Higher Tate traces of Chow motives

TL;DR

This paper classifies Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups using a new invariant called the Tate trace, and generalizes motivic equivalence through higher Tits p-indexes.

Contribution

It introduces the Tate trace as a new motivic invariant and extends the classification of motives and motivic equivalence to broader classes of algebraic groups and varieties.

Findings

Complete classification of Chow motives for p-inner semi-simple groups.

Introduction of the Tate trace as a maximal pure Tate summand.

Generalization of motivic equivalence via higher Tits p-indexes.

Abstract

We establish the complete classification of Chow motives of projective homogeneous varieties for $p$-inner semi-simple algebraic groups, with coefficients in $\mathbb{Z}/p\mathbb{Z}$. Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov-Gille-Merkurjev decompositions and their interpretation through Bialynicki-Birula-Hesselink-Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits $p$-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

Equations

$$\operatorname{End}(M\{i\})=\operatorname{End}(M)=\pi\operatorname{End}\bigl{(}M(X)\bigr{)}\pi\subset\operatorname{End}\bigl{(}M(X)\bigr{)}.$$

$$\alpha \in {\mathrm{Ch}}_k(X)\simeq {\mathrm{Ch}}^{d-k}(\mathrm{Spec}F\times X)\text{mbox}and\beta \in {\mathrm{Ch}}_{d-k}(X)\simeq {\mathrm{Ch}}^k(X\times \mathrm{Spec}F).$$

$$\begin{gathered} \mathrm{rk}(M+N)=\mathrm{rk}(M)+\mathrm{rk}(N),h(M+N)=\min \{h(M),h(N)\}, \\ \mathrm{rk}(M\otimes N)=\mathrm{rk}(M)\mathrm{rk}(N)\text{mbox}andh(M\otimes N)=h(M)+h(N). \end{gathered}$$

$$\begin{gathered} M(Q)\simeq \mathbb{Z}/2\mathbb{Z}\{0\}+\cdots +\mathbb{Z}/2\mathbb{Z}\{i_0-1\}+M(Q_{\mathrm{an}})\{i_0\} \\ +\mathbb{Z}/2\mathbb{Z}\{2m-1-i_0\}+\cdots +\mathbb{Z}/2\mathbb{Z}\{2m-2\}. \end{gathered}$$

$$\operatorname{Tr}(M(Q))\simeq\sum_{k=0}^{i_{0}-1}\bigl{(}{\mathbb{Z}}/2{\mathbb{Z}}\{k\}+{\mathbb{Z}}/2{\mathbb{Z}}\{2m-2-k\}\bigr{)}.$$

$$\operatorname{Tr}(M(X_{\sigma}))\simeq\left\{\begin{array}[]{ll}0&\mbox{ if $A$ is non split},\\ \sum_{k=0}^{i_{w}(\sigma)-1}\bigl{(}{\mathbb{Z}}/2{\mathbb{Z}}\{k\}+{\mathbb{Z}}/2{\mathbb{Z}}\{2m-2-k\}\bigr{)}&\mbox{ if $A$ is split.}\end{array}\right.$$

$$M\otimes N\simeq\sum_{1\leq k\leq r,\ 1\leq\ell\leq s}\bigl{(}U_{X_{k}}\otimes U_{Y_{\ell}}\bigr{)}\{i_{k}+j_{\ell}\}.$$

$$X_1{⪰}_{\Lambda }{Y}_{{\ell }_1}{⪰}_{\Lambda }{X}_{k_1}{⪰}_{\Lambda }{Y}_{{\ell }_2}{⪰}_{\Lambda }\cdots {⪰}_{\Lambda }{X}_{k_n},$$

$$M(X_D\times X_{D^{\prime }})\simeq \sum _{i=0}^{n-1}M(X_D)\{i\}\text{mbox}in{\mathrm{Mot}}_F(\Lambda ).$$

$$coef{f}_{\overline{\Lambda }/\Lambda }:{\mathrm{Mot}}_F\Lambda ⟶{\mathrm{Mot}}_F\overline{\Lambda }\text{mbox}andcoef{f}_{\overline{\Lambda }/(\mathbb{Z}/p\mathbb{Z})}:{\mathrm{Mot}}_F(\mathbb{Z}/p\mathbb{Z})⟶{\mathrm{Mot}}_F(\overline{\Lambda }).$$

$$coef{f}_{\overline{\Lambda }/\Lambda }:{\mathrm{Mot}}_F^{\mathrm{I}}(\Lambda )⟶{\mathrm{Mot}}_F^{\mathrm{I}}(\overline{\Lambda })\text{mbox}andcoef{f}_{\overline{\Lambda }/(\mathbb{Z}/p\mathbb{Z})}:{\mathrm{Mot}}_F^{\mathrm{I}}(\mathbb{Z}/p\mathbb{Z})⟶{\mathrm{Mot}}_F^{\mathrm{I}}(\overline{\Lambda })$$

$$f:\operatorname{End}(P)\longrightarrow\operatorname{End}(P)\otimes_{\Lambda}\overline{\Lambda}=\operatorname{End}\bigl{(}coe\!f\!\!f_{\overline{\Lambda}/\Lambda}(P)\bigr{)}.$$

$$0⟶\mathcal{N}\otimes K⟶\mathrm{End}(coef{f}_{K/{\mathbb{F}}_p}(U_X))\stackrel{{\mathrm{mult}}_K}{⟶}K⟶0$$

$$P\bigl{(}M(X_{\Theta,G}),t\bigr{)}=\sum_{i\geq 0}a_{i}t^{i},$$

$$\phi :\Delta (G)⟶\Delta (G^{\prime }),$$

$$M(X_{\Theta ,G_E})\simeq \underset{\delta }{⨁}M(Z_{\delta ,G_{E,\Theta }})\{l(\delta )\}$$

$$M(X_{\Theta ,G_E})\simeq \underset{\delta }{⨁}M(Z_{\delta ,G_{E,\Theta }})\{l(\delta )\}\simeq \underset{\delta }{⨁}M(Z_{\delta ,G_{E,\phi (\Theta )}^{\prime }})\{l(\delta )\}\simeq M(X_{\phi (\Theta ),G_E^{\prime }}).$$

$$\sum_{k=0}^{i-1}\,\biggl{(}{\mathbb{Z}}/2{\mathbb{Z}}\{k\}+{\mathbb{Z}}/2{\mathbb{Z}}\{2m-2-k\}\biggr{)},$$

$$F\hookrightarrow F(X_{k_0}^0)\hookrightarrow F(X_{k_1}^1)\hookrightarrow ...\hookrightarrow F(X_{k_s}^s)$$

$$K_p(X_{k_0}^0)$$

Full text

1. Introduction

Chow motives provide a powerful tool to study the geometry of varieties without rational points, including projective homogeneous varieties, such as quadrics and Severi-Brauer varieties. In this article we introduce a new family of motivic invariants, the Tate traces, which measures the isotropy level of motives. More precisely, the Tate Trace of a motive is a pure Tate direct summand of maximal rank. Through the Tate traces, we obtain a complete classification for the motives of projectives homogeneous varieties for $p$-inner semisimple algebraic groups, with coefficients in a finite local ring with residue characteristic $p$.

The theory of motives of projective homogeneous varieties was brought to the forefront of modern algebra in the nineties, with Voevodsky’s proof of Milnor’s conjecture [39] which relied on the Rost motive. This inspired an extensive study of the motives of projective quadrics by Vishik, Karpenko, Merkurjev and others, leading to numerous breakthroughs for classical problems of the algebraic theory of quadratic forms [24, 25, 36, 38]. Among these results is the classification of motives of projective quadrics [36, Thm. 4.18] (see [13, Thm. 93.1] for the result in arbitrary characteristic).

Theorem 1.1** (Vishik’s criterion).**

Let $Q$ and $Q^{\prime}$ be smooth projective quadrics, given by quadratic forms $q$ and $q^{\prime}$ of the same dimension over a field $F$.

The motives of $Q$ and $Q^{\prime}$ (with coefficients in ${\mathbb{Z}}$ or ${\mathbb{Z}}/2{\mathbb{Z}}$) are isomorphic if and only if for all field extensions $E/F$, the Witt indexes of $q_{E}$ and $q^{\prime}_{E}$ are equal.

Inspired by one of the main tools used by Vishik, the Čech simplicial schemes, Karpenko introduced in [22] their avatars for projective homogeneous varieties, the upper motives. Upper motives are easier to handle than motives of varieties, yet they encode deep geometric information, such as their canonical $p$-dimension [21], and were used by Karpenko to classify motives of Severi-Brauer varieties (see [9, 26] for other applications). The classification of motives for more general flag varieties of inner type $A_{n}$ is given by [9, Theorème 19].

Criteria for motivic isomorphisms were achieved in several other cases such as involution varieties [12], Borel varieties [33] and for some exceptional projective homogeneous varieties [14, 31, 32]. However, all the above cases deal with projective homogeneous varieties of the same type, and with respect to semisimple groups with the same Dynkin diagram. Theorem 3.15 extends these results with finite coefficients for arbitrary projective homogeneous varieties for $p$-inner semisimple algebraic groups. Moreover, its proof does not rely on the classification of semisimple groups. The class of $p$-inner semisimple groups is quite large: all absolutely simple groups are $p$-inner for some prime $p$, except the trialitarian groups of type ${}^{6}\!D_{4}$. Products of $p$-inner semisimple groups are $p$-inner, as well as Weyl restrictions of such, with respect to separable extensions of degree a power of $p$. Note that considering direct summands of motives of projective quadrics in Voevodsky’s category of motives $\mathrm{DM}(F;\mathbb{F}_{2})$, Tate traces corresponds to Vishik’s isotropic motivic realization [37] and are given by Bachmann’s generalized geometric fixed point functor [3, 4].

We actually show more: from Theorem 3.15, higher Tate traces characterize isomorphism classes of objects in the additive category $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ generated by twists of upper motives of geometrically split irreducible varieties satisfying the nilpotence principle (Definition 3.11). The result thus applies to arbitrary direct summands of projective homogeneous varieties for $p$-inner semisimple groups, our main focus in this article, but actually holds for a broader class of motives, containing Artin-Tate motives with respect to separable $p$-primary field extensions, motives of projective pseudo-homogeneous varieties and much more. Our results therefore highlight the question of determining the varieties suited for the theory of upper motives, that is, the varieties whose motive is given by direct sum of initial motives.

We derive from Theorem 3.15 the following applications:

1. (1)

computing the Tate trace of tensor products of motives, we prove a cancellation property in the category $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ (Corollary 3.22); 2. (2)

as Tate traces of projective homogeneous varieties for $p$-inner groups are determined by the Tits indexes of the underlying semisimple group, we show that their motives are determined by cohomological invariants in numerous situations (Corollaries 4.5 and 4.8); 3. (3)

considering $G$, $G^{\prime}$ two $p$-inner semisimple groups of the same type, we relate motivic isomorphisms of projective homogeneous varieties for $G$ and $G^{\prime}$ and their Tits indexes over field extensions (Theorem 4.12). This generalizes the main result of [9], expressing motivic equivalence of semisimple algebraic groups through their higher Tits $p$-indexes. The proof of Theorem 4.12 relies on the theory of upper motives, Chernousov-Gille-Merkurjev motivic decompositions [7] and their interpretation through Bialynicki-Birula-Hesselink-Iversen filtrations due to Brosnan [6]. 4. (4)

we introduce a notion of motivic splitting pattern and motivic splitting towers (Definition 5.6), which comprises the classical splitting pattern and Knebusch’s tower of quadratic forms [28].

Acknowledgments. We are grateful to Skip Garibaldi, Stefan Gille, Nikita Karpenko, Srimathy Srinivasan and Alexander Vishik for their comments on a preliminary version of this paper, and insightful discussions.

1.1. Notation

Our main reference for the basics of Chow groups with coefficients $\operatorname{Ch}_{\ast}(\cdot)$ and Chow motives is [13]. The base field is denoted by $F$. Varieties over $F$, that is separated scheme of finite type over $\operatorname{Spec}F$, are always assumed to be smooth and projective. We denote by $\operatorname{Mot}_{F}(\Lambda)$ the category of Chow motives with coefficients in a commutative ring $\Lambda$. Except in a few places where we consider motives with coefficients in ${\mathbb{Z}}$, we will assume that $\Lambda$ is a finite local ring with residue field of characteristic $p>0$. In particular, $\Lambda$ is connected.

2. Preliminary results

A pure Tate motive is a direct sum of Tate motives, that is $M=\Sigma_{i\in I}\Lambda\{k_{i}\}$ for a finite set of integers $\{k_{i},\ i\in I\}$, where $\Lambda\{k\}$ stands for $M(\operatorname{Spec}F)\{k\}$. A motive $M\in\operatorname{Mot}_{F}(\Lambda)$ is called geometrically split if there exists a field extension $E/F$ such that $M_{E}$ is a pure Tate motive. A variety $X$ over $F$ is called geomerically split if $M(X)$ is geometrically split. An irreducible variety $X$ over $F$ is called generically split if $M(X)_{F(X)}$ is a pure Tate motive. For instance, projective homogeneous varieties under the action of an affine semisimple algebraic group $G$ are geometrically split by [30]; see also [13, 66.5, 66.7] for split examples. If in addition $G$ has inner type, the variety of Borel subgroups of $G$ is generically split.

Let $M\in\operatorname{Mot}_{F}(\Lambda)$ be a motive. A decomposition of $M$ as a direct sum of motives is called complete if all summands are indecomposable motives. We say that the Krull-Schmidt property holds for $M$ if any decomposition of $M$ can be refined into a complete one, and $M$ admits a unique complete decomposition, up to permutation and isomorphism of the summands. By [36], the Krull-Schmidt property holds for $M(X)\in\operatorname{Mot}_{F}({\mathbb{Z}})$ if $X$ is a projective quadric. Nevertheless, it does not hold in general for motives with coefficients in ${\mathbb{Z}}$, even for projective homogeneous varieties. See [8, Ex. 32] for an explicit example using Severi-Brauer varieties. If we now consider Chow motives with finite coefficients, the Krull-Schmidt property holds for a large class of varieties, as we proceed to recall.

We say that a variety $X$ satisfies the nilpotence principle if for any field extension $E/F$ and any ring of coefficients $\Lambda$, an endomorphism $f$ of the motive $M(X)$ which maps to zero in $\operatorname{End}(M(X_{E}))$ for some field extension $E/F$ is nilpotent, see [34]. For example, projective homogeneous varieties under the action of an affine semisimple algebraic group satisfy the nilpotence principle [7, Thm 8.2].

By [22, §2.I], if $X$ is geometrically split and satisfies the nilpotence principle, then the Krull-Schmidt property holds for $M(X)$. More precisely, we have the following:

Proposition 2.1**.**

[22*, Cor. 2.6]**

Recall $\operatorname{Mot}_{F}(\Lambda)$ is the category of Chow motives with coefficients in a finite local ring $\Lambda$. Let $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ be the full additive subcategory generated by direct summands of shifts of motives of geometrically split varieties satisfying the nilpotence principle. The Krull-Schmidt property holds for all objects in $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$.*

Proof.

We reproduce the argument for the reader’s convenience. By definition, all objects of $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ are finite sums of indecomposable ones, and an indecomposable object is isomorphic to $M\{i\}$, for some $i\in{\mathbb{Z}}$, and some indecomposable summand $M$ of $M(X)$, where $X$ is a geometrically split variety satisfying the nilpotence principle. Therefore, by [2, Thm 3.6 of Chapter I], we need to prove that the endomorphism rings of indecomposable objects are local rings. So, consider an indecomposable motive $M\{i\}$ as above and let $\pi\in\operatorname{End}(M(X))$ be the corresponding projector. We have

[TABLE]

In particular, by [22, Cor. 2.2], for any $f\in\operatorname{End}(M\{i\})$, there exists a non-zero integer $n$ such that $f^{n}$ is a projector. Since $M\{i\}$ is indecomposable, it follows that $f^{n}\in\{0,1\}$. Therefore, all non-invertible elements of $\operatorname{End}(M\{i\})$ are nilpotent, hence it is a local ring. ∎

Assume now that $X$ and $Y$ are two irreducible varieties of respective dimensions $d$ and $d^{\prime}$. By definition, a morphism $M(X)\rightarrow M(Y)$ in the category $\operatorname{Mot}_{F}(\Lambda)$ is a correspondence $\alpha:X\rightsquigarrow Y$, that is a cycle $\alpha\in\operatorname{Ch}_{d}(X\times Y)$. The projection $p$ on the first component induces a morphim $p_{*}:\,\operatorname{Ch}_{d}(X\times Y)\mapsto\operatorname{Ch}_{d}(X)\simeq\Lambda\cdot[X]$. The multiplicity of $\alpha$ is the element $\operatorname{mult}(\alpha)\in\Lambda$ defined by $p_{*}(\alpha)=\operatorname{mult}(\alpha)\cdot[X]$.

A composition of correspondences $\alpha:X\rightsquigarrow Y$ and $\beta:Y\rightsquigarrow Z$ has multiplicity $\operatorname{mult}(\beta\circ\alpha)=\operatorname{mult}(\beta)\operatorname{mult}(\alpha)$, see [18, Cor. 1.7]. In particular, the multiplicity of a projector $\pi:X\rightsquigarrow X$ is an idempotent of $\Lambda$, hence [math] or $1$ since $\Lambda$ is connected. As explained in [22, §2.II], if the diagonal $\Delta_{X}$ decomposes as a sum of pairwise orthogonal projectors, one has multiplicity $1$ and all others have multiplicity [math]. It follows that in a given direct sum decomposition of $M(X)$, there is a unique summand defined by a projector of mutiplicity $1$, which is called upper. If in addition $X$ is geometrically split and satisfies the nilpotence principle, then $M(X)$ satisfies the Krull-Schmidt property, so that $M(X)$ contains a unique (up to isomorphism) indecomposable upper summand, called the upper motive of $X$ and denoted by $U_{X}$.

The variety $X$ is called isotropic with coefficients in $\Lambda$ if $X$ admits a [math]-cycle of degree $1$ (or, equivalently, a [math]-cycle with invertible degree $\delta\in\Lambda^{\times}$). Such a cycle $\alpha\in\operatorname{Ch}_{0}(X)$ may also be viewed as a correspondence $\operatorname{Spec}F\rightsquigarrow X$ of multiplicity $1$. We get the following characterisation of isotropy:

Lemma 2.2**.**

Let $X$ be an irreducible geometrically split variety. The following are equivalent :

1. (i)

$X$* is isotropic with coefficients in $\Lambda$;* 2. (ii)

$M(X)$* contains a summand isomorphic to $\Lambda\{0\}$;* 3. (iii)

$M(X)$* contains a summand isomorphic to $\Lambda\{k\}$ for some $k\in{\mathbb{Z}}$.*

Proof.

Assume $X$ is isotropic with coefficients in $\Lambda$ and let $\alpha\in\operatorname{Ch}_{0}(X)$ be a cycle with $\deg(\alpha)=1$. Consider the product $[X]\times\alpha\in\operatorname{Ch}_{d}(X\times X)$. Viewed as a correspondence $X\rightsquigarrow X$, one may check it is idempotent, so it defines a summand $(X,[X]\times\alpha)$ of $M(X)$. Moreover, its multiplicity is $\operatorname{mult}([X]\times\alpha)=\deg(\alpha)=1$, and the correspondences $\alpha:\operatorname{Spec}F\rightsquigarrow X$ and $[X]:X\rightsquigarrow\operatorname{Spec}F$ induce isomorphisms between $(X,[X]\times\alpha)$ and $\Lambda\{0\}$, and we get (ii). The implication $(ii)\,\Rightarrow\,(iii)$ is obvious, and it remains to prove $(iii)\,\Rightarrow\,(i)$. Assume $M(X)$ contains a summand $(X,\pi)$ isomorphic to $\Lambda\{k\}$, and let $\alpha:\,\Lambda\{k\}\rightsquigarrow X$ and $\beta:X\rightsquigarrow\Lambda\{k\}$ be correspondences inducing an isomorphism between $(X,\pi)$ and $\Lambda\{k\}$. We have

[TABLE]

Moreover, the intersection product of their pull-backs to $\operatorname{Spec}F\times X\times\operatorname{Spec}F$, along the projections $p_{12}$ and $p_{23}$, respectively, is a cycle $\gamma\in\operatorname{Ch}^{d}(X)\simeq\operatorname{Ch}_{0}(X)$. Since $\beta\circ\alpha=1\in\operatorname{Ch}_{0}(\operatorname{Spec}F)\simeq\Lambda$, the [math]-cycle $\gamma$ has degree $1$ and this proves that $X$ is isotropic with coefficients in $\Lambda$. ∎

Let $X$ and $Y$ be two irreducible varieties. There is a correspondence $\alpha:X\rightsquigarrow Y$ of multiplicity $1$ if and only if $Y_{F(X)}$ is isotropic with coefficients in $\Lambda$, see for instance [13, Cor. 57.11, Lem. 75.1]. When these conditions are satisfied, we say that $X$ dominates $Y$ with coefficients in $\Lambda$, and we write $X\succeq_{\Lambda}Y$, see [9, Def. 4]. The domination relation is a preorder and the associated equivalence relation is called equivalence with coefficients in $\Lambda$ and denoted by $\approx_{\Lambda}$.

Example 2.3*.*

Assume $X$ and $Y$ are quadrics. By Springer’s theorem [13, Cor. 71.3], $Y_{F(X)}$ is isotropic with coefficients in ${\mathbb{Z}}/2{\mathbb{Z}}$ if and only if there is a rational map $X\dashrightarrow Y$. Hence, for quadrics, the domination relation with coefficients in ${\mathbb{Z}}/2{\mathbb{Z}}$ coincides with the relation considered by Bruno Kahn in [17, §I.7], and the corresponding equivalence relation is the stable birational equivalence. Therefore, we may view the equivalence relation with coefficients in $\Lambda$ as a motivic version of stable birational equivalence.

By definition, $X\approx_{\Lambda}Y$ if and only if there are correspondences $X\rightsquigarrow Y$ and $Y\rightsquigarrow X$ of multiplicity $1$, or equivalently, $X_{F(Y)}$ and $Y_{F(X)}$ are isotropic with coefficients in $\Lambda$. For the varieties we are interested in, the following result, which is due to Karpenko, provides another characterization in terms of upper motives:

Lemma 2.4**.**

[22*, Cor. 2.15]**

Let $X$ and $Y$ be two irreducible geometrically split $F$-varieties satisfying the nilpotence principle. The varieties $X$ and $Y$ are equivalent with coefficients in $\Lambda$ if and only if the upper motives $U_{X}$ and $U_{Y}$ are isomorphic in $\operatorname{Mot}_{F}(\Lambda)$.*

3. Tate trace and initial motives

Recall the ring of coefficients $\Lambda$ is a finite local ring. Let $M$ be a motive in $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$. By definition, $M$ is geometrically split; a field $E/F$ such that $M_{E}$ is a pure Tate motive is called a splitting field of $M$. The decomposition as a sum of Tate motives of $M_{E}\in\mathrm{Mot}^{\mathrm{KS}}_{E}({\Lambda})$ does not depend on the choice of a splitting field $E$. Hence, we may give the following definition:

Definition 3.1**.**

1. (1)

The rank $\operatorname{rk}(M)$ of a motive $M\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ is the rank of $\operatorname{Ch}_{*}(M_{E})$ as a $\Lambda$-module, where $E/F$ is a splitting field for $M$. 2. (2)

The hook $h(M)$ of a non-trivial motive $M\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ is the smallest integer $i\in{\mathbb{Z}}$ such that $\operatorname{Ch}_{i}(M_{E})$ is non-zero, where $E/F$ is a splitting field for $M$.

Example 3.2*.*

Assume $T$ is a non trivial pure Tate motive. The rank of $T$ is the number of Tate summands in a complete decomposition of $T$, while its hook $h(T)$ is the smallest integer $i$ such that $\Lambda\{i\}$ is a direct summand of $T$. This follows from the fact that $\operatorname{Hom}_{\operatorname{Mot}_{F}(\Lambda)}(\Lambda\{i\},M)=\operatorname{Ch}_{i}(M)$ and $\operatorname{Hom}_{\operatorname{Mot}_{F}(\Lambda)}(\Lambda\{i\},\Lambda\{k\})$ is [math] is $i\not=k$ and $\Lambda$ otherwise.

Example 3.3*.*

Let $X$ be an irreducible geometrically split variety satisfying the nilpotence principle. The hook of the motive of $X$ is h\bigl{(}M(X)\bigr{)}=0.

Moreover, by Lemma 2.2, the upper summands of $M(X)$ are precisely the summands $N$ such that $h(N)=0$. In particular, we have $h(U_{X})=0$.

Example 3.4*.*

Given two motives $M,N\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$, one may easily check that

[TABLE]

In particular, for all $i\in{\mathbb{Z}}$, we have $h(M\{i\})=h(M)+i$, and for all irreducible geometrically split variety $X$ satisfying the nilpotence principle, $h(U_{X}\{i\})=i$.

Let $M\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ be a motive. By the Krull-Schmidt property, see Prop 2.1, two pure Tate summands of $M$ of maximal rank are isomorphic. Therefore, we may give the following definition:

Definition 3.5**.**

A pure Tate motive is called the Tate trace of $M\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ if it is isomorphic to a pure Tate summand of maximal rank of $M$. The Tate trace of $M$ is uniquely defined up to isomorphism and denoted by $\operatorname{Tr}(M)$.

Example 3.6*.*

A motive $M$ is a pure Tate motive if and only if $\operatorname{Tr}(M)\simeq M$.

By Lemma 2.2, if $X$ is an irreducible geometrically split variety satisfying the nilpotence principle, then \operatorname{Tr}\bigl{(}M(X)\bigr{)}\not\simeq 0 if and only if $X$ is isotropic with coefficients in $\Lambda$. In particular, the Tate trace does not commute with scalar extension.

Example 3.7*.*

Let $q=i_{0}{\mathbb{H}}+q_{\mathrm{an}}$ be a $2m$-dimensional quadratic form with Witt index $i_{0}$ and anisotropic part $q_{an}$. Denote by $Q$ and $Q_{\mathrm{an}}$ the corresponding projective quadrics. By [34], the motive $M(Q)\in\operatorname{Mot}_{F}({\mathbb{Z}}/2{\mathbb{Z}})$ satisfies:

[TABLE]

By Springer’s theorem [13, 71.3], $Q_{\mathrm{an}}$ is anisotropic with coefficients in ${\mathbb{Z}}/2{\mathbb{Z}}$ so that $M(Q_{\mathrm{an}})$ has trivial Tate trace. Therefore, the Tate trace of $M(Q)$ is given by

[TABLE]

In view of this example, we call a motive $M\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ isotropic if it has non-trivial Tate trace and anisotropic otherwise. By the Krull-Schmidt property, $M$ decomposes in a unique way as $M=\operatorname{Tr}(M)+M_{an}$ for some anisotropic motive $M_{an}$ called the anisotropic part of $M$, and uniquely defined up to isomorphism. Therefore, the Tate trace of a motive measures its splitting.

Example 3.8*.*

Let $A$ be an $F$-central simple algebra endowed with an orthogonal involution $\sigma$. The Witt index $i_{w}(\sigma)$ of the involution $\sigma$ is the reduced dimension of a maximal isotropic right ideal in $A$ [29, §6.A]. The variety of isotropic right ideals of reduced dimension $1$ is called the involution variety and denoted by $X_{\sigma}$. It is a $\mathrm{PGO}^{+}(A,\sigma)$-projective homogeneous variety, hence $M(X_{\sigma})\in\mathrm{Mot}^{\mathrm{KS}}_{F}({{\mathbb{Z}}/2{\mathbb{Z}}})$. We claim that its Tate trace is given by

[TABLE]

Indeed, the algebra $A$ contains a right ideal of reduced dimension $1$ if and only if it is split. Moreover, since $A$ admits an orthogonal involution, it has exponent $2$ and its Schur index is invariant under any odd degree extension of the base field. Therefore, if $A$ is non-split, then the variety $X_{\sigma}$ is anisotropic with coefficients in ${\mathbb{Z}}/2{\mathbb{Z}}$ and this proves the first equality. Assume now that $A$ is split. Then $\sigma$ is adjoint to a quadratic form $q$ over $F$ and it follows from the definitions that $X_{\sigma}$ is isomorphic to the corresponding quadric $Q$ and $i_{w}(\sigma)$ is equal to the Witt index of $q$. Hence the second equality follows from Example 3.7.

We may characterize the domination relation with coefficients in $\Lambda$ in terms of Tate traces as follows:

Proposition 3.9**.**

Let $X$ and $Y$ be two irreducible geometrically split varieties satisfying the nilpotence principle. Then $X\succeq_{\Lambda}Y$ if and only if \operatorname{Tr}\bigl{(}M(Y_{E})\bigr{)}\not\simeq 0 for all field extensions $E/F$ such that \operatorname{Tr}\bigl{(}M(X_{E})\bigr{)}\not\simeq 0.

Proof.

Assume first that $X\succeq_{\Lambda}Y$. By definition there exists a correspondence $\alpha:X\rightsquigarrow Y$ of multiplicity $1$. Hence, any correspondence $Z\rightsquigarrow X$ provides, by composition with $\alpha$, a correspondence $Z\rightsquigarrow Y$ of the same multiplicity. So, $Y_{E}$ is isotropic with coefficients in $\Lambda$ for all $E/F$ such that $X_{E}$ is isotropic with coefficients in $\Lambda$. By example 3.6, this proves the first implication. To prove the converse, it suffices to consider the field $E=F(X)$. Since $X_{F(X)}$ is isotropic with coefficients in $\Lambda$, the corresponding motive $M(X_{F(X)})$ has non-zero Tate trace. Therefore \operatorname{Tr}\bigl{(}M(Y_{F(X)})\bigr{)}\not=0 and this proves $Y_{F(X)}$ is isotropic with coefficients in $\Lambda$. So $X\succeq_{\Lambda}Y$, and this finishes the proof. ∎

Therefore, we may extend the domination relation to motives as follows:

Definition 3.10**.**

Let $M,N\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ be two motives.

1. (1)

We say that $M$ dominates $N$, and we write $M\succeq_{\Lambda}N$, if $\operatorname{Tr}(N_{E})\not\simeq 0$ for all field extension $E/F$, such that $\operatorname{Tr}(M_{E})\not\simeq 0$. 2. (2)

We say that $M$ is equivalent to $N$, and we write $M\approx N$ if $M\succeq N$ and $N\succeq M$.

Two motives are equivalent if and only if they are isotropic over the same field extensions of the base field. In particular, two isotropic motives are equivalent.

We now consider a subcategory of $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$, which, as recalled below, contains the motives we are interested in, and for which we will be able to prove an isomorphism criterion for objects using the Tate trace (see Theorem 3.15).

Definition 3.11**.**

1. (1)

A motive $P\in\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ is called initial if there exists an irreducible geometrically split variety $X$ satisfying the nilpotence principle such that $P$ is isomorphic to $U_{X}\{i\}$ for some $i\in{\mathbb{Z}}$, where $U_{X}$ denotes upper motive of $X$. 2. (2)

The full additive subcategory of $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$ generated by initial motives is denoted by $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$.

By definition, an object in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ is a finite direct sum of initial motives. Assume $G$ is a semi-simple affine algebraic group over $F$ of $p$-inner type, i.e. $G_{E}$ is inner for some field extension $E/F$ of $p$-power degree. By [23, Thm. 1.1], the motive of a projective $G$-homogeneous variety with coefficients in a finite local ring $\Lambda$ of residue characteristic $p$ decomposes as a direct sum of twists of upper motives, for various quasi-$G$-homogeneous varieties. In other words, motives of projective $G$-homogeneous varieties belong to $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$.

Remark 3.12*.*

Let $X$ be a geometrically irreducible and geometrically split variety satisfying the nilpotence principle. Denote by $U_{X}$ the indecomposable upper summand of $M(X)$. The motive ${(U_{X})}_{L}$ is still an upper summand of $M(X_{L})$ but it need not be indecomposable anymore. Therefore, $U_{(X_{L})}$ is a direct summand of ${(U_{X})}_{L}$ but in general they are not isomorphic. Hence, given a motive $M\in\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$, it is not known in general whether or not $M_{L}\in\mathrm{Mot}^{\mathrm{I}}_{L}({\Lambda})$.

A nice property of $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ is that the hook of an element is easy to compute: by example 3.4, if $M=\sum_{k=1}^{r}U_{X_{k}}\{i_{k}\}$, then $h(M)={\mathrm{min}}\{i_{k},\ k\in\llbracket 1,r\rrbracket\}$. In addition, we have the following isomorphism criterion for indecomposable motives:

Lemma 3.13**.**

Let $M=U_{X}\{i\}$ and $N=U_{Y}\{j\}$ be two initial motives. Then $M\simeq N$ if and only if $i=j$ and $X\approx_{\Lambda}Y$.

Proof.

If $M$ and $N$ are isomorphic, they have the same hook, so we have $i=j$. Hence, $M$ and $N$ are isomorphic if and only if $U_{X}$ and $U_{Y}$ are, and Lemma 2.4 finishes the proof. ∎

We may also compute the Tate trace of a tensor product of motives in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$:

Lemma 3.14**.**

For all $N,M\in\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$, we have $\operatorname{Tr}(M\otimes N)=\operatorname{Tr}(M)\otimes\operatorname{Tr}(N)$.

Proof.

Starting from decompositions $M=\sum_{k=1}^{r}U_{X_{k}}\{i_{k}\}$ and $N=\sum_{\ell=1}^{s}U_{Y_{\ell}}\{j_{\ell}\}$ as sums of initial motives, we get

[TABLE]

Given two irreducible geometrically split varieties satisfying the nilpotence principle $X$ and $Y$, we claim that the Tate trace of $U_{X}\otimes U_{Y}$ is non zero if and only if both factors are Tate motives. Indeed, $U_{X}\otimes U_{Y}$ is a direct summand of $M(X\times Y)$. Therefore, if $\operatorname{Tr}(U_{X}\otimes U_{Y})\not\simeq 0$, then $X\times Y$ is isotropic with coefficients in $\Lambda$. Hence both $X$ and $Y$ are isotropic with coefficients in $\Lambda$, and by Lemma 2.2, we get that $U_{X}\simeq\Lambda\{0\}\simeq U_{Y}$.

As a consequence, the Tate factors in a complete decomposition of $M\otimes N$ all occur as a tensor product of a Tate factor in a complete decomposition of $M$ and a Tate factor in a complete decomposition of $N$. ∎

The main result of this section is the following:

Theorem 3.15**.**

Consider two motives $M$ and $N$ in the full subcategory $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ of the category of Chow motives with coefficients in a finite local ring $\Lambda$. Then, $M$ and $N$ are isomorphic if and only if $M_{L}$ and $N_{L}$ have isomorphic Tate traces for all field extensions $L/F$.

Remark 3.16*.*

When the right hand side condition holds, we say that $M$ and $N$ have isomorphic higher Tate trace.

Proof.

Since $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ is a full subcategory of $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$, where the Tate trace is defined, one implication is clear. To prove the converse, let us assume that $M$ and $N$ are objects of $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$, and that they have the same higher Tate trace. In particular, $\operatorname{Tr}(M)\simeq\operatorname{Tr}(N)$, hence $M$ and $N$ are isomorphic if and only if their anisotropic parts are isomorphic. So we may assume $M$ and $N$ are anisotropic.

Consider a field $E/F$ which is a splitting field for both $M$ and $N$. We have $M_{E}\simeq\operatorname{Tr}(M_{E})\simeq\operatorname{Tr}(N_{E})\simeq N_{E}$. In particular, $M$ and $N$ have the same rank $r$ and hook $h$. Tensoring $M$ and $N$ with $\Lambda\{-h\}$, we may assume they both have hook [math]. We now proceed by induction on $r$. The result trivially holds if $r=0$, so we may assume $M$ and $N$ have rank $r\geq 1$.

We let $M_{0}$ (respectively $N_{0}$) be the sum of all indecomposable summands in a complete decomposition of $M$ (respectively $N$) having hook [math]. Since $M$ and $N$ have hook [math], the motives $M_{0}$ and $N_{0}$ are non-trivial. They admit decompositions $M_{0}=\sum_{k=1}^{s}U_{X_{k}}$ and $N_{0}=\sum_{\ell=1}^{t}U_{Y_{\ell}}$ for some non-zero integers $s$ and $t$ and some irreducible geometrically split varieties $X_{k}$ and $Y_{\ell}$ satisfying the nilpotence principle. Consider the function field $F_{1}=F(X_{1})$. The variety $X_{1}$ becomes isotropic with coefficients in $\Lambda$ over $F_{1}$, therefore by Lemma 2.2, $(M_{0})_{F_{1}}$ contains a summand isomorphic to $\Lambda\{0\}$. Since $M$ and $N$ have the same higher Tate trace, it follows that $\Lambda\{0\}$ also is a direct summand of $N_{F_{1}}$. By definition of $N_{0}$, we have $N=N_{0}+N_{1}$ with $h(N_{1})\geq 1$, so that $\Lambda\{0\}$ actually is a summand of $(N_{0})_{F_{1}}$. It follows $\Lambda\{0\}$ is a summand of $(U_{Y_{\ell_{1}}})_{F_{1}}$ for some $\ell_{1}\in\llbracket 1,t\rrbracket$, hence of $M(Y_{\ell_{1}})_{F_{1}}$. Applying again Lemma 2.2, we get that $(Y_{\ell_{1}})_{F_{1}}$ is isotropic with coefficients in $\Lambda$, that is $X_{1}$ dominates $Y_{\ell_{1}}$ with coefficients in $\Lambda$. The same argument applied to scalar extension to $F(Y_{\ell_{1}})$ shows that $Y_{\ell_{1}}$ dominates $X_{k_{1}}$ with coefficients in $\Lambda$ for some $k_{1}\in\llbracket 1,s\rrbracket$. The integer $k_{1}$ need not be equal to $1$; but iterating the process, we get a decreasing chain of varieties

[TABLE]

for some integers $k_{i}\in\llbracket 1,s\rrbracket$ and $\ell_{i}\in\llbracket 1,t\rrbracket$. Since the set of varieties $X_{k}$ and $Y_{\ell}$ are finite, there exists $k\in\llbracket 1,s\rrbracket$ and $\ell\in\llbracket 1,t\rrbracket$ such that $X_{k}\succeq_{\Lambda}Y_{\ell}\succeq_{\Lambda}X_{k}$, hence, $X_{k}\approx_{\Lambda}Y_{\ell}$. By Lemma 3.13, it follows that $U_{X_{k}}\simeq U_{Y_{\ell}}$, so we may write $M\simeq U_{X_{k}}+M^{\prime}$ and $N\simeq U_{X_{k}}+N^{\prime}$ for some motives $M^{\prime}$ and $N^{\prime}$ in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$. Since $M$ and $N$ have the same higher Tate trace, the same holds for $M^{\prime}$ and $N^{\prime}$ and this concludes the proof by induction. ∎

As we already explained, by Karpenko [22, Thm. 3.5], [23, Thm. 1.1], our theorem applies to the motives of projective homegeneous varieties for groups of inner type for all finite local $\Lambda$, and groups of $p$-inner type for $\Lambda={\mathbb{Z}}/p{\mathbb{Z}}$. In this setting, Theorem 3.15 recovers numerous known results, see the following and § 4. For quadrics, it provides a new proof of Vishik’s criterion for motivic equivalence:

Corollary 3.17** (Vishik [36]).**

Let $Q$ and $Q^{\prime}$ be smooth projective quadrics corresponding to quadratic forms $q$ and $q^{\prime}$ of the same dimension. The motives of $Q$ and $Q^{\prime}$ are isomorphic in $\operatorname{Mot}_{F}({\mathbb{Z}}/2{\mathbb{Z}})$ if and only if the quadratic forms $q_{L}$ and $q^{\prime}_{L}$ have the same Witt index for all field extension $L/F$.

Proof.

As $Q$ and $Q^{\prime}$ are of the same dimension, Rost’s decomposition [34, Prop. 2] states that for any field extension $L/F$, the Tate trace of the motive $M(Q_{L})$ (resp. $M(Q^{\prime}_{L})$) determines and is determined by the Witt index of $q_{L}$ (resp. of $q^{\prime}_{L}$). The corollary then follows immediately from Theorem 3.15. ∎

The same argument applied to involution varieties, whose Tate traces are described in Example 3.8, leads to the following generalization:

Corollary 3.18**.**

Let $A$ be a central simple $F$-algebra and $\sigma$ and $\tau$ two orthogonal involutions on $A$. The motives of the corresponding involution varieties $X_{\sigma}$ and $X_{\tau}$ are isomorphic in $\operatorname{Mot}_{F}({\mathbb{Z}}/2{\mathbb{Z}})$ if and only if for all field extension $L/F$ such that $A_{L}$ is a split algebra, we have $i_{w}(\sigma_{L})=i_{w}(\tau_{L})$.

Remark 3.19*.*

Corollary 3.18, combined with Karpenko’s theorem on anisotropy of orthogonal involutions after generic splitting [18], provides a new proof that involution varieties are critical, see [12, Thm 3.6].

Example 3.20*.*

Consider two central simple $F$-algebras $A$ and $B$ of the same degree. The corresponding Severi-Brauer varieties $\operatorname{SB}(A)$ and $\operatorname{SB}(B)$ have isomorphic motives with coefficients in ${\mathbb{Z}}/p{\mathbb{Z}}$ if and only if their $p$-primary components $A_{p}$ and $B_{p}$ generate the same subgroup in the Brauer group of $F$. This can be deduced from the results of [22] or shown as follows. By [20], the motives of $\operatorname{SB}(A)$ and $\operatorname{SB}(B)$ decompose, respectively, as direct sums of twists of the Severi-Brauer varieties of $A_{p}$ and $B_{p}$. It follows that given a field extension $E/F$, the motive $M(\operatorname{SB}(A_{E}))$ (resp. $M(\operatorname{SB}(B_{E}))$) is either pure Tate or anisotropic, depending on whether the $p$-primary component $(A_{p})_{E}$ (resp. $(B_{p})_{E}$) is split or not. The conclusion is implied by Theorem 3.15, since two central simple algebras are split by the same field extensions of $F$ if and only if they generate the same subgroup of the Brauer group of $F$, by [1, Theorem 9.3].

Remark 3.21*.*

As opposed to Vishik’s criterion (Cor. 3.17), Theorem 3.15 does not hold for motives with ${\mathbb{Z}}$ coefficients. Indeed, consider two central division $F$-algebras $D$ and $D^{\prime}$ of degree $n$ and let $\operatorname{SB}(D)$ and $\operatorname{SB}(D^{\prime})$ be their respective Severi-Brauer varieties. Assume $D$ and $D^{\prime}$ generate the same subgroup of the Brauer-group, and yet $D$ is isomorphic neither to $D^{\prime}$ nor to its opposite. By [19, Criterion 7.1], $M(X_{D})$ and $M(X_{D^{\prime}})$ are not isomorphic in $\operatorname{Mot}_{F}({\mathbb{Z}})$; nevertheless, by theorems of Châtelet and Amitsur (see [15, 5.1.3, 5.4.1]), for any field extension $E/F$, either $M(X_{D})$ and $M(X_{D^{\prime}})$ are anisotropic over $E$, or both are pure Tate motives isomorphic to $M({\mathbb{P}}_{E}^{n-1})$.

We conclude this section with a cancellation property in the category $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$:

Corollary 3.22**.**

Let $M,N,N^{\prime}\in\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ be three non-trivial motives.

If $N\succeq M$, $N^{\prime}\succeq M$, and $M\otimes N\simeq M\otimes N^{\prime}$, then $N\simeq N^{\prime}$.

Proof.

Consider $M$, $N$ and $N^{\prime}$ as above and let $E/F$ be a field extension. Since $M\otimes N\simeq M\otimes N^{\prime}$, they have the same higher Tate trace. By Lemma 3.14, we get that $\operatorname{Tr}(M_{E})\otimes\operatorname{Tr}(N_{E})\simeq\operatorname{Tr}(M_{E})\otimes\operatorname{Tr}(N^{\prime}_{E})$.

Assume first that $\operatorname{Tr}(N_{E})=0$. Then $\operatorname{Tr}(M_{E})\otimes\operatorname{Tr}(N^{\prime}_{E})=0$. Hence, either $\operatorname{Tr}(M_{E})=0$ or $\operatorname{Tr}(N^{\prime}_{E})=0$. Since $N^{\prime}\succeq M$, we actually get $\operatorname{Tr}(N^{\prime}_{E})=0$ in both cases.

Assume now that $\operatorname{Tr}(N_{E})\not=0$. Since $N\succeq M$, we also have $\operatorname{Tr}(M_{E})\not=0$. In particular, $\operatorname{Tr}(M_{E})$ has non-zero rank, and it follows that $\operatorname{Tr}(N_{E})$ and $\operatorname{Tr}(N^{\prime}_{E})$ have the same rank $r$, see Example 3.4. Let $T$ be a pure Tate motive of maximal rank which is a direct summand of both $\operatorname{Tr}(N_{E})$ and $\operatorname{Tr}(N^{\prime}_{E})$. We may decompose $\operatorname{Tr}(N_{E})=T+S$ and $\operatorname{Tr}(N^{\prime}_{E})=T+S^{\prime}$ for some pure Tate motives $S$ and $S^{\prime}$ that contain no isomorphic Tate summand. By the Krull-Schmidt property, we have $\operatorname{Tr}(M_{E})\otimes S\simeq\operatorname{Tr}(M_{E})\otimes S^{\prime}$. Assume for the sake of contradiction that $rk(T)<r$, so that $S$ and $S^{\prime}$ are non-zero. Then the hook of $\operatorname{Tr}(M_{E})\otimes S\simeq\operatorname{Tr}(M_{E})\otimes S^{\prime}$ is $h(\operatorname{Tr}(M_{E}))+h(S)=h(\operatorname{Tr}(M_{E}))+h(S^{\prime})$. It follows that $S$ and $S^{\prime}$ have the same hook $h$. Since they are pure Tate motives, they both contain a summand isomorphic to $\Lambda\{h\}$ and this contradicts the maximality of $T$.

So, we have proved that $N$ and $N^{\prime}$ have the same higher Tate trace, and by Theorem 3.15, we get that $N\simeq N^{\prime}$. ∎

Remark 3.23*.*

Let $D$ and $D^{\prime}$ be two division algebras of degree $n$. Assume the Brauer class of $D^{\prime}$ belongs to the subgroup of the Brauer group generated by the class of $D$. Then the projection $X_{D}\times X_{D^{\prime}}\rightarrow X_{D}$ is a projective bundle, and for an arbitrary ring $\Lambda$, we have

[TABLE]

It follows that $M(X_{D}\times X_{D^{\prime}})\simeq M(X_{D}\times X_{D})$ in $\operatorname{Mot}_{F}({\mathbb{Z}})$. Applied to $D$ and $D^{\prime}$ as in Remark 3.21, this shows that the Corollary 3.22 does not hold with coefficients in ${\mathbb{Z}}$. We also have $M(X_{D}\times X_{D})\simeq M(X_{D}\times{\mathbb{P}}_{F}^{n-1})$ in $\operatorname{Mot}_{F}(\Lambda)$, hence, the domination conditions are necessary in Corollary 3.22.

4. Application to projective homogeneous varieties

Throughout this section, $\Lambda$ is a finite local ring with residue field of characteristic $p$. As we already mentioned, the motive of a projective homogeneous variety under the action of an affine semi-simple algebraic group of $p$-inner type belongs to $\mathrm{Mot}^{\mathrm{I}}_{F}({{\mathbb{Z}}/p{\mathbb{Z}}})$ ( [22, Thm. 3.5], [23, Thm. 1.1]). In this section, we explore consequences of Theorem 3.15 for those. Another benefit of working in the category generated by initial motives is that we may replace $\Lambda$ by ${\mathbb{Z}}/p{\mathbb{Z}}$, as we proceed to show.

4.1. Change of coefficients

Recall $\Lambda$ is a finite local ring, and its residue field $\bar{\Lambda}$ has characteristic $p$. The tensor product with $\overline{\Lambda}$ on Chow groups induce an additive change of coefficients functors

[TABLE]

We prove the following :

Proposition 4.1**.**

The change of coefficients induce functors

[TABLE]

which preserve motivic decompositions and Tate traces, and induce a bijection between isomorphism classes of objects.

Proof.

Let $P$ be an indecomposable object in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ and consider the surjection of rings

[TABLE]

Given a projector $\pi$ in \operatorname{End}\bigl{(}coe\!f\!\!f_{\overline{\Lambda}/\Lambda}(P)\bigr{)}, pick $\alpha$ in $\operatorname{End}(P)$ is such that $f(\alpha)=\pi$. As $\Lambda$ is finite, and $P$ is indecomposable, there is a power of $\alpha$ which is an idempotent of $\operatorname{End}(P)$, that is [math] or $1$, see [22, Cor. 2.2]. Hence $\operatorname{End}(coe\!f\!\!f_{\overline{\Lambda}/\Lambda}(P))$ is connected, that is [math] and $1$ are its only idempotents, and it follows $coe\!f\!\!f_{\overline{\Lambda}/\Lambda}(P)$ is indecomposable.

By definition, $P$ is an initial motive, $P\simeq U_{X}\{i\}$ for some irreducible geometrically split variety $X$ satisfying the nilpotence principle. The image of $U_{X}$ in $\mathrm{Mot}^{\mathrm{KS}}_{F}({\overline{\Lambda}})$ is an upper summand of $M(X)$, hence the upper motive of $X$, since it is indecomposable. Therefore, $coe\!f\!\!f_{\overline{\Lambda}/\Lambda}(U_{X}\{i\})$ also is an initial motive. This already proves that $coe\!f\!\!f_{\overline{\Lambda}/\Lambda}$ induces a functor between the subcategories generated by initial motives.

We claim that two initial motives are isomorphic with coefficients in $\Lambda$ if and only if they are isomorphic with coefficients in ${\overline{\Lambda}}$. Let $X$ and $Y$ be two irreducible geometrically split varieties satisfying the nilpotence principle. A [math]-cycle of degree $1$ on the variety $Y_{F(X)}$ with coefficients in $\overline{\Lambda}$ lifts to a [math]-cycle of degree $\delta\in\Lambda$ with $\bar{\delta}=1$. Since $\Lambda$ is local, $\delta$ is invertible, hence this provides a [math]-cycle of degree $1$ with coefficients in $\Lambda$. Therefore, the equivalence relations with coefficients in $\Lambda$ and $\overline{\Lambda}$ are the same, and the claim follows by Lemma 3.13. Since the functor is additive and the Krull-Schmidt property holds in both categories, this proves the proposition for this functor.

We proceed in a similar way for the second change of coefficients. We first prove that $coe\!f\!\!f_{\overline{\Lambda}/({\mathbb{Z}}/p{\mathbb{Z}})}(U_{X}\{i\})$ are indecomposable for all initial motives $U_{X}\{i\}$ in $\mathrm{Mot}^{\mathrm{I}}_{F}({{\mathbb{Z}}/p{\mathbb{Z}}})$. The argument is borrowed from [10], where this is proved if $X$ is projective homogeneous under the action of a group of $p$-inner type, using reduced Chow motives. We may assume $i=0$; so consider the upper motive $U_{X}$ of an irreducible geometrically split variety satisfying the nilpotence principle. Since $U_{X}$ is defined by a projector of multiplicity $1$, the multiplicity induces a ring homomorphism $\operatorname{mult}_{{\mathbb{Z}}/p{\mathbb{Z}}}:\operatorname{End}(U_{X})\longrightarrow{\mathbb{Z}}/p{\mathbb{Z}}$. As $\operatorname{End}(U_{X})$ is local, the set $\mathcal{N}$ of its nilpotent elements is precisely the kernel of $\operatorname{mult}_{{\mathbb{Z}}/p{\mathbb{Z}}}$. It follows that we have a short exact sequence

[TABLE]

with $\mathcal{N}\otimes K$ nilpotent. Now if $\pi\in\operatorname{End}(coe\!f\!\!f_{K/\mathbb{F}_{p}}(U_{X}))$ is a projector, then $\operatorname{mult}_{K}(\pi)$ is either [math] or $1$. We may replace $\pi$ by $1-\pi$ and so assume that $\pi$ has trivial multiplicity, which implies that $\pi$ is nilpotent, hence trivial. It follows that $coe\!f\!\!f_{K/\mathbb{F}_{p}}(U_{X}\{i\})$ is indecomposable, and it only remains to prove that the equivalence relations modulo $\overline{\Lambda}$ and modulo ${\mathbb{Z}}/p{\mathbb{Z}}$ are the same. This comes from the fact that a [math]-cycle of degree $1$ with coefficients in $\overline{\Lambda}$ is a linear combination of closed points, with at least one of them of prime to $p$ degree in ${\mathbb{Z}}$. ∎

Remark 4.2*.*

In the above proposition, we actually prove that $coe\!f\!\!f_{\overline{\Lambda}/\Lambda}(P)$ are indecomposable for all indecomposable motives in $\mathrm{Mot}^{\mathrm{KS}}_{F}({\Lambda})$. The same does not hold in general for $coe\!f\!\!f_{\overline{\Lambda}/({\mathbb{Z}}/p{\mathbb{Z}})}$. An explicit counter-example using some Artin-Tate motives can be provided as in [10]: if $L/F$ is a cubic Galois field extension, then M\bigl{(}\operatorname{Spec}(L)\bigr{)} in $\mathrm{Mot}^{\mathrm{KS}}_{F}({{\mathbb{Z}}/2{\mathbb{Z}}})$ contains an indecomposable summand $N$ with decomposable image $coe\!f\!\!f_{{\mathbb{F}}_{4}/({\mathbb{Z}}/2{\mathbb{Z}})}(N)$ in $\mathrm{Mot}^{\mathrm{KS}}_{F}({{\mathbb{F}}_{4}})$. In particular, this proves that $M(\operatorname{Spec}(L))$ does not belong to $\mathrm{Mot}^{\mathrm{I}}_{F}({{\mathbb{Z}}/2{\mathbb{Z}}})$. Note that the $F$-variety $\operatorname{Spec}(L)$ is a direct summand of the motive of a projective homogeneous variety under the action of the Weil restriction of $(\mathrm{PGL}_{2})_{L}$, which is not a $2$-inner group.

4.2. Motives of projectives homogeneous varieties

We now explore consequences of our main theorem in the setting of projective homogeneous varieties. Let $G$ be semi-simple algebraic group over $F$, $T\subset G$ a maximal torus, and $\Delta(G)$ the Dynkin diagram of $G$. Its set of vertices, also denoted by $\Delta(G)$, is given by a set of simple roots of the root system of $G_{F_{sep}}$ with respect to $T_{F_{sep}}$. The absolute Galois group $\operatorname{Gal}(F)=\operatorname{Gal}(F_{sep}/F)$ acts on the Dynkin diagram $\Delta(G)$ via the $\ast$-action. We assume $G$ is of $p$-inner type, that is the $\ast$-action becomes trivial over a field extension of $p$-power degree.

A projective homogeneous variety $X$ is a $G$-variety over $F$, isomorphic over a separable closure $F_{sep}$ to a quotient $G_{F_{sep}}/P$ by a parabolic subgroup. Given a subset $\Theta$ of $\Delta(G)$, there is a parabolic subgroup $P_{\Theta}$ of $G_{F_{sep}}$ whose Levi component has Dynkin diagram $\Delta(G)\setminus\Theta$. Note that in the literature, there are two opposite conventions for parabolic subgroups of type $\Theta$; here, a Borel subgroup has type $\Delta(G)$. The variety $G_{F_{sep}}/P_{\Theta}$ is defined over $F$ if and only if $\Theta$ is $\operatorname{Gal}(F)$-invariant, leading to a bijection $\Theta\leftrightarrow X_{\Theta,G}$ between $\ast$-invariant subsets of $\Delta(G)$ and isomorphism classes of projective homogeneous varieties for $G$ defined over $F$. For all such $\Theta$, $M(X_{\Theta,G})$ belongs to $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ if $G$ is of $p$-inner type, by [23, Thm. 1.1]. Therefore, Theorem 3.15 applies to direct summands of such motives, and we get:

Corollary 4.3**.**

Let $\Lambda$ be a finite local ring with residue field of characteristic $p$. Let $X$ and $X^{\prime}$ be two projective homogeneous varieties, for some $p$-inner semisimple groups $G$ and $G^{\prime}$ over $F$.

Two direct summands $M$ and $M^{\prime}$ of $M(X)$ and $M(X^{\prime})$ are isomorphic in $\operatorname{Mot}_{F}(\Lambda)$ if and only if their higher Tate traces are isomorphic.

Remark 4.4*.*

Corollary 4.3 extends to the case of projective pseudo-homogeneous varieties (that is, for non-reduced parabolic subgroups) by [35].

Assume that $M(X_{\Theta,G})$ is a pure Tate motive for some $\operatorname{Gal}(F)$-invariant subset $\Theta\subset\Delta(G)$. The $\Lambda$-module $\operatorname{Ch}(X_{\Theta,G})$ is free and the isomorphism class of $M(X_{\Theta,G})$ is determined by the Poincaré polynomial

[TABLE]

where $a_{i}:=\dim(\operatorname{Ch}_{i}(X_{\Theta,G}))$. The motive $M(X_{\Theta,G})$ being pure Tate, this polynomial is invariant under field extensions and equal to $P(M(X_{\Theta,G_{F_{sep}}}),t)$. Hence, it does not depend on the coefficient ring $\Lambda$ and as explained in [30], it is described solely through the Weyl group of $G$ and the length function; we thus call it the Poincaré polynomial of $X_{\Theta,G_{F_{sep}}}$. The next corollary applies, notably but not exclusively, to varieties of Borel subgroups of a semi-simple group of inner type.

Corollary 4.5**.**

Let $\Lambda$ be a finite local ring with residue field of characteristic $p$. Consider a $G$-projective homogeneous variety $X$ and a $G^{\prime}$-projective homogeneous variety $X^{\prime}$, where $G$ and $G^{\prime}$ are of inner type. We assume in addition that $G_{F(X)}$ and $G^{\prime}_{F(X^{\prime})}$ are split. The following are equivalent :

1. (i)

The motives of $X$ and $X^{\prime}$ are isomorphic in $\operatorname{Mot}_{F}(\Lambda)$; 2. (ii)

The Poincaré polynomials of $X_{F_{sep}}$ and $X^{\prime}_{F_{sep}}$ are equal and for all field extensions $E/F$, $G_{E}$ is split over a prime to $p$ extension of $E$ if and only if $G^{\prime}_{E}$ is split over a (possibly different) prime to $p$ extension of $E$.

Proof.

As $G$ and $G^{\prime}$ are of inner type, both $M(X)$ and $M(X^{\prime})$ belong to $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ [22]. Moreover, since $G_{F(X)}$ is split, $X$ is generically split, and for all field extensions $E/F$, the motive $M(X_{E})$ is either anisotropic or a pure Tate motive. By Proposition 4.1, we may assume $\Lambda={\mathbb{Z}}/p{\mathbb{Z}}$. Hence, $M(X_{E})$ is a pure Tate motive if and only if $X_{E}$ is isotropic with coefficients in ${\mathbb{Z}}/p{\mathbb{Z}}$, or equivalently, $G_{E}$ is split by a prime-to-$p$ field extension. Indeed, if $L/E$ a field extension of prime-to-$p$ degree such that $X_{L}$ has a rational point, then the extension $L(X)/L$ is purely transcendental, and thus $G$ splits over $L$. The converse is obvious. These assertions also hold for $X^{\prime}$, and with this in hand, we prove the corollary as follows.

$(i)\Rightarrow(ii)$ Assuming $M(X)$ and $M(X^{\prime})$ are isomorphic, the Poincaré polynomials $P(X_{F_{sep}},t)$ and $P(X^{\prime}_{F_{sep}},t)$ are clearly equal and the upper motives of $X$ and $X^{\prime}$ are isomorphic. So $X$ and $X^{\prime}$ are equivalent with coefficients in ${\mathbb{Z}}/p{\mathbb{Z}}$, and for all field extensions $E/F$, $X_{E}$ is isotropic with coefficients in ${\mathbb{Z}}/p{\mathbb{Z}}$ if and only if $X^{\prime}_{E}$ also is. The conclusion follows by the previous observations.

$(ii)\Rightarrow(i)$ Let $E/F$ be a field extension. The second part of condition (ii) guarantees that $M(X_{E})$ and $M(X^{\prime}_{E})$ are both anisotropic or both pure Tate motives. If they are pure Tate motives, they are isomorphic since they have the same Poincaré polynomial. The motives $M(X)$ and $M(X^{\prime})$ are thus isomorphic by Theorem 3.15, as their higher Tate traces are isomorphic. ∎

Remark 4.6*.*

If $G$, $G^{\prime}$ are inner forms of the same quasi-split group and $X$ and $X^{\prime}$ are of the same type, we may drop the assumption on the Poincaré polynomials of $X_{F_{sep}}$ and $X^{\prime}_{F_{sep}}$. This applies for instance to Severi-Brauer varieties of the same dimension, or varieties of Borel subgroups for two semi-simple groups of the same type and rank.

Example 4.7*.*

Let $G$ be a semisimple group of type $G_{2}$ (for instance, the automorphism group of a Cayley algebra [29, §31]) and set $p=2$. The projective homogeneous varieties $X_{\{1\},G}$ and $X_{\{2\},G}$ both have Poincaré polynomial $\sum_{i=0}^{5}t^{i}$ over $F_{sep}$. Furthermore, both are generically split by [11, Proposition 9] and are isotropic over a field extension $E/F$ if and only if $G_{E}$ is split by an odd degree field extension of $E$. By Corollary 4.5, the motives $M(X_{\{1\},G})$ and $M(X_{\{2\},G})$ are thus isomorphic in $\operatorname{Mot}_{F}(\Lambda)$ with coefficients in any finite local ring with residue field of characteristic $2$. This was originally noticed by Bonnet [5].

Assorted with the description of the Tits $p$-indexes of semisimple groups, Corollary 4.5 produces numerous motivic isomorphisms through cohomological invariants of semisimple groups (see [11, Def. 4, §IV.3] for the definitions of the invariants $b(.)$ and $f_{3}(.)$).

Corollary 4.8**.**

Let $\Lambda$ be a finite local ring with residue field of characteristic $p$. Assume that $G$ and $G^{\prime}$ are of the same below types and that $X$, $X^{\prime}$ are anisotropic projective homogeneous varieties of the same type for $G$ and $G^{\prime}$, respectively.

• •

($F_{4}$, $p=3$ or $E_{8}$, $p=5$) $M(X)\simeq M(X^{\prime})$ if and only if $b(G)$ and $b(G^{\prime})$ generate the same subgroup of $H^{3}(F,\mathbb{Z}/p\mathbb{Z}(2))$;

• •

($G_{2}$, $p=2$) $M(X)\simeq M(X^{\prime})$ if and only if $G$ and $G^{\prime}$ are isomorphic;

• •

(​ ${}^{1}\!E_{6}$, $p=2$) $M(X)\simeq M(X^{\prime})$ if and only if $f_{3}(G)=f_{3}(G^{\prime})$;

• •

($E_{7}$, $p=3$) $M(X)\simeq M(X^{\prime})$ if and only if $b(G)=\pm b(G^{\prime})\in H^{3}(F,\mathbb{Z}/3\mathbb{Z}(2))$.

Proof.

As $X$ and $X^{\prime}$ are of the same type, the Poincaré polynomials of $X_{F_{sep}}$ and $X^{\prime}_{F_{sep}}$ are equal. Furthermore the analysis of the Tits $p$-indexes of [11] shows that in all the case considered, $G_{F(X)}$ and $G^{\prime}_{F(X^{\prime})}$ are split.

Corollary 4.5 thus states that $M(X)$ and $M(X^{\prime})$ are isomorphic if and only if for any field extension $E/F$, $G_{E}$ is split by a prime-to-$p$ field extension if and only if $G^{\prime}_{E}$ is split by a (possibly different) prime-to-$p$ extension. The result boils down to cohomological characterizations of this property given in [11], namely Corollary 10 for types $G_{2}$, $F_{4}$ and $E_{8}$; Corollary 14 for type ${}^{1}\!E_{6}$ and Proposition 15 for type $E_{7}$. ∎

4.3. Motivic equivalence of semisimple algebraic groups

The Tits index of a semisimple algebraic group $G$ over $F$ is a fundamental invariant, which describes the isotropy of $G$. For classical groups, it encompasses classical invariants of algebraic structures over fields, such as the Schur index of central simple algebras or the Witt index of quadratic forms. It has been known for a while that the isomorphism classes of motives of $G$-projective homogeneous varieties determine the Tits-index of $G$, up to prime-to-$p$ base change. The main result of [9] provides a converse. Theorem 4.12 below generalizes this result. The proof, based on Theorem 3.15, is independant of [9].

We use the same notations as in [11, §I.1]. In particular, $\Delta(G)$ denotes the Dynkin diagram of $G$ as well as its set of vertices. The distinguished vertices form a subset denoted by $\delta_{0}(G)$. It consists of a union of orbits for the $\ast$-action; more precisely, an orbit $\Theta\subset\Delta(G)$ is distinguished if and only if projective homogeneous varieties of type $\Theta$ have a rational point. Hence, $\delta_{0}(G)$ is empty when $G$ is anisotropic, and coincides with $\Delta(G)$ when $G$ is quasi-split.

Definition 4.9**.**

Let $G$ and $G^{\prime}$ be semisimple algebraic groups over a field $F$, which are inner forms of the same quasi-split group. We say that $G$ and $G^{\prime}$ are motivic equivalent with coefficients in $\Lambda$ if there is an isomorphism of diagrams

[TABLE]

compatible with the $\ast$-actions, such that for any $Gal(F_{sep}/F)$-invariant subset $\Theta$ of $\Delta(G)$, the motives $M(X_{\Theta,G})$ and $M(X_{\varphi(\Theta),G^{\prime}})$ are isomorphic with coefficients in $\Lambda$.

Example 4.10*.*

Given two central simple algebras $A$ and $A^{\prime}$ of the same dimension and setting $\varphi=id$, the groups $SL(A)$ and $SL(A^{\prime})$ are motivic equivalent with coefficients in $\Lambda$ if and only if for any sequence of integers $d_{1}<...<d_{k}$, the motives of the varieties of flags of right ideals in $A$ and $A^{\prime}$ of reduced dimensions $d_{1},..,d_{k}$ are isomorphic in $\operatorname{Mot}_{F}(\Lambda)$. The same holds for special orthogonal groups associated to quadratic forms of the same dimension, replacing flags of ideals by flags of isotropic subspaces.

Motivic equivalence of semisimple algebraic groups aims at giving a classification of these with respect to the motives of their respective projective homogeneous varieties. In the view of Corollary 4.3, these motives are determined up to isomorphism by their higher Tate trace. Isotropy of semisimple groups is controlled by the Tits indexes but in order to prove results in $\operatorname{Mot}_{F}(\Lambda)$, one should rather consider their values over $p$-special fields, the Tits $p$-indexes.

A field is $p$-special if any of its finite extension has degree a power of $p$. For any field $F$ there is a $p$-special closure of $F$, that is a minimal $p$-special field extension $F_{p}/F$. The $p$-special closures of $F$ are algebraic over $F$ and isomorphic [13, §101.B]. The Tits $p$-index of a semisimple algebraic group $G$ over $F$ is defined as the Tits index of $G_{F_{p}}$, where $F_{p}$ is a $p$-special closure of $F$. The admissible values of the Tits $p$-indexes over fields are determined in [11].

Lemma 4.11**.**

Let $M\in\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ and $F_{p}/F$ be a $p$-special closure of $F$. The Tate trace of $M$ is isomorphic over $F_{p}$ to the Tate trace of $M_{F_{p}}$.

Proof.

As $M$ is isomorphic to a direct sum of initial motives, the result boils down to showing that a variety $X$ over $F$ has a [math]-cycle of prime-to-$p$ degree if and only if $X_{F_{p}}$ has a rational point.

Assume that $X_{F_{p}}$ has a rational point $x$. Replacing $X_{F_{p}}$ by an affine open subscheme containing $x$, we may assume that $X_{F_{p}}=\operatorname{Spec}(A_{F_{p}})$ for some finitely generated $F$-algebra $A$ so that $x$ corresponds to a morphism $A_{F_{p}}\longrightarrow F_{p}$ of $F$-algebras. The image of $A$ is a finite $F$-algebra contained in $F_{p}$, hence a finite field extension $L$ of $F$. The variety $X_{L}$ has a rational point, and by definition of $p$-special closure, $L/F$ is of prime-to-$p$ degree, giving rise to the needed [math]-cycle on $X$. The converse is obvious. ∎

In particular, if $M$ and $N$ in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ have isomorphic Tate traces after scalar extension to a $p$-special closure of $F$, they already have isomorphic Tate traces over $F$. We now prove the main result of this section:

Theorem 4.12**.**

Let $G$ and $G^{\prime}$ be semisimple algebraic groups over a field $F$, which are inner twisted forms of the same quasi-split group of $p$-inner type. Let $\Theta_{0}$ be a Galois-invariant subset of $\Delta(G)$, and consider an isomorphism of Dynkin diagrams

[TABLE]

*compatible with the $\ast$-actions. The following are equivalent:

1. (i)

For any projective homogeneous variety $X_{\Theta,G}$ with $\Theta\supset\Theta_{0}$, the motives $M(X_{\Theta,G})$ and $M(X_{\varphi(\Theta),G^{\prime}})$ are isomorphic in $\operatorname{Mot}_{F}(\Lambda)$. 2. (ii)

For any $p$-special field extension $E/F$, $\Theta_{0}$ is distinguished in $\Delta(G_{E})$ if and only if $\varphi(\Theta_{0})$ is distinguished in $\Delta(G^{\prime}_{E})$. Moreover, when this holds, $\varphi$ induces a bijection between $\delta_{0}(G_{E})$ and $\delta_{0}(G^{\prime}_{E})$.

Proof.

$(i)\Rightarrow(ii)$ By assumption the motives $M(X_{\Theta_{0},G})$ and $M(X_{\varphi(\Theta_{0}),G^{\prime}})$ are isomorphic, hence so are $M(X_{\Theta_{0},G_{E}})$ and $M(X_{\varphi(\Theta_{0}),G^{\prime}_{E}})$ for all field extensions $E/F$. It follows their upper motives are isomorphic. This holds with coefficients in $\Lambda$, hence also with coefficients in ${\mathbb{Z}}/p{\mathbb{Z}}$ by Proposition 4.1. For any field extension $E/F$, $X_{\Theta_{0},G_{E}}$ thus has a [math]-cycle of prime-to-$p$ degree if and only if $X_{\varphi(\Theta_{0}),G^{\prime}_{E}}$ as well. For $p$-special $E/F$, this corresponds to these varieties having a rational point and $\Theta_{0}$ is distinguished for $\Delta(G_{E})$ if and only if $\varphi(\Theta_{0})$ is for $\Delta(G^{\prime}_{E})$.

Assume now that $E/F$ is a $p$-special field and $\Theta_{0}$ is distinguished in $\Delta(G_{E})$, which means $\delta_{0}(G_{E})$ contains $\Theta_{0}$. With the same reasoning as above, assumption $(i)$ with $\Theta=\delta_{0}(G_{E})$ implies that $\varphi(\delta_{0}(G_{E}))$ is distinguished in $\Delta(G^{\prime}_{E})$, therefore $\varphi(\delta_{0}(G_{E}))\subset\delta_{0}(G^{\prime}_{E})$.

Now, we showed that $\varphi(\Theta_{0})$ is distinguished in $\Delta(G^{\prime}_{E})$; that is $\varphi(\Theta_{0})$ is contained in $\delta_{0}(G^{\prime}_{E})$. The subset $\varphi^{-1}(\delta_{0}(G^{\prime}_{E}))$ contains $\Theta_{0}$, so we may apply assumption $(i)$ with $\Theta=\varphi^{-1}(\delta_{0}(G^{\prime}_{E}))$. We get that $\varphi^{-1}(\delta_{0}(G^{\prime}_{E}))\subset\delta_{0}(G_{E})$, whence $\delta_{0}(G^{\prime}_{E})=\varphi(\delta_{0}(G_{E}))$. The isomorphism $\varphi$ then identifies the Tits indexes of $G_{E}$ and $G^{\prime}_{E}$.

$(ii)\Rightarrow(i)$ Let $\Theta$ be a $\operatorname{Gal}(F)$-invariant subset of $\Delta(G)$ containing $\Theta_{0}$. We now show that the higher Tate traces of $X_{\Theta,G}$ and $X_{\varphi(\Theta),G^{\prime}}$ are isomorphic. By Lemma 4.11, we only need to work over $p$-special fields containing $F$.

Let $E/F$ be a $p$-special field. Since $\Theta$ contains $\Theta_{0}$, if $\Theta$ is distinguished in $\Delta(G_{E})$, then $\Theta_{0}$ also is and the second part of condition (ii) holds. Therefore, $\Theta$ is contained in $\delta_{0}(G_{E})$ if and only if $\varphi(\Theta)$ is contained in $\delta_{0}(G^{\prime}_{E})$, that is $X_{\Theta,G_{E}}$ has a rational point if and only if $X_{\varphi(\Theta),G^{\prime}_{E}}$ also does. Since $E$ is $p$-special, having a rational point is equivalent to being isotropic modulo $p$ and we get that the varieties $X_{\Theta,G_{E}}$ and $X_{\varphi(\Theta),G^{\prime}_{E}}$ have non-trivial Tate traces over the same $p$-special fields $E/F$.

We now proceed by induction on the common rank of $G$ and $G^{\prime}$ (that is, the number of vertices of their Dynkin diagrams). If $G$ and $G^{\prime}$ are of rank [math], the result is obvious. Let $E/F$ be a $p$-special field extension such that $X_{\Theta,G_{E}}$ has a rational point. From Chernousov, Gille, Merkurjev motivic decompositions of isotropic projective homogeneous varieties (see [7], [6, Theorem 7.4]), there is an isomorphism

[TABLE]

where $G_{E,\Theta}$ is the semisimple part of a Levi component of $G_{E}$ associated to $\Theta$, the $\delta$’s are the orbits for the $\ast$-action on minimal length cosets representatives for some double cosets of the Weyl group of $G$, and $l(.)$ is the length. This data is uniquely determined by the combinatorics of $G$ and does not depend on the choice of a set of simple roots of $G$.

The semisimple algebraic groups $G_{E,\Theta}$ and $G^{\prime}_{E,\varphi(\Theta)}$ are inner forms of the same quasisplit group: their Dynkin diagrams, together with the $\ast$-actions, are obtained from the ones of $G$ and $G^{\prime}$ by removing the subsets $\Theta$, $\varphi(\Theta)$, respectively. By assumption $(ii)$, for any $p$-special field extension $K/E$, $\varphi$ thus identifies the Tits indexes of $G_{K}$ and $G^{\prime}_{K}$. Induction hypothesis with $\Theta_{0}=\emptyset$ then gives that $G_{E,\Theta}$ and $G^{\prime}_{E,\Theta}$ are motivic equivalent with coefficients in $\Lambda$, so that

[TABLE]

It follows from this isomorphism that $X_{\Theta,G_{E}}$ and $X_{\varphi(\Theta),G^{\prime}_{E}}$ have isomorphic Tate traces for all $p$-special fields $E/F$. Casting Theorem 3.15, the motives of $X_{\Theta,G}$ and $X_{\varphi(\Theta),G^{\prime}}$ are isomorphic as well. ∎

In the special case $\Theta_{0}=\emptyset$, we get the following statement, which is a reformulation of [9, Thm. 16].

Corollary 4.13**.**

Two semisimple algebraic groups $G$ and $G^{\prime}$, inner forms of the same quasi-split group of $p$-inner type, are motivic equivalent with coefficients in $\Lambda$ relatively to an isomorphism of $\varphi:\Delta(G)\longrightarrow\Delta(G^{\prime})$ if and only if $\varphi$ induces an isomorphism between the Tits indexes of $G_{E}$ and $G^{\prime}_{E}$, for any $p$-special field extension $E/F$.

5. Partial $p$-splitting fields and Motivic splitting patterns

Throughout this section, $\Lambda$ is a finite local ring and $p$ is the characteristic of its residue field. Theorem 3.15 states that two motives of $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ are isomorphic if they have isomorphic Tate traces over all field extensions of the base field. In this section, we prove that it is enough to compare Tate traces over a finite set of function fields. We also define a motivic version of the splitting pattern for motives of projective homogeneous varieties.

Definition 5.1**.**

Let $M$ be a motive in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$ and write $M\simeq\sum_{k=1}^{r}U_{X_{k}}\{i_{k}\}$ as a direct sum of initial motives. The function fields $F(X_{1}),...,F(X_{r})$ are called partial $p$-splitting fields of the motive $M$, and we say that the set $\mathcal{C}=\{F(X_{1}),...,F(X_{r})\}$ is a partial $p$-splitting family of $M$.

By Lemma 3.13, given two partial $p$-splitting families $\mathcal{C}=\{F(X_{1}),...,F(X_{r})\}$ and $\mathcal{C}^{\prime}=\{F(Y_{1}),...,F(Y_{r})\}$ of a motive $M$, and renumbering one of them if necessary, we may assume that $X_{i}$ and $Y_{i}$ are equivalent with coefficients in $\Lambda$ for all $1\leq i\leq r$.

Example 5.2*.*

Let $X$ be a projective homogeneous variety, for a semisimple algebraic group $G$ of $p$-inner type. Fix a field extension $E/F$ such that $G_{E}$ is of inner type. By [23, Thm 1.1], the motive $M(X)\in\operatorname{Mot}_{F}(\Lambda)$ has a $p$-splitting family given by function fields of some corestrictions of projective $G_{L}$-homogeneous varieties, for several $E/L/F$.

We now prove that Tate traces over partial $p$-splitting fields detect isomorphisms in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$.

Proposition 5.3**.**

Given two objects $M$ and $N$ in $\mathrm{Mot}^{\mathrm{I}}_{F}({\Lambda})$, the following assertions are equivalent:

1. (a)

the motives $M$ and $N$ are isomorphic; 2. (b)

$M$* and $N$ have isomorphic Tate traces over all partial $p$-splitting fields of $M$ and all partial $p$-splitting fields of $N$;* 3. (c)

there is a $p$-splitting family $\mathcal{C}_{M}$ of $M$ and a $p$-splitting family $\mathcal{C}_{N}$ of $N$ such that $M$ and $N$ have isomorphic Tate traces over all fields in $\mathcal{C}_{M}\cup\mathcal{C}_{N}$.

Proof.

The outline of the proof is the same as for Theorem 3.15. The implications $(a)\Rightarrow(b)$ and $(b)\Rightarrow(c)$ are clear and we only need to prove that $(c)\Rightarrow(a)$. We proceed by induction on the maximum of the rank of $M$ and the rank of $N$. Assume condition (c) holds for some partial $p$-splitting families ${\mathcal{C}}_{M}$ and ${\mathcal{C}}_{N}$ given by the direct sums $M\simeq\sum_{k=1}^{r}U_{X_{k}}\{i_{k}\}$ and $N\simeq\sum_{\ell=1}^{s}U_{Y_{\ell}}\{j_{\ell}\}$. We may assume the summands are ordered by increasing hooks, so that $h(M)=i_{1}$ and $h(N)=j_{1}$. Since $M$ contains a summand isomorphic to $\Lambda\{i_{1}\}$ over the field $F(X_{1})$ which belongs to ${\mathcal{C}}_{M}$, condition (c) guarantees that $h(N)=j_{1}\leq i_{1}$. Using a similar process with $F(Y_{1})$, we get that $M$ and $N$ have the same hook and we may assume it is equal to [math]. By the same argument as in the proof of Theorem 3.15, where we only compared Tate traces over fields that belong to $\mathcal{C}_{M}\cup\mathcal{C}_{N}$, we may write $M\simeq U_{X_{k}}+M^{\prime}$ and $N\simeq U_{Y_{\ell}}+N^{\prime}$, with $U_{X_{k}}\simeq U_{Y_{\ell}}$. The motives $M^{\prime}$ and $N^{\prime}$ have partial $p$-splitting families contained in ${\mathcal{C}}_{M}$ and ${\mathcal{C}}_{N}$ and we conclude by induction. ∎

For projective homogeneous varieties, we get the following by Example 5.2.

Corollary 5.4**.**

Let $X$ and $Y$ be two projective homogeneous varieties for $p$-inner semisimple groups $G$ and $G^{\prime}$, respectively. Let $M$ be a direct summand of $M(X)$ and $N$ be a direct summand of $M(Y)$ in $\operatorname{Mot}_{F}(\Lambda)$. Fix a minimal field extension $E/F$ such that both $G_{E}$ and $G^{\prime}_{E}$ are of inner type.

The motives $M$ and $N$ are isomorphic if and only if their Tate traces are isomorphic over the function fields of the corestrictions to $F$ of projective homogeneous varieties for $G_{L}$ and $G^{\prime}_{L}$, for all intermediate field extensions $E/L/F$.

Remark 5.5*.*

The family of fields considered in the corollary is not minimal; but its definition depends only on the groups $G$ and $G^{\prime}$, and not on a decomposition of $M$ and $N$ as direct sums of indecomposable summands.

Recall that the splitting pattern of a quadratic form $q$ over $F$ is defined as the set of values taken by the Witt index of $q$ over all field extensions, see [16],  [13, §25]. It is thus a finite set of integers, bounded by half the dimension of $q$. As shown by Knebusch [28], the splitting pattern of $q$ is obtained considering a generic splitting tower of $q$. More precisely, the splitting pattern is equal to the set of Witt indices $\{i_{w}(q_{F_{0}}),\dots,i_{w}(q_{F_{h}})\}$, where $F_{0}=F$, for $i\geq 1$, $F_{i}$ is the function field of the anisotropic part of $q_{F_{i-1}}$ and $h$ is the height of $q$, that is the smallest integer such that $q_{F_{h}}$ has anisotropic part of dimension at most $1$.

We define a motivic version of the splitting pattern for projective homogeneous varieties and their direct summands as follows:

Definition 5.6**.**

Let $X$ be a projective homogeneous variety for some $p$-inner semisimple group $G$ and let $M$ be a direct summand of $M(X)$ in $\operatorname{Mot}_{F}(\Lambda)$. We say that a pure Tate motive $T$ belongs to the splitting pattern of $M$ if there is a field extension $K/F$ such that $T_{K}$ is isomorphic to the Tate trace of $M_{K}$. The set of such pure Tate motives is the motivic splitting pattern of $M$.

Example 5.7*.*

Let $Q$ be a projective quadric defined by a $2m$ dimensional quadratic form $q$. In view of Example 3.7, the motivic splitting pattern of $Q$ with coefficients in ${\mathbb{Z}}/2{\mathbb{Z}}$ consists of the pure Tate motives

[TABLE]

where $i$ runs through the (classical) splitting pattern of $q$.

The motivic splitting pattern of $M$ may be described using partial $p$-splitting fields, as we now proceed to show.

Let $M$ be a direct summand of the motive of a projective homogeneous variety over $F$ for a $p$-inner semisimple group $G$, with coefficients in $\Lambda$. Fix a $p$-splitting family $\mathcal{C}_{M}=\{F(X^{0}_{1}),...,F(X^{0}_{r_{0}})\}$ of $M$, where the varieties $X^{0}_{k}$ are corestrictions of projective $G_{L}$-homogeneous varieties (see Example 5.2).

If $M$ is not a pure Tate motive, there exists an integer $k_{0}$ such that $X^{0}_{k_{0}}$ is anisotropic with coefficients in $\Lambda$. The Tate trace of the motive $M_{F(X^{0}_{k_{0}})}$ has greater rank than the Tate trace of $M$. Write $\mathcal{C}_{M_{F(X^{0}_{k_{0}})}}=\{F(X^{1}_{1}),...,F(X^{1}_{r_{1}})\}$ for a $p$-splitting family of $M_{F(X^{0}_{k_{0}})}$. If $M_{F(X^{0}_{k_{0}})}$ is not pure Tate, we pick again an anisotropic $X^{1}_{k_{1}}$ and extend scalars to $F(X^{1}_{k_{1}})$. After a finite number of such field extensions, the motive $M$ becomes a pure Tate motive, as the rank of the Tate trace increases at each step. The corresponding sequence of field extensions

[TABLE]

is called a splitting tower of $M$.

Remark 5.8*.*

Let $Q$ be a projective quadric, defined by a quadratic form $q$. Assume $q$ is non-split, and denote by $Q_{\mathrm{an}}$ the quadric defined by its anisotropic part. By [34], see also Example 3.7, the motive $M(Q)$ contains a shift of $M(Q_{\mathrm{an}})$, hence of its upper motive $U_{Q_{\mathrm{an}}}$, as a direct summand. Therefore, we may take $X^{0}_{k_{0}}=Q_{\mathrm{an}}$ in the process above. Iterating the process, this proves that the generic splitting tower of $q$ introduced by Knebusch in [28] is a splitting tower for the motive $M(Q)$.

Proposition 5.9**.**

Let $G$ be a $p$-inner semisimple group and $X$ a projective $G$-homogeneous variety. Consider a direct summand $M$ of the motive $M(X)$ in $\operatorname{Mot}_{F}({\Lambda})$. A pure Tate motive $T$ belongs to the motivic splitting pattern of $M$ if and only if $T_{E}$ is isomorphic to the Tate trace of $M_{E}$, for some field extension $E/F$ appearing in a splitting tower of $M$.

Proof.

We proceed by induction on the anisotropic rank of $M$, that is the integer \operatorname{rk}(M)-\operatorname{rk}\bigr{(}\operatorname{Tr}(M)\bigl{)}. If this is [math], $M$ is a pure Tate motive, $\operatorname{Tr}(M)$ is the only element in the motivic splitting pattern of $M$ and we may take $E=F$. Assume now that \operatorname{rk}(M)>\operatorname{rk}\bigl{(}\operatorname{Tr}(M)\bigr{)} and consider a pure Tate motive $T$ which is in the motivic splitting pattern of $M$. If $T=\operatorname{Tr}(M)$, again we may take $E=F$, so we assume $T$ has rank strictly larger than $\operatorname{Tr}(M)$. By definition of the motivic splitting pattern, $\operatorname{Tr}(M_{K})$ is isomorphic to $T_{K}$ for some field extension $K/F$. Consider a $p$-splitting family of $M$, $\mathcal{C}_{M}=\{F(X^{0}_{1}),...,F(X^{0}_{r_{0}})\}$, consisting of corestrictions of projective homogeneous varieties given by a decomposition of $M$ as a direct sum of indecomposable summands $M=\sum_{k=1}^{r_{0}}U_{X^{0}_{k}}\{i_{k}\}$. Since $\operatorname{Tr}(M_{K})$ has rank strictly larger than $\operatorname{Tr}(M)$, there is an integer $k_{0}$ such that $X^{0}_{k_{0}}$ is anisotropic and becomes isotropic over $K$, with coefficients in $\Lambda$. As $G$ is $p$-inner, we may write $X_{k_{0}}^{0}$ as a corestriction $\mathrm{corr}_{L/F}(X)$ for some projective $G_{L}$-homogeneous variety $X$, with respect to a finite separable extension $L/F$ of degree a power of $p$. Consider the following diagram of fields

[TABLE]

where $K_{p}$ denotes the $p$-special closure of $K$. Since $X^{0}_{k_{0}}$ is isotropic over $K$, it has a rational point over its $p$-special closure $K_{p}$. The $L$-variety $X$ thus has a rational point over $K_{p}$ and $K_{p}(X^{0}_{k_{0}})$ is purely transcendental over $K_{p}$, by [27, Thm. 3.10]. Therefore, the Tate trace of $M$ is preserved under this extension, and combining with Lemma 4.11, we get that $\operatorname{Tr}(M_{K_{p}(X^{0}_{k_{0}})})=T_{K_{p}(X^{0}_{k_{0}})}$. In particular, $T$ belongs to the splitting pattern of $M_{F(X^{0}_{k_{0}})}$. We get by induction a field extension $E/F(X^{0}_{k_{0}})$ appearing in a splitting tower of $M_{F(X^{0}_{k_{0}})}$ such that $T_{E}$ is isomorphic to the Tate trace of $M_{E}$. Completing this tower of extensions with $F(X^{0}_{k_{0}})/F$, $E$ belongs to a splitting tower of $M$ as well. ∎