K)_{\mathbb{Q}} \text{ is } \begin{cases} \text{mono} & n=1 \\ \text{iso} & n \ge 2. \end{cases} $$

Note in particular that $H_{\mathbb{D}}^{3,2}(\mathcal{O}_K)$ is necessarily 0 according to the first isomorphism. As a summary, we have obtained the lower K-groups as follows:

$$ K_0(K)_{\mathbb{Q}} = H_{\mathbb{D}}^{0,0}(K) = \mathbb{Q} $$

$$ K_0(\mathcal{O}_K)_{\mathbb{Q}} = H_{\mathbb{D}}^{0,0}(\mathcal{O}_K) \oplus H_{\mathbb{D}}^{2,1}(\mathcal{O}_K) = \mathbb{Q} \oplus \mathrm{Pic}(\mathcal{O}_K)_{\mathbb{Q}} $$

$$ K_1(K)_{\mathbb{Q}} = H_{\mathbb{D}}^{1,1}(K) = K^{\times} \otimes_{\mathbb{Z}} \mathbb{Q} $$

$$ K_1(\mathcal{O}_K)_{\mathbb{Q}} = H_{\mathbb{D}}^{1,1}(\mathcal{O}_K) = \mathcal{O}_K^{\times} \otimes_{\mathbb{Z}} \mathbb{Q} $$

<footnote n="29">²⁹The second one is a direct sum of instances of the first one, but it is practical to use both forms.</footnote>
Thus, we are reduced to compute $K_n(\mathcal{O}_K)_\mathbb{Q}$ for $n \ge 2$, and to determine its graduation for the $\gamma$-filtration to get the above calculation. As already mentioned, the first task was achieved by Borel, in his famous paper [Bor74]. The second task follows from the comparison of the Borel and the Beilinson regulators, as made explicit in [BG02] (see also [Rap88] for an earlier account). In the remaining of this section, we will explicitly recall the computations of Borel and show how to give a direct argument for the determination of the $\gamma$-filtration.

**5.1.2. Rational K-theory and homology of groups.**$^{30}$ Given any commutative ring $A$ and an integer $n > 1$, one gets the following classical computation:

$$K_n(A) \simeq \pi_n(\mathrm{BGL}(A)^+) \simeq \pi_n(\mathrm{BSL}(A)^+)$$

where $\mathrm{GL}(A)$ (resp. $\mathrm{SL}(A)$) is the infinite general (resp. special) linear group associated with $A$, and $X^+$ is Quillen $+$-construction applied to a space $X$ (see also Example 7.0.2). The first isomorphism is Quillen’s comparison of the plus and $\mathbb{Q}$ constructions ([Gra76], [God] with more details). To get the second isomorphism one considers the obvious exact sequence of groups:

$$1 \to \mathrm{SL}(A) \to \mathrm{GL}(A) \xrightarrow{\mathrm{det}} A^\times \to 1.$$

After applying the classifying space construction, followed by the plus construction, one gets a fibration sequence in the homotopy category:$^{31}$

$$\mathrm{BSL}(A)^+ \to \mathrm{BGL}(A)^+ \to \mathrm{B}(A^\times).$$

Thus the associated exact sequence of homotopy groups allows to conclude as $A^\times$ is a finite abelian group.

Next we recall that $\mathrm{BSL}(A)^+$ is an h-space$^{32}$, therefore the Milnor-Moore/Cartan-Serre theorem implies that the Hurewicz map

$$\pi_n(\mathrm{BSL}(A)^+) \to H_n^{\mathrm{sing}}(\mathrm{BSL}(A), \mathbb{Z})$$

induces an isomorphism after tensoring with $\mathbb{Q}$

$$(5.1.2.a) \qquad K_n(A)_\mathbb{Q} \xrightarrow{\sim} P_n H_*(\mathrm{SL}(A), \mathbb{Q})$$

where $P_n$ denotes the primitive part in degree $n$ of the Hopf algebra:

$$H_*(\mathrm{SL}(A), \mathbb{Q}) = H_*^{\mathrm{sing}}(\mathrm{BSL}(A), \mathbb{Q}) \simeq H_*^{\mathrm{sing}}(\mathrm{BSL}(A)^+, \mathbb{Q}).$$

<footnote>$^{30}$These facts are well-known, but we still recall them for completeness, and for introducing our notation.</footnote>

<footnote>$^{31}$Note that the $+$-construction respects fibration sequences, or equivalently homotopy fibers, as it is a localization functor.</footnote>

<footnote>$^{32}$in fact, the $+ = \mathbb{Q}$ theorem of Quillen shows it is a loop space.</footnote>
**5.1.3.** We next fix an integer $r > 0$, consider the arithmetic group $\Gamma_r = SL_r(\mathcal{O}_K)$, which can be seen as a subgroup of the real Lie group$^{33}$

$$G_r = \mathrm{SL}_r(K \otimes_{\mathbb{Q}} \mathbb{R}) = \mathrm{SL}_r(\mathbb{R})^{r_1} \times \mathrm{SL}_r(\mathbb{C})^{r_2}.$$

We let $\nu_r : \Gamma_r \to G_r$ be the natural inclusion, where $\Gamma_r$ is seen with the discrete topology. Viewing $\mathbb{R}$ as a discrete module over $G_r$ (resp. $\Gamma_r$), one can define the continuous cohomology $H_{ct}^*(-, \mathbb{R})$ with coefficient in $\mathbb{R}$ (see e.g. [BW00, IX, §1]). As an application of the main result of [Bor74], one gets the following fundamental computation.

**Theorem 5.1.4.** *The restriction map along $\nu_r$ in continuous cohomology induces an isomorphism for any $n \le r/4$:*

$$H_{ct}^n(G_r, \mathbb{R}) \xrightarrow{\nu_r^*} H_{ct}^n(\Gamma_r, \mathbb{R}) = H^n(\Gamma_r, \mathbb{R}).$$

*Proof.* As it is very involved, we indicate how to deduce the proof of the above theorem from [Bor74], Th. 7.5.

One considers the maximal compact open subgroup $K_r = SO_r^{r_1} \times SU_r^{r_2}$ of $G_r$ and the associated symmetric space $X_r = K_r \backslash G_r$. The group $\Gamma_r$ acts properly discontinuously and almost freely on $X_r$ so that the topological quotient $O_r = X_r/\Gamma_r$ is an orbifold (see e.g. [Thu22]). The canonical map $B\Gamma_r = X_r \backslash G_r \to O_r$ is a $\mathbb{Q}$-equivalence (Definition 7.0.1).$^{34}$ The de Rham isomorphism for orbifolds (see [Sat56]) implies that one can compute the singular cohomology of $O_r$ with real coefficients, and therefore that of $B\Gamma_r$, as the cohomology of the differential graded $\mathbb{R}$-algebra $\Omega^*(O_r) \simeq \Omega^*(X_r)^{\Gamma_r}$. On the other hand, according to the (topological version) of the Poincaré lemma, the continuous cohomology of $G_r$ can be identified with the cohomology of the $\mathbb{R}$-dga $\Omega^*(X_r)^{G_r}$. The Mastushima map considered in [Bor74, §3.1] is nothing else than the inclusion map of real differential graded algebras

$$j_{\Gamma_r} : \Omega^*(X_r)^{G_r} \to \Omega^*(X_r)^{\Gamma_r}.$$

The latter is therefore a quasi-isomorphism in degree less than $r/4$ according to the conjunction of Th. 7.5 and §9.5, formula (3) of [Bor74]. Finally, it is straightforward to check that $j_{\Gamma_r}$ models the restriction map $\nu_r^*$ on continuous cohomology. $\square$

**Corollary 5.1.5.** *Consider the inclusion of topological groups*

$$\nu : \mathrm{SL}(\mathcal{O}_K) \to \mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}).$$

<footnote>$^{33}$The latter can also be seen as the real points of the algebraic $\mathbb{Q}$-group scheme obtained by Weil scalar restriction of the $K$-group scheme $\mathrm{SL}_{r,K}$.</footnote>

<footnote>$^{34}$According to Selberg's lemma, there existe a finite index normal torsion-free subgroup $\Gamma'_r \subset \Gamma_r$. Then $X_r/\Gamma'_r$ is a manifold and a model for $B\Gamma_r$. But now the canonical map $X_r/\Gamma'_r \to O_r$ is a finite branched cover, and therefore a $\mathbb{Q}$-equivalence.</footnote>
Then the pullback along $\nu$ induces an isomorphism:

$$H_{ct}^{*}(\mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}), \mathbb{R}) \xrightarrow{\nu^{*}} H^{*}(\mathrm{SL}(\mathcal{O}_{K}), \mathbb{R})$$

of graded $\mathbb{R}$-algebras, whose graded pieces are all finite dimensional over $\mathbb{R}$.

Indeed, for any $n \le r/4$, $H_{ct}^n(G_r, \mathbb{R})$ has finite dimension over $\mathbb{R}$, and does not depend on $r$.<fnref n="35" /> It follows that both the source and the target of $\nu_r^*$ satisfies the Mittag-Leffler condition, and this implies that $\nu^*$ can be identified with the projective limit of the $\nu_r^*$. Thus the preceding theorem allows us to conclude.

**5.1.6.** Recall that according to the universal coefficient theorem, one always gets a canonical isomorphism:

$$H^n(\mathrm{SL}(\mathcal{O}_K), \mathbb{R}) = H_n(\mathrm{SL}(\mathcal{O}_K), \mathbb{R})^{\vee}$$

where $\vee$ means the dual of (discrete) $\mathbb{R}$-vector spaces. Therefore, as this $\mathbb{R}$-vector space is finite dimensional according to the preceding corollary, one deduces a canonical isomorphism

$$H_n(\mathrm{SL}(\mathcal{O}_K), \mathbb{R}) \to H_n(\mathrm{SL}(\mathcal{O}_K), \mathbb{R})^{\vee\vee} \simeq H^n(\mathrm{SL}(\mathcal{O}_K), \mathbb{R})^{\vee}.$$

We will define the following $\mathbb{R}$-valued continuous homology<fnref n="36" />:

$$H_n^{ct}(\mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}), \mathbb{R}) := H_{ct}^n(\mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}), \mathbb{R})^{\vee}$$

Therefore, the preceding corollary can be formulated by saying that the restriction map $\nu$ induces an isomorphism of $\mathbb{R}$-coalgebras:

$$H_*(\mathrm{SL}(\mathcal{O}_K), \mathbb{R}) \xrightarrow{\nu_*} H_*^{ct}(\mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}), \mathbb{R}).$$

Therefore, according to (5.1.2.a), one gets the following computation:

**Corollary 5.1.7.** Consider the above notation. Then for any integer $n \ge 2$, the Hurewicz map composed with the above isomorphism $\nu_*$ induces an isomorphism of $\mathbb{R}$-vector spaces:

$$K_n(\mathcal{O}_K) \otimes_{\mathbb{Q}} \mathbb{R} \xrightarrow{\sim} P_n H_*^{ct}(\mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}), \mathbb{R}).$$

This is a nice formula, which already allows to recover the rank of K-theory. However, we do not know how to determine the gamma filtration on the right hand-side directly, though it exists naturally (see below). Therefore we have to come back to the known strategy and use the compact form of the ind-Lie group $\mathrm{SL}(K \otimes_{\mathbb{Q}} \mathbb{R})$.

<footnote n="35">³⁵This follows from the van Est isomorphism, which allows to identify the latter cohomology groups with the singular cohomology group $H_{sing}^n(G_r^u, \mathbb{R})$ of the compact real form $G_r^u$ associated with the semi-simple real Lie group $G_r$. See below.</footnote>

<footnote n="36">³⁶Because of finite dimensionality, it coincides with other possible definition of continuous homology via derived functors.</footnote>
**5.1.8.** We now recall the necessary computation from Lie groups and Lie algebras. To the ind-Lie group $G = \text{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}) = \text{SL}(\mathbb{R})^{r_1} \times \text{SL}(\mathbb{C})^{r_2}$, one associates using Cartan duality the following objects:

<table><thead><tr><th>ind-object of</th><th>non ind-compact type</th><th>ind-compact dual</th></tr></thead><tbody><tr><td>Lie groups</td><td>$G = \text{SL}(\mathbb{R})^{r_1} \times \text{SL}(\mathbb{C})^{r_2}$</td><td>$G^u = \text{SU}^{r_1} \times (\text{SU} \times \text{SU})^{r_2}$</td></tr><tr><td>max. compact</td><td>$K = \text{SO}^{r_1} \times \text{SU}^{r_2}$</td><td>same</td></tr><tr><td>sym. spaces</td><td>$X = (\text{SL}(\mathbb{R})/\text{SO})^{r_1} \times (\text{SL}(\mathbb{C})/\text{SU})^{r_2}$</td><td>$X^u = (\text{SU}/\text{SO})^{r_1} \times \text{SU}^{r_2}$</td></tr></tbody></table>

The duality is defined via the associated ind-Lie algebras: starting with the Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ associated with $G$ (and its Killing form), we consider the *compact form* of $\mathfrak{g}$ as the Lie algebra $\mathfrak{g}^u = \mathfrak{k} \oplus \mathfrak{p}^u$ where $\mathfrak{p}^u = i_{\mathfrak{p}} \subset \mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C}$. Then $\mathfrak{g}^u$ is the ind-Lie algebra of $G^u$.

As explained in the proof of Theorem 5.1.4, the continuous cohomology of $G$ can be identified with the cohomology of the differential graded $\mathbb{R}$-algebra $\Omega^*(X)^G$ of $G$-invariant differential forms on $X$.<fnref n="37" /> Further, the classical Lie cohomology theory translates into the following composite of isomorphisms or homotopy equivalence:

$$ \Omega^*(X)^G \xrightarrow{\sim} C^*(\mathfrak{g}, \mathfrak{k}; \mathbb{R}) \xrightarrow{\sim} C^*(\mathfrak{g}^u, \mathfrak{k}; \mathbb{R}) \xrightarrow{\sim} \Omega^*(X^u)^{G^u} \xrightarrow{\sim} \Omega^*(X^u) $$

where (1) and (3) are the isomorphisms obtained by evaluating a differential form on $X$ or $X_u$ at the base point (class of Id), (2) is the isomorphism which in degree $n$ is multiplication with $i^n$, and (4) is a homotopy equivalence, the retraction of the obvious inclusion, given by integration $\omega \mapsto \int_{G^u} g^*\omega dg$ where $dg$ is the Haar measure on $G^u$.

Using the de Rham isomorphism for the ind-smooth variety $X^u$, we finally get a composite isomorphism:

$$ \varphi : H_{ct}^*(G, \mathbb{R}) \xrightarrow{\sim_{(1234)}} H_{dR}^*(X^u) \xrightarrow{\sim} H_{sing}^*(X^u, \mathbb{R}). $$

Let us pause the exposition with the following definition.

**Definition 5.1.9.** For any integer $n \ge 2$, we have obtained an isomorphism:

$$ \rho_B : K_n(\mathcal{O}_K) \otimes_{\mathbb{Q}} \mathbb{R} \xrightarrow{\nu_*} P_n H_*^{ct} (\text{SL}(K \otimes_{\mathbb{Q}} \mathbb{R}), \mathbb{R}) \xrightarrow{\varphi^\vee} P_n H_*^{sing} ((\text{SU}/\text{SO})^{r_1} \times \text{SU}^{r_2}, \mathbb{R}) $$
called the *Borel regulator* associated with the number field $K$ (in degree $n$).

**Remark 5.1.10.** (1) Beware that Burgos has explained with a remarkable precision in [BG02, §9.5] that one needs to renormalize the above regulator in order to compare it with the Beilinson regulator (up to multiplying by 2 on the factors corresponding to the $r_1$ real places). In our approach, this problem is irrelevant.

<footnote n="37">³⁷Recall that the differentials of this complex are trivial, as the $G$-invariant differential forms are in fact harmonic.</footnote>
(2) The only non-canonical isomorphism that composes the Borel regulator is the isomorphism (2) mentioned above: it is noted, however, that it is degree-wise a multiplication by a scalar within the ambient complex ind-Lie algebra $\mathfrak{g}_c$.

**5.1.11.** We are now in position to elucidate the behavior of the Borel regulator with respect to the canonical $\lambda$-structure on the left hand-side, and an appropriate computable $\lambda$-structure on the right hand-side.

To define the $\lambda$-operations, we use the method of Soulé in [Sou85, §1]. In our case, this leads to consider the exterior power of matrices. After taking the obvious base on $\Lambda^r(\mathbb{Z}^n)$, this gives a morphism of schemes

$$ \Lambda_n^r : \mathrm{SL}_n \to \mathrm{SL}(n) $$

and as $n$ goes to $\infty$, a morphism of ind-group schemes $\Lambda^r : \mathrm{SL} \to \mathrm{SL}$. According to the construction of loc. cit., this morphism evaluated at $\mathcal{O}_K$ induces by functoriality a morphism on $K_n(\mathcal{O}_K) = \pi_n(\mathrm{BSL}(\mathcal{O}_K)^+)$ which is nothing else than the operation $\lambda^k$ defined in loc. cit.

By the same formula, we obtain operations $\Lambda^k$ on SU and SO respectively, and therefore by functoriality an operation on the target of the Borel regulator $\rho_B$. It follows from the construction, as we have detailed, that $\rho_B$ commutes with the action just constructed on both its source and target.

It remains to determine the induced structure on $H_*^{sing}((\mathrm{SU}/\mathrm{SO})^{r_1} \times \mathrm{SU}^{r_2}, \mathbb{R})$ and on its primitive part.

For this, we recall the theory of characteristic classes for notation. Given a topological group $\mathfrak{G}$, we know that there exists universal principal topological bundle $\mathfrak{E}\mathfrak{G}$ classifying respectively the special (i.e. with trivial determinant) complex and real vector bundles (see e.g. [Mil56]). One deduces a fibration sequence:

$$ \mathfrak{G} \to \mathfrak{E}\mathfrak{G} \xrightarrow{p} \mathfrak{B}\mathfrak{G} $$

where the classifying space $\mathfrak{B}\mathfrak{G}$ is pointed by the trivial principal bundle. This implies the existence of the (classical) homotopy equivalence $\mathfrak{G} \simeq \Omega\mathfrak{B}\mathfrak{G}$ with target the indicated loop space. By adjunction, one deduces the so-called *suspension map*

$$ s : \Sigma\mathfrak{G} \to \mathfrak{B}\mathfrak{G} $$

and the induced map in cohomology:

$$ H_{\mathrm{sing}}^{n+1}(\mathfrak{B}\mathfrak{G}, \mathbb{R}) \xrightarrow{s^*} H_{\mathrm{sing}}^{n+1}(\Sigma\mathfrak{G}, \mathbb{R}) \simeq H_{\mathrm{sing}}^n(\mathfrak{G}, \mathbb{R}). $$

Note that this map is natural with respect to the topological group $\mathfrak{G}$. In our case, where $\mathfrak{G} = \mathrm{SU}, \mathrm{SO}$ is an ind-Lie group, we deduce from the Serre spectral sequence associated with the fibration $p$ that the suspension map is an isomorphism. By taking real duals, one deduces the following homological suspension isomorphisms
which fit into the commutative diagram:

$$
\begin{array}{c}
(5.1.11.a) \qquad \begin{array}{r@{\;}c@{\;}l@{\quad}l}
& H_n^{\mathrm{sing}}(\mathrm{SU}, \mathbb{R}) & \xrightarrow{\substack{s_* \\ \sim}} & H_{n+1}^{\mathrm{sing}}(\mathrm{B}\,\mathrm{SU}, \mathbb{R}) \\
& \multicolumn{1}{@{}c@{}}{\scriptstyle \iota_* \uparrow} & & \multicolumn{1}{@{}c@{}}{\scriptstyle \uparrow \iota_*} \\
& H_n^{\mathrm{sing}}(\mathrm{SO}, \mathbb{R}) & \xrightarrow{\substack{s_* \\ \sim}} & H_{n+1}^{\mathrm{sing}}(\mathrm{B}\,\mathrm{SO}, \mathbb{R})
\end{array} \\
\text{where } \iota: \mathrm{SO} \to \mathrm{SU} \text{ is the natural inclusion. Now, the theory of characteristic}
\end{array}
$$

where $\iota: \mathrm{SO} \to \mathrm{SU}$ is the natural inclusion. Now, the theory of characteristic classes (see e.g. [Swi02, Chap. 16]) applies and gives:

$$
H_{\mathrm{sing}}^{*}(\mathrm{B}\,\mathrm{SU}, \mathbb{R}) = \mathbb{R}[c_2, \dots, c_i, \dots]
$$

$$
H_{\mathrm{sing}}^{*}(\mathrm{B}\,\mathrm{SO}, \mathbb{R}) = \mathbb{R}[p_2, \dots, p_j, \dots]
$$

where the Chern (resp. Pontryagin) classes $c_i$ (resp. $p_j$) are in degree $2i$ (resp. $4j$) and in weight $i$ (resp. $j$) for the natural $\lambda$-structure, induced as above. One can compute $\iota^*$ via complexification of real vector bundles, and one deduces that

$$
\iota^*(c_i) = \begin{cases} 0 & i \text{ odd,} \\ (-1)^j (2j-1)!.p_j + \text{lower degrees} & i = 2j. \end{cases}
$$

Therefore, the kernel of $\iota^*$ in rational cohomology is generated by the odd Chern classes. One should be careful also that $\iota^*$ divides weights by 2.<fnref n="38" />

Therefore the singular homology of B SU is the free symmetric algebra generated by the $c_i^\vee$ in even degree $2i$ and weight $i$ for $i \ge 2$. The operation of taking the cokernel of $\iota_*$ kills the classes $c_i^\vee$ for $i$ even. According to the construction of $\lambda$-operations recalled above, the suspension isomorphisms in diagram (5.1.11.a) shift the degree by +1 and preserves the weight. Thus one finally obtains the following key computation:

**Proposition 5.1.12.** The real singular homology with its weight graduation coming from the preceding $\lambda$-structure

$$
H_{*}^{\mathrm{sing}}((\mathrm{SU} / \mathrm{SO})^{r_1} \times \mathrm{SU}^{r_2}, \mathbb{R})
$$

is the free exterior real algebra generated by $r_2$ classes of degree $2i - 1$ and weight $i$ for any $i \ge 2$ and by $r_1$ classes of degree $2j - 1$ and weight $j$ for odd $j \ge 2$.

This follows from the preceding discussion and the (dual) Künneth formula. Let us draw the explicit corollary, which achieves the proof of Theorem 5.1.4.

**Corollary 5.1.13.** Given a number field $K$ with $r_1$, $r_2$ real and complex places respectively, a basis for the $K$-theory graded $\mathbb{Q}$-vector space $K_*(\mathcal{O}_K)_{\mathbb{Q}}$ restricted to degrees $* \ge 2$ is given by:

* $r_2$ elements in degree $2i - 1$ and weight $i$ for $i \ge 2$,

* $r_1$ elements in degree $2j - 1$ and weight $j$ for odd $j \ge 2$.

<footnote n="38">³⁸All this comes from the relation $p(E) = c(E \otimes_{\mathbb{R}} \mathbb{C}) \cdot c(E \otimes_{\mathbb{R}} \mathbb{C})$ for a real vector bundle $E$, between the total Pontryagin and Chern classes. Thus $\iota^*$ can be thought of as extracting a square root.</footnote>
**5.2. The effect of transcendental elements.** We will now deduce from the computations of generic motives obtained in the preceding section some information about motivic cohomology of fields. All the next results are based on formula (5.0.1.a). First, as a corollary of Proposition 4.1.2 and Theorem 5.1.4, we get the following computation:

**Proposition 5.2.1.** Let $K$ be a number field. Then for any integers $(n, i) \in \mathbb{Z}^2$,

$$ H_{\mathbb{D}}^{n,i}(K(t)) = \begin{cases} \mathbb{Q} & n = i = 0 \\ K^{\times} \oplus Z_0(\mathbb{A}_K^1)_{\mathbb{Q}} & n = i = 1 \\ H_{\mathbb{D}}^{1,i}(K) & n = 1, i > 1 \\ \bigoplus_{x \in \mathbb{A}_{K,(0)}^1} H_{\mathbb{D}}^{1,i-1}(\kappa_x) & n = 2, i \ge 2 \\ 0 & \text{otherwise.} \end{cases} $$

We have denoted by $Z_0(-)$ the group of algebraic 0-cycles.

As there are infinitely many points $x \in \mathbb{A}_{K,(0)}^1$, whose residue field $\kappa_x$ has at least one complex place, one deduces from Theorem 5.1.4 that for all $i \ge 2$, the abelian group $H_{\mathbb{D}}^{n,i}(K(t))$ has infinite rank.

This picture can be confirmed in higher transcendence degree. Indeed, applying now Proposition 4.3.1, one obtains the following proposition.

**Proposition 5.2.2.** Let $K$ be a field of characteristic 0 such that $\text{degtr } K \ge d > 0$.

Then for any $n \in [2, d+1]$ and any $i \ge n$, the abelian group $H_{\mathbb{D}}^{n,i}(K)$ has infinite rank.

*Proof.* Indeed, by assumptions, $K$ contains the field of rational functions $\mathbb{Q}(t_1, ..., t_d)$. According to Corollary 4.1.4, one can assume that $K = \mathbb{Q}(t_1, ..., t_d)$. According to Proposition 4.3.1, one deduces that $H_{\mathbb{D}}^{n,i}(K)$ contains $\bigoplus_{x \in \mathbb{A}_{\mathbb{Q},(0)}^1} H_{\mathbb{D}}^{1,i-n+1}(\kappa_x)$ which is infinite according to Theorem 5.1.4 and the fact there exists an infinity of finite monogeneous extensions of $\mathbb{Q}$ with at least on complex place. $\square$

*Remark 5.2.3.* According to the splitting $\varphi$ given in Proposition 4.1.2, it is easy to construct infinite free families of elements of $H_{\mathbb{D}}^{n,i}(\mathbb{Q}(t_1, ..., t_n))$ under the assumptions of the preceding proposition. As a function field $K/k$ of transcendence degree $d = n$ is a finite extension of $\mathbb{Q}(t_1, ..., t_n)$ one deduces infinite free families of $H_{\mathbb{D}}^{n,i}(K)$. We let these constructions as an exercise to the reader.<fnref n="39" />

One can even push this argument further to get non-finiteness results of motivic cohomology.

<footnote n="39">³⁹As a hint, consider product of symbols extracted from some transcendental basis of $K/\mathbb{Q}$, elements coming from the Borel regulator on an appropriate number field $E$ and eventually restriction or corestriction along finite field extensions on motivic cohomology. All these operations can be realized geometrically as well by taking appropriate smooth models.</footnote>
**Proposition 5.2.4.** Let $K$ be an extension field of $\mathbb{Q}$ with infinite transcendence degree. Then for any integers $i \ge n \ge 2$, the abelian group $H_{\mathbb{Q}}^{n,i}(K)$ has rank equal to $\mathrm{card}(K)$.

An important remark is that this result extends an earlier theorem due to Springer, which concerns the Milnor K-theory part $n = i$: see [BT73, 5.10] (and below for more discussion).

*Proof.* According to formula (5.0.1.b), we get $\mathrm{card}(H_{\mathbb{Q}}^{n,i}(K)) \le \mathrm{card}(K)$.

We next consider the other inequality. By assumption, there exists a subfield $F \subset K$ and a finite morphism $F(t_1, ..., t_n) \to K$. Using Corollary 4.1.4, it suffices to treat the case of $K = F(t_1, ..., t_n)$. Then according to Proposition 4.3.1, one deduces that $H_{\mathbb{Q}} n, i(K)$ contains as a direct factor the abelian group $\oplus_{x \in \mathbb{A}_{F,(0)}^1} H_{\mathbb{Q}}^{1, i-n+1}(\kappa_x)$. Applying again Corollary 4.1.4, the latter groups contains for example $\oplus_{x \in F} H_{\mathbb{Q}}^{1, i-n+1}(\mathbb{Q}[i])$ whose rank is bigger or equal to $\mathrm{card}(F) = \mathrm{card}(K)$. $\square$

**5.2.5.** As a conclusion, there is a range, controlled by the transcendence degree over $\mathbb{Q}$, in which all motivic cohomology groups of fields are infinite. However, a tempting conjecture emerges that we will now formulate. We will directly state it in arbitrary characteristics though we have focused on the characteristic 0 case. Recall one defines the Kronecker dimension of a field $K$ as the integer

$$ \delta(K) = \begin{cases} \mathrm{degtr}(K/\mathbb{Q}) + 1 & \mathrm{char}(K) = 0 \\ \mathrm{degtr}(K/\mathbb{F}_p) & \mathrm{char}(K) = p > 0. \end{cases} $$

**Conjecture.** Let $K$ be a field. Then,

$$ H_{\mathbb{Q}}^{n,i}(K) = 0 $$

unless $(n, i) = (0, 0)$ or $n \in [1, \delta(K)]$.

There is few evidences for this conjecture. Nevertheless, it holds for the fields $K$ and $K(t)$ such that $\delta(K) \le 1$: as we have seen, the case of number fields follows from Theorem 5.1.4 and Proposition 5.2.1. The case of characteristic $p > 0$ is a consequence of [Qui72] and [Har77] (see below). We will end-up this section with some further evidences.

**5.2.6.** *The Beilinson-Soulé conjecture.* Actually the above conjecture in the range $n < 1$ is already well-known as the Beilinson-Soulé conjecture, in its “strong” form. This terminology appears in the introduction of [Lev93]. By opposition, the “weak” form of the Beilinson-Soulé conjecture asks for the vanishing of motivic cohomology in degree $< 0$ (see e.g. [SV00, §3, Be2]).$^{40}$ We will simply say *Beilinson-Soulé conjecture* in the sequel for the strong form.

<footnote>$^{40}$The integral version is equivalent to the rational one: see [Kah05, Lem. 24].</footnote>
**Remark 5.2.7.** We have seen in Proposition 5.2.1 that the line $n = 1$ on motivic cohomology of fields has a special behavior: it does not seem to grow with transcendental elements. In general, one conjectures (after Beilinson) that for any $i > 1$, $H_{\mathbb{D}}^{1,i}(K) = H_{\mathbb{D}}^{1,i}(K_0)$ where $K_0 \subset K$ is the field of constants (i.e. the algebraic closure of the prime subfield of $K$). This is sometimes called the *rigidity* conjecture. For more discussion on this and the Belinson-Soulé conjecture, the reader is advised to look at [Kah05, §4.3.4, Lem. 24].$^{41}$

**5.2.8.** *The Milnor K-theory part.* The above conjecture in the restricted case $n = i$ was already asked in [BT73, Question after the proof of 5.10]. Even in that restricted case, nothing is known beyond the case $\delta(K) \le 1$ already mentioned.

**5.2.9.** Recall that the Beilinson-Soulé conjecture is equivalent to the existence of the motivic t-structure on (rational) Tate motives over $K$ with a weight filtration such that $\mathbb{1}(n)$ is of weight $-2n$ (see [Lev93, Th. 4.2]).$^{42}$

In that respect, the novel of our question is rather about the vanishing above the Kronecker dimension of the field. Moreover, under the Beilinson-Soulé conjecture, the vanishing in this particular range is equivalent to ask that the category of Tate motives has cohomological dimension $\delta(K)$ (see again [Lev93, Cor. 4.3] in the case $\delta(K) = 1$).

**5.2.10.** *The positive characteristic case.* When $K$ is a field of characteristic $p > 0$, the above conjecture is in fact a consequence of two known conjectures. And in fact, one hopes for a stronger vanishing of motivic cohomology: the conjecture of Beilinson and Parshin implies that $H_{\mathbb{D}}^{n,i}(K) = 0$ if $n \neq i$ (see [Gei98, Th. 3.4]). Therefore, the previous conjecture would be a consequence of the Beilinson-Parshin conjecture and a positive answer to the question of Bass and Tate recalled above.

**5.3.** **Motivic cohomology with coefficients in abelian varieties.** We finish this paper by mentioning an interesting cohomology theory that comes out of Voevodsky’s theory of motivic complexes.

**Definition 5.3.1.** Let $k$ be a perfect field, and $A$ be an abelian variety over $k$. For an integer $i \in \mathbb{Z}$, we consider the motivic complex in $\mathrm{DM}(k)$:

$$ \underline{\mathcal{A}}(i) = \Sigma^{\infty}(\underline{\mathcal{A}}) \otimes \mathbb{1}(i). $$

Then, given any smooth $k$-scheme of finite type, we define the motivic cohomology of $X$ with coefficients in $A$, degree $n$ and twists $i$, as the abelian group:

$$ H^{n,i}(X, \underline{\mathcal{A}}) = \mathrm{Hom}_{\mathrm{DM}(k)}(\mathrm{M}(X), \underline{\mathcal{A}}(i)). $$

This new cohomology theory inherits all the properties of the motive $\hat{M}(X)$:

<footnote>$^{41}$Note that the conjectures 16 and 17 of loc. cit. are now a theorem due to Voevodsky, with important contributions of Rost.</footnote>

<footnote>$^{42}$Actually, the assertion that $\mathbb{1}(n)$ is in the heart of the motivic t-structure and of weight $(-2n)$ for all $n \in \mathbb{Z}$ is equivalent to the strong form of the Beilinson-Soulé conjecture for $K$.</footnote>
* It is contravariant with respect to arbitrary morphisms in X, and covariant up to twists and shift with respect to proper smoothable morphisms (use Gysin morphisms).

* It satisfies the localization property with respect to a smooth closed pair $(X, Z)$ (use the Gysin triangle).

* It satisfies the projective bundle formula as well as the blow-up formula with smooth center.

According to Proposition 4.2.4, $\underline{A}(-i) = 0$ for $i > 0$. One deduces from the cancellation theorem that for $i \ge 0$:

$$H^{n,i}(X, \underline{A})) \simeq H_{\mathrm{Nis}}^{n}(X, \underline{A}(i))$$

where the left hand-side is the Nisnevich cohomology of the motivic complexes $\underline{A} \otimes 1(i) \in \mathrm{DM}^{\mathrm{eff}}(k)$. According to the property of the derived tensor product in $\mathrm{DM}^{\mathrm{eff}}(k)$, one deduces that $\underline{A}(i)$ is a complex of Nisnevich sheaves concentrated in cohomological degree $\le i$. One deduces:

$$H^{n,i}(X, \underline{A}) = \begin{cases} \underline{A}(X) & n = i = 0, \\ 0 & i < 0 \text{ or } n > 2i \text{ or } n - i > \dim(X). \end{cases}$$

Finally, one gets for any function field $K/k$ the following computation:

$$H^{n,n}(K, \underline{A}) \simeq K_{n}^{SM}(K; A)$$

with the notation of Proposition 4.2.4. Moreover, it follows from Theorem 3.3.7 that $K_*^{SM}(-; A) \simeq \hat{\underline{A}}_*$ is a cycle module over $k$. One deduces a differential from Rost's cycle complex with coefficients in the latter cycle module:

$$\bigoplus_{y \in X^{(n-1)}} K_1^{SM}(\kappa_y; A) \xrightarrow{\mathrm{div}_A} \bigoplus_{x \in X^{(n)}} A(\kappa_x).$$

**Definition 5.3.2.** Consider the above notation. One defines the codimension $n$ algebraic cycles of $X$ with coefficients in $A$ as the abelian group:

$$Z^n(X; A) = \bigoplus_{x \in X^{(n)}} A(\kappa_x).$$

The map $\mathrm{div}_A$ defined above will be called the divisor class map. One defines the Chow group of $X$ in codimension $n$ and coefficients in $A$ as the cokernel:

$$\mathrm{CH}^n(X; A) := \mathrm{coKer}(\mathrm{div}_A) = Z^n(X; A) / \sim_A.$$

Here $\sim_A$ is the obvious equivalence relation, that will be called the rational equivalence. As a consequence of the coniveau spectral sequence together with the previous computation, one gets:

**Proposition 5.3.3.** Let $X$ be a smooth $k$-scheme and $A$ an abelian variety. Then for any integer $n \ge 0$, there exists a canonical isomorphism:

$$H^{2n,n}(X, \underline{A}) \simeq \mathrm{CH}^n(X; A).$$
**Remark 5.3.4.** (1) By considering motives of the form $A \otimes B$ in $\mathrm{DM}^{\mathrm{eff}}(k)$, one can define exterior products of motivic cohomology with coefficients in abelian varieties.

(2) Up to inverting the characteristic exponent of $k$, it can be extended to singular $k$-schemes in a contravariant way. The extension satisfies cdh-descent.

(3) One can define a compactly supported version by using motive with compact supports.

**5.3.5.** The preceding definition is useful to state the computation of motivic cohomology of function fields that comes from Proposition 4.2.2.

**Proposition 5.3.6.** Let $K$ be a number field, and $\bar{C}/K$ be a smooth projective curve with base point $x_0 \in \bar{C}_{(0)}$. We let $L$ be the function field of $\bar{C}$, and $A$ be the jacobian of pointed curve $(\bar{C}, x_0)$.

Then, there are canonical morphisms

$$ \pi_{L/K}^{n,i} : \mathrm{H}_{M}^{n-2,i-1}(K, \underline{A}) \to \mathrm{H}_{M}^{n,i}(L). $$

which are isomorphisms for any pair $(n, i)$ such that $i \ge n, n \in \mathbb{Z} - [1, 3]$.

For $i > 1$, the following map is an isomorphism:

$$ \mathrm{H}_{M}^{-1,i-1}(K, \underline{A}) \oplus \mathrm{H}_{M}^{1,i}(K) \xrightarrow{(\pi_{L/K}^{1,i}, \varphi^{*})} \mathrm{H}_{M}^{1,i}(L) $$

Moreover, for $i \ge 3$, the following sequence of abelian groups is exact:

$$ 0 \to \mathrm{H}_{M}^{0,i-1}(K, \underline{A}) \xrightarrow{\pi_{L/K}^{2,i}} \mathrm{H}_{M}^{2,i}(L) \xrightarrow{\Sigma_x \partial_x} \bigoplus_{x \in C_{(0)}} \mathrm{H}_{M}^{1,i-1}(\kappa_x) \to \mathrm{H}_{M}^{1,i-1}(K, \underline{A}) \xrightarrow{\pi_{L/K}^{2,i}} \mathrm{H}_{M}^{3,i}(L) \to 0. $$

*Proof.* This is a faithful translation of the homotopy exact sequence of Proposition 4.2.2, with the duality result of Lemma 4.2.6, and using the vanishing of motivic cohomology of the fields $K$ and $\kappa_x$ from Theorem 5.1.4. $\square$

**5.3.7.** One can therefore interpret the preceding conjecture for the function field $L$ in terms of properties of the cohomology groups $\mathrm{H}_{M}^{n,i}(K, \underline{A})$, which are attached to an abelian variety over a number field. These data might be easier to grasp than the transcendental extension $L$.

# APPENDIX

## 6. CATEGORICAL PRESENTATION OF CYCLE MODULES

One can describe Rost's cycle premodules as covariant functors from a category whose morphisms are defined by generators and relations. We reproduce below the definition from [Dég08b, §4.1] for the reader's convenience.
**Definition 6.0.1.** . We let $\tilde{\mathcal{E}}_k$ be the category whose objects are given by couples $(E, n)$ where $E/k$ is a function field, and $n \in \mathbb{Z}$ an integer. The morphisms of $\tilde{\mathcal{E}}_k$ are abelian groups defined by the following generators $D*$ and relations $R*$ as displayed below, where maps are always assumed to be $k$-algebra morphisms, and the valuations are always understood as (geometric) valuation of function fields:
**Generators:**

D1: $\varphi_* : (E, n) \to (L, n)$ for $\varphi : E \to L, n \in \mathbb{Z}$.

D2: $\varphi' : (L, n) \to (E, n)$, for $\varphi : E \to L$ finite, $n \in \mathbb{Z}$.

D3: $\gamma_x : (E, n) \to (E, n+r)$, for $x \in K_r^M(E)$, $n \in \mathbb{Z}$.

D4: $\partial_v : (E, n) \to (\kappa(v), n-1)$, for $(E, v)$ valued function field over $k, n \in \mathbb{Z}$.

**Relations:**

R0: For all $x, y \in K_*^M(E)$, $\gamma_x \circ \gamma_y = \gamma_{x.y}$.

R1a: $(\psi \circ \varphi)_* = \psi_* \circ \varphi_*$.

R1b: $(\psi \circ \varphi)' = \varphi' \circ \psi'$.

R1c: Let $\varphi : K \to E$, $\psi : K \to L$ be finite. For any $z \in \text{Spec}(E \otimes_K L)$, let $\bar{\varphi}_z : L \to E \otimes_K L/z$ and $\bar{\psi}_z : E \to E \otimes_K L/z$ be the induced morphisms:
$$ \psi_* \varphi' = \sum_{z \in \text{Spec}(E \otimes_K L)} \lg(E \otimes_K L_z) \cdot (\bar{\varphi}_z)' (\bar{\psi}_z)_* $$
where $\lg(A)$ is the length of an Artin ring $A$.

R2a: For all $\varphi : E \to L$, $x \in K_*^M(E)$, $\varphi_* \circ \gamma_x = \gamma_{\varphi*(x)} \circ \varphi_*$.

R2b: For $\varphi : E \to L$ finite and all $x \in K_*^M(E)$, $\varphi' \circ \gamma_{\varphi*(x)} = \gamma_x \circ \varphi'$.

R2c: For $\varphi : E \to L$ finite and all $y \in K_*^M(L)$, $\varphi' \circ \gamma_y \circ \varphi_* = \gamma_{\varphi!(y)}$.

R3a: Let $\varphi : E \to L$ be a morphism, $v$ and $w$ be valuations $L/k$ and $E/k$ respectively, and $e > 0$ an integer such that $v|_{E^\times} = e.w$. Let $\bar{\varphi} : \kappa(w) \to \kappa(v)$ be the induced morphism:
$$ \partial_v \circ \varphi_* = e.\bar{\varphi}_* \circ \partial_w. $$

R3b: Let $\varphi : E \to L$ be finite, and $v$ a valuation on $E/k$. Given a valuation $w$ on $L$ which extends $v$, we let $\bar{\varphi}_w : \kappa(w) \to \kappa(v)$ be the induced morphism on the respective residue fields:
$$ \partial_v \circ \varphi' = \sum_{w/v} \bar{\varphi}_w' \circ \partial_w. $$

R3c: Let $\varphi : E \to L$, $v$ a valuation on $L/k$ which is zero on $E^\times$: $\partial_v \circ \varphi_* = 0$.

R3d: Let $(E, v)$ be a valued function field over $k$ and $\pi$ uniformizer on $v$: $\partial_v \circ \gamma_{\{-\pi\}} \circ \varphi_* = \bar{\varphi}_*.$

R3e: Let $(E, v)$ be a valued function field over $k$, $u \in E^\times$ such that $v(u) = 0$: $\partial_v \circ \gamma_{\{u\}} = -\gamma_{\{\bar{u}\}} \circ \partial_v$.

According to [Ros96, Def. 1.1], a cycle premodule over $k$ is simply a (covariant) functor $M : \tilde{\mathcal{E}} \to \mathcal{A}b$.

**6.0.2. Higher order valuations.**— The preceding category admits a more elegant description due to Rost. We introduce some notation to state it.
Let $F$ be field with an arbitrary valuation (non necessarily discrete)

$$ v : F^\times \to \Gamma $$
where $\Gamma$ is a totally ordered abelian group. We refer the reader to [Vaq06] about arbitrary valuation rings.

As usual, we denote by:

$$ \mathcal{O}_v = \{x \in F \mid v(x) \geq 0\} $$

$$ \mathcal{M}_v = \{x \in F \mid v(x) > 0\} $$

the associated valuation ring, and the maximal ideal of the latter, where we extend $v$ as usual by setting $v(0) = \infty$, for $\Gamma_\infty = \Gamma \cup \{\infty\}$, obtained by adding a maximal element $\infty$. We also let $\kappa_v = \mathcal{O}_v/\mathcal{M}_v$ be the residue (class) field.

Recall that the rank of $v$ is the rank of the abelian group $\mathrm{Im}(v)$ (loc. cit. Definition after Th. 1.7), or equivalently the (Krull) dimension of the valuation ring $\mathcal{O}_v$ (loc. cit. Corollary after Th. 1.7). If $v$ has finite rank $r$, then $\Gamma \simeq \mathbb{Z}^r$ and $v$ can be defined as a composition of $r$ valuations $(v_1, \dots, v_r)$, $v_{i+1}$ being a valuation on the residue field $\kappa_{v_i}$ (see loc. cit. Rem. 1.7).

**6.0.3. Milnor K-theory of valuation rings.**– We consider an arbitrary valuation $v : F \to \Gamma$. Using the above notation, we have an exact sequence of abelian groups:

$$ (6.0.3.a) \qquad 1 \to (1 + \mathcal{M}_v) \to \mathcal{O}_v^\times \to \kappa_v^\times \to 1 $$

Following [Ros96, Rem. 1.6], we define a $\mathbb{Z}\mathbb{Z}$-graded ring associated to $v$ by the formula:

$$ K_*^M(v) = K_*^M(F)/(1 + \mathcal{M}_v). $$

Considering the obvious projection map $p : K_*^M(F) \to K_*^M(v)$, there exists a morphism $i : K_*^M(\kappa_v) \to K_*^M(v)$ of graded ring uniquely defined by the universal property$^{43}$:

$$ \begin{array}{ccc} K_*^M(\mathcal{O}_v) & \xrightarrow{p} & K_*^M(v) \\ \bigdownarrow & & \biguparrow \\ K_*^M(\kappa_v) & & \end{array} $$

where the vertical map is induced by the obvious surjection. The following proposition was stated in [Ros96, Rem. 1.10].

**Proposition 6.0.4.** Let $k$ be an arbitrary field, and $E, F$ be function fields over $k$. Given an integer $r \geq 0$, we consider the following set:

$$ \mathcal{I}_r(F, E) = \left\{ (v_*, L) \mid \begin{array}{l} v_1, \dots, v_r \text{ geom. valuations on } F/k, v = v_1 \circ \dots v_r \\ L = \kappa_x, x \in \mathrm{Spec}(E \otimes_k \kappa_v), E \xrightarrow{\varphi} L \xleftarrow{\psi} \kappa_v, \varphi \text{ finite} \end{array} \right\} $$

<footnote>$^{43}$This follows from the exact sequence (6.0.3.a) by induction on the degree of symbols in Milnor K-theory</footnote>
Then the following map:

$$ \bigoplus_{r \ge 0, (v_*, L) \in \mathcal{I}_r(F, E)} K_*^M(L) \otimes_{K_*^M(\kappa_v)} K_*^M(v) \to \operatorname{Hom}_{\tilde{\mathcal{E}}_k}((F, 0), (L, *) ) \\ \sigma \otimes \bar{\tau} \mapsto \varphi'^{!} \circ \gamma_{\sigma} \circ \psi_* \circ \partial_{v_1}, \circ \dots \circ \partial_{v_r} \circ \gamma_{\tau} $$

where $\tau \in K_*^M(\mathcal{O}_v)$ is an arbitrary element such that $p(\tau) = \bar{\tau}$, is well-defined and induces an isomorphism of graded abelian groups.

Obviously, given that the functor $(E, 0) \mapsto (E, 1)$ is an auto-equivalence of the category $\tilde{\mathcal{E}}_k$, this completely describes the morphism in the latter category. The proof follows by applying the relations (R*) to put morphisms in $\tilde{\mathcal{E}}_k$ in the above normalized form. A key point is Lemma 1.9 of [Ros96].

## 7. RECALL ON BOUSFIELD LOCALIZATIONS

Recall the following definition due to Bousfield (originally phrased in terms of model category [Bou75]):

**Definition 7.0.1.** Let $R$ be a ring. A map $f: X \to Y$ in the homotopy category $\mathcal{H}$, will be called an $R$-equivalence if the induced map $f_*: H_*(X, R) \to H_*(Y, R)$ is an isomorphism. An object $T$ of $\mathcal{H}$ is called $R$-local if for any $R$-equivalence $f: X \to Y$, the induced map $f_*: \mathrm{Map}(Y, P) \to \mathrm{Map}(X, P)$ is an equivalence.

We let $\mathcal{H}_R$ be the localization of the $\infty$-category $\mathcal{H}$ with respect to $R$-equivalences, and call it the *Bousfield $R$-local homotopy category*. This is a presentable $\infty$-category, that can be identified with the full $\infty$-subcategory of $\mathcal{H}$ spanned by $R$-local objects (see [Lur09]). Moreover, there exists an adjunction of $\infty$-categories

$$ \pi_R : \mathcal{H} \stackrel{\sim}{\to} \mathcal{H}_R : L'_R $$

It is customary to let $L_R = L'_R \circ \pi_R$ and call this the $R$-localization functor. For any space $X$, there exists a canonical map

$$ X \to L_R(X) $$

which is an $R$-equivalence and such that $L_R(X)$ is $R$-local. This map is universal.

**Example 7.0.2.** (1) When $R = \mathbb{Z}$, $L_{\mathbb{Z}}(X) = X^+$ for any space $X$ where $X^+$ denotes Quillen's plus construction. This actually follows from the construction of Quillen: see [Hoy].

(2) When $R = \mathbb{Q}$, and $X$ is simply connected $L_{\mathbb{Q}}(X)$ can be computed as either the Sullivan or the Quillen model of rationalization of simply connected spaces. See [Iva22].

## REFERENCES

[AM69] M. Artin and B. Mazur, *Etale homotopy*, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969.
[Ayo07] J. Ayoub, *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique*. I, Astérisque (2007), no. 314, x+466.

[Ayo14] —, *La réalisation étale et les opérations de Grothendieck*, Ann. Sci. Éc. Norm. Supér. (4) **47** (2014), no. 1, 1–145.

[Bei02] A. Beilinson, Remarks on $n$-motives and correspondences at the generic point, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 35–46.

[BG02] J. I. Burgos Gil, *The regulators of Beilinson and Borel*, CRM Monograph Series, vol. 15, American Mathematical Society, Providence, RI, 2002.

[BHH17] I. Barnea, Y. Harpaz, and G. Horel, *Pro-categories in homotopy theory*, Algebr. Geom. Topol. **17** (2017), no. 1, 567–643.

[Bi84] A. A. Beilinson, Higher regulators and values of L-functions, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238.

[Blo86] S. Bloch, *Algebraic cycles and higher K-theory*, Adv. in Math. **61** (1986), no. 3, 267–304.

[Blo94] —, *The moving lemma for higher Chow groups*, J. Algebraic Geom. **3** (1994), no. 3, 537–568.

[Bor74] A. Borel, *Stable real cohomology of arithmetic groups*, Ann. Sci. École Norm. Sup. (4) **7** (1974), 235–272 (1975).

[Bou75] A. K. Bousfield, *The localization of spaces with respect to homology*, Topology **14** (1975), 133–150.

[BT73] H. Bass and J. Tate, *The Milnor ring of a global field*, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 342, Springer, Berlin-New York, 1973, pp. 349–446.

[BVK16] L. Barbieri-Viale and B. Kahn, *On the derived category of 1-motives*, Astérisque (2016), no. 381, xi+254.

[BW00] A. Borel and N. Wallach, *Continuous cohomology, discrete subgroups, and representations of reductive groups*, second ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000.

[CD15] D.-C. Cisinski and F. Déglise, *Integral mixed motives in equal characteristics*, vol. Extra volume: Alexander S. Merkurjev’s sixtieth birthday, 2015.

[CD16] —, *Étale motives*, Compos. Math. **152** (2016), no. 3, 556–666.

[CD19] —, *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer, Cham, 2019.

[Cis19] D.-C. Cisinski, *Higher categories and homotopical algebra*, Cambridge Studies in Advanced Mathematics, vol. 180, Cambridge University Press, Cambridge, 2019.

[Dég02] F. Déglise, *Modules homotopiques avec transferts et motifs génériques*, Ph.D. thesis, Univ. Paris VII, 2002, http://deglise.perso.math.cnrs.fr/docs/avant2003/these_deglise.pdf.

[Dég07] —, *Finite correspondences and transfers over a regular base*, Algebraic cycles and motives. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 343, Cambridge Univ. Press, Cambridge, 2007, pp. 138–205.

[Dég08a] —, *Around the Gysin triangle II*, Doc. Math. **13** (2008), 613–675.

[Dég08b] —, *Motifs génériques*, Rend. Semin. Mat. Univ. Padova **119** (2008), 173–244.

[Dég11] —, *Modules homotopiques*, Doc. Math. **16** (2011), 411–455.

[Dég12] —, *Coniveau filtration and motives*, Regulators, Contemporary Mathematics, vol. 571, 2012, pp. 51–76.
[Del79] P. Deligne, *Valeurs de fonctions L et périodes d'intégrales*, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Amer. Math. Soc., Providence, RI, 1979, With an appendix by N. Koblitz and A. Ogus, pp. 313–346.

[DP24] F. Déglise and R. Pawar, *Coniveau spectral sequences in unstable homotopy theory*, in preparation, 2024.

[Gal23] M. Gallauer, *∞-categories: a first course*, https://homepages.warwick.ac.uk/staff/Martin.Gallauer/teaching/23icats/icats.pdf, 2023.

[EGA4] A. Grothendieck and J. Dieudonné, *Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV*, Publ. Math. IHES 20, 24, 28, 32 (1964-1967).

[Gei98] T. Geisser, *Tate's conjecture, algebraic cycles and rational K-theory in characteristic p, K-Theory* 13 (1998), no. 2, 109–122.

[GLVdB21] F. Genovese, W. Lowen, and M. Van den Bergh, *t-structures and twisted complexes on derived injectives*, Adv. Math. 387 (2021), Paper No. 107826, 70.

[God] P. Godfard, *K-théorie algébrique*, Master's thesis, https://github.com/PierreGodfard/memoire/blob/master/memoire.pdf.

[Goe96] P. G. Goerss, *The homology of homotopy inverse limits*, J. Pure Appl. Algebra 111 (1996), no. 1-3, 83–122.

[Gra76] Daniel Grayson, *Higher algebraic K-theory. II (after Daniel Quillen)*, Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Lecture Notes in Math., Vol. 551, Springer, Berlin-New York, 1976, pp. 217–240.

[Har77] G. Harder, *Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern*, In-vent. Math. 42 (1977), 135–175.

[Hoy] F. Hoyois, *On quillen's plus construction*, https://hoyois.app.uni-regensburg.de/papers/acyclic.pdf.

[Iva22] Sergei O. Ivanov, *An overview of rationalization theories of non-simply connected spaces and non-nilpotent groups*, Acta Math. Sin. (Engl. Ser.) 38 (2022), no. 10, 1705–1721.

[Kah05] B. Kahn, *Algebraic K-theory, algebraic cycles and arithmetic geometry*, Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 351–428.

[KMP07] B. Kahn, J. P. Murre, and C. Pedrini, *On the transcendental part of the motive of a surface*, Algebraic cycles and motives. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 344, Cambridge Univ. Press, Cambridge, 2007, pp. 143–202.

[KS17] B. Kahn and R. Sujatha, *Birational motives, II: Triangulated birational motives*, Int. Math. Res. Not. IMRN (2017), no. 22, 6778–6831.

[KY13] B. Kahn and T. Yamazaki, *Voevodsky's motives and Weil reciprocity*, Duke Math. J. 162 (2013), no. 14, 2751–2796.

[Lev93] M. Levine, *Tate motives and the vanishing conjectures for algebraic K-theory*, Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167–188.

[Lev94] __________, *Bloch's higher Chow groups revisited, K-theory* (Strasbourg, 1992), no. 226, 1994, pp. 10, 235–320.

[Lur09] J. Lurie, *Higher topos theory*, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009.

[Lur17] __________, *Higher algebra*, available at the author's homepage, http://math.ias.lurie, version 09/2017, 2017.
[Mil56] J. Milnor, *Construction of universal bundles*. II, Ann. of Math. (2) **63** (1956), 430–436.

[MNP13] J. P. Murre, J. Nagel, and C. A. M. Peters, *Lectures on the theory of pure motives*, University Lecture Series, vol. 61, American Mathematical Society, Providence, RI, 2013.

[Mur90] J. P. Murre, *On the motive of an algebraic surface*, J. Reine Angew. Math. **409** (1990), 190–204.

[MVW06] C. Mazza, V. Voevodsky, and C. Weibel, *Lecture notes on motivic cohomology*, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI, 2006.

[Org04] F. Orgogozo, *Isomotifs de dimension inférieure ou égale à un*, Manuscripta Math. **115** (2004), no. 3, 339–360.

[Par21] D. Park, *Construction of triangulated categories of motives using the localization property*, Kodai Math. J. **44** (2021), no. 2, 195–264.

[Qui72] D. Quillen, *On the cohomology and K-theory of the general linear groups over a finite field*, Ann. of Math. (2) **96** (1972), 552–586.

[Qui73a] ——, *Finite generation of the groups $K_i$ of rings of algebraic integers*, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. Vol. 341, Springer, Berlin-New York, 1973, pp. 179–198.

[Qui73b] ——, *Higher algebraic K-theory. I*, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. Vol. 341, Springer, Berlin-New York, 1973, pp. 85–147.

[Rap88] M. Rapoport, *Comparison of the regulators of Beilinson and of Borel*, Beilinson’s conjectures on special values of L-functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988, pp. 169–192.

[Rob15] M. Robalo, *K-theory and the bridge from motives to noncommutative motives*, Adv. Math. **269** (2015), 399–550.

[Ros96] M. Rost, *Chow groups with coefficients*, Doc. Math. **1** (1996), No. 16, 319–393.

[Sat56] I. Satake, *On a generalization of the notion of manifold*, Proc. Nat. Acad. Sci. U.S.A. **42** (1956), 359–363.

[Som90] M. Somekawa, *On Milnor K-groups attached to semi-abelian varieties*, K-Theory **4** (1990), no. 2, 105–119.

[Sou85] C. Soulé, *Opérations en K-théorie algébrique*, Canad. J. Math. **37** (1985), no. 3, 488–550.

[Spi18] M. Spitzweck, *A commutative $P^1$-spectrum representing motivic cohomology over Dedekind domains*, Mém. Soc. Math. Fr. (N.S.) (2018), no. 157, 110.

[SS03] M. Spießand T. Szamuely, *On the Albanese map for smooth quasi-projective varieties*, Math. Ann. **325** (2003), no. 1, 1–17.

[SV00] A. Suslin and V. Voevodsky, *Bloch-Kato conjecture and motivic cohomology with finite coefficients*, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 117–189.

[Swi02] R. M. Switzer, *Algebraic topology—homotopy and homology*, Classics in Mathematics, Springer-Verlag, Berlin, 2002, Reprint of the 1975 original [Springer, New York; MR0385836 (52 #6695)].

[Thu22] W. P. Thurston, *The geometry and topology of three-manifolds*. Vol. IV, American Mathematical Society, Providence, RI, [2022] ©2022, Edited and with a preface by Steven P. Kerckhoff and a chapter by J. W. Milnor.
[Vaq06] M. Vaquié, *Valuations and local uniformization*, Singularity theory and its applications, Adv. Stud. Pure Math., vol. 43, Math. Soc. Japan, Tokyo, 2006, pp. 477–527.

[Voe92] V. Voevodsky, *Homology of schemes and covariant motives*, Ph.D. thesis, Harvard University, 1992, http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9228294.

[Voe10] ——, *Cancellation theorem*, Doc. Math. (2010), no. Extra vol.: Andrei A. Suslin sixtieth birthday, 671–685.

[Voe11] ——, *On motivic cohomology with Z/l-coefficients*, Ann. of Math. (2) **174** (2011), no. 1, 401–438.

[VSF00] V. Voevodsky, A. Suslin, and E. M. Friedlander, *Cycles, transfers and motivic homology theories*, Annals of Mathematics Studies, vol. 143, Princeton Univ. Press, 2000.