2.1 Correspondences of degree $0$

Let ${\mathrm {PSm}}_{k}$ be the category of smooth and projective k-schemes. Given two such schemes X and Y, a correspondence of degree 0 between X and Y is an element $\alpha $ in

(2.1) $$ \begin{align} \bigoplus\limits_{i=1}^{l}{\mathrm{CH}}_{\dim X_{i}}(X_{i}\times_{k} Y)\, =\, \bigoplus\limits_{i=1}^{l}\bigoplus\limits_{j=1}^{m}{\mathrm{CH}}^{\dim Y_{j}}(X_{i}\times_{k}Y_{j}), \end{align} $$

where $X_{1},\ldots ,X_{l}$ are the connected components of X and $Y_{1},\ldots ,Y_{m}$ the ones of Y. We write then $\alpha :X\leadsto Y$ . Given $\alpha :X\leadsto Y$ and $\beta :Y\leadsto Z,$ their composition is defined as

$$ \begin{align*}\beta\circ\alpha\, :=\; p_{XZ\,\ast}\big(\, p_{XY}^{\ast}(\alpha)\cdot p_{YZ}^{\ast}(\beta)\,\big), \end{align*} $$

where $p_{XY},p_{XZ}$ , and $p_{YZ}$ are the respective projections from $X\times _{k}Y\times _{k}Z$ to $X\times _{k}Y$ , $X\times _{k}Z$ , and $Y\times _{k}Z$ . This product is associative and the class of the image of the diagonal morphism acts as identity. Hence, we have an additive category of correspondences of degree 0 over k, denoted ${\mathrm {Corr}}^{0}(k)$ , whose objects are the smooth projective k-schemes, the morphisms are correspondences of degree $0$ , and the direct sum is the disjoint union.

2.2 Chow motives

The idempotent completion of ${\mathrm {Corr}}^{0} (k)$ is the category of (effective) Chow motives over k, denoted $\mathfrak {Chow} (k)$ . The objects of $\mathfrak {Chow} (k)$ are pairs $(X,p)$ , where $p:X\leadsto X$ is a correspondence of degree $0$ satisfying $p\circ p=p$ , i.e., p is an idempotent morphism in ${\mathrm {Corr}}^{0} (k)$ , and the morphisms are given by

$$ \begin{align*}\operatorname{\mathrm{Hom}}_{k}\big( (X,p),(Y,q)\big)\, :=\; q\circ\operatorname{\mathrm{Mor}}_{{\mathrm{Corr}}^{0}(k)}(X,Y)\circ p\;\;\subseteq\operatorname{\mathrm{Mor}}_{{\mathrm{Corr}}^{0}(k)}(X,Y). \end{align*} $$

If X is a smooth projective k-scheme, we denote by the same symbol X its motive in $\mathfrak {Chow} (k)$ . We also set $\operatorname {\mathrm {End}}_{k}(X,p):=\operatorname {\mathrm {Hom}}_{k}((X,p),(X,p))$ . The Cartesian product induces a “tensor product” on $\mathfrak {Chow} (k)$ , denoted $(X,p)\otimes (Y,q)$ .

There is a covariant functor ${\mathrm {PSm}}_{k}\longrightarrow \mathfrak {Chow} (k)$ , which is the identity on objects and sends a morphism $f:X\longrightarrow Y$ to the class of its graph $\Gamma _{f}$ .

The Tate motive and its (nonnegative) twists are denoted by $\underline {\mathbb {Z}} (i):=(\underline {\mathbb {Z}}(1))^{\otimes i}$ , $i\in \mathbb {N}\cup \{ 0\}$ , and if X is a smooth projective k-scheme, we set $X(i):=X\otimes \underline {\mathbb {Z}}(i)$ . Recall that the Tate motive is the complement of the motive of the point in the projective line: $\mathbb {P}^{1}_{k}\simeq \underline {\mathbb {Z}}\oplus \underline {\mathbb {Z}}(1)$ in $\mathfrak {Chow} (k)$ .

A motive is called split if it is a direct sum of twists of Tate motives, and geometrically split if this is the case over the algebraic closure of the base field. Examples of geometrically split motives are the motives of projective quadrics, or more generally of projective homogeneous varieties, see Köck [Reference Köck11], and the motives of geometrically rational surfaces, see, e.g., [Reference Gille8].

Replacing ${\mathrm {CH}}_{i}(\, -\, )$ by ${\mathrm {CH}}_{i}(\, -\, )_{R}:=R\otimes _{\mathbb {Z}}{\mathrm {CH}}_{i}(\, -\, )$ for a commutative ring R (with $1$ ) we get Chow motives with R as ring of coefficients, which we denote by $\mathfrak {Chow} (k,R)$ . In this case, we use $\operatorname {\mathrm {Hom}}_{k}((X,p),(Y,q))_{R}$ and $\operatorname {\mathrm {End}}_{k}(X,p)_{R}$ for the homomorphism groups and the endomorphism rings, respectively, which are R-modules, respectively, R-algebras. The Tate motives and its twists will then be denoted by $\underline {R}$ and $\underline {R}(i)$ , and we set $X(i)=X\otimes \underline {R}(i)$ for $i\geq 0$ .

We fix in the following a commutative coefficient ring R with $1$ .

2.3 The restriction functor

Let $E\supseteq k$ be a field extension. Then, $X\mapsto X_{E}$ and $\alpha \mapsto \alpha _{E}$ induce a restriction morphism

$$ \begin{align*}{\mathrm{res}}_{E/k}\, :\;\operatorname{\mathrm{Hom}}_{k}(M,N)_{R}\,\longrightarrow\operatorname{\mathrm{Hom}}_{E}(M_{E},N_{E})_{R} \end{align*} $$

for $M,N\in \mathfrak {Chow} (k,R)$ . This defines a contravariant functor $\mathfrak {Chow} (k,R)\longrightarrow \mathfrak {Chow} (E,R)$ , called restriction, mapping a motive M in $\mathfrak {Chow} (k,R)$ onto $M_{E}\in \mathfrak {Chow} (E,R)$ . As in the introduction, we denote the kernel of

$$ \begin{align*}{\mathrm{res}}_{E/k}\, :\;\operatorname{\mathrm{End}}_{k}(M)_{R}\,\longrightarrow\,\operatorname{\mathrm{End}}_{E}(M_{E})_{R} \end{align*} $$

by ${\mathcal I}_{E/k}^{R}(M)$ for all $M\in \mathfrak {Chow} (k,R)$ and all field extensions $E\supseteq k$ .

2.4 Chow motives relative to a base

In Section 4.4, we make use of the following generalization of Chow motives over a field, which has been introduced by Vishik and Zainoulline [Reference Vishik and Zainoulline18].

Let V be an essentially smooth scheme over k, i.e., a localization of a smooth k-scheme. The objects of the category of relative correspondences over V with coefficients in R are smooth and projective morphisms $X\longrightarrow V$ . The R-module of morphisms between two objects $X\longrightarrow V$ and $Y\longrightarrow V$ is defined as

$$ \begin{align*}\bigoplus\limits_{j=1}^{m}{\mathrm{CH}}^{d_{j}}(X\times_{V}Y_{j})_{R}, \end{align*} $$

where $Y_{1},\ldots ,Y_{m}$ are the irreducible components of Y and $d_{j}$ is the relative dimension of $Y_{j}$ over V for all $1\leq j\leq m$ . As in ${\mathrm {Corr}}^{0}(k),$ the composition of two morphisms $\alpha : [X\longrightarrow V]\leadsto [Y\longrightarrow V]$ and $\beta :[Y\longrightarrow V]\leadsto [Z\longrightarrow V]$ is defined by

$$ \begin{align*}\beta\circ\alpha\, :=\, p_{XZ\,\ast}\big( p_{XY}^{\ast}(\alpha)\cdot p_{YZ}^{\ast}(\beta)\big), \end{align*} $$

where $p_{XY}$ , $p_{XZ}$ , and $p_{YZ}$ are the projections from $X\times _{V}Y\times _{V}Z$ to $X\times _{V}Y$ , $X\times _{V}Z$ , and $Y\times _{V}Z$ , respectively.

The idempotent completion of this additive category is the category of Chow motives relative to V, denoted $\mathfrak {Chow} (V,R)$ . Clearly, if $V=\operatorname {\mathrm {Spec}} k,$ we get back the category $\mathfrak {Chow} (k,R)$ of Chow motives over k with coefficients in R, see (2.1) for the agreement of the groups of morphisms.

We denote the R-module of morphism in $\mathfrak {Chow} (V,R)$ between $[X\longrightarrow V]$ and $[Y\longrightarrow V]$ by $\operatorname {\mathrm {Hom}}_{V}([X\longrightarrow V],[Y\longrightarrow V])_{R}$ , and the R-algebra of endomorphisms of $[X\longrightarrow V]$ by $\operatorname {\mathrm {End}}_{V}([X\longrightarrow V])_{R}$ .

If $f:W\longrightarrow V$ is a morphism of essentially smooth k-schemes, the map

$$ \begin{align*}[X\longrightarrow V]\,\longmapsto\, [X\times_{V}W\longrightarrow W] \end{align*} $$

induces a restriction functor $\mathfrak {Chow} (V,R)\longrightarrow \mathfrak {Chow} (W,R)$ , which maps a correspondence $\alpha : [X\longrightarrow V]\leadsto [Y\longrightarrow V]$ onto

$$ \begin{align*}(\operatorname{\mathrm{id}}_{X\times_{V}Y}\times f)^{\ast}(\alpha)\in\operatorname{\mathrm{Hom}}_{W}([X_{W}\longrightarrow W],[Y_{W}\longrightarrow W])_{R}. \end{align*} $$

Here, we have (following [Reference Vishik and Zainoulline18]) identified $(X\times _{V}W)\times _{W}(Y\times _{V}W)\simeq X\times _{V}Y\times _{V}W$ .

In particular, we have for every $[X\longrightarrow V]\in \mathfrak {Chow} (V,R),$ an R-algebra homomorphism

$$ \begin{align*}{\mathrm{res}}_{W/V}\, :\;\operatorname{\mathrm{End}}_{V}([X\longrightarrow V])_{R}\,\longrightarrow\,\operatorname{\mathrm{End}}_{W}([X_{W}\longrightarrow W])_{R}, \end{align*} $$

whose kernel is denoted by ${\mathcal I}_{W/V}^{R}([X\longrightarrow V])$ .

The following lemma is due to Vishik and Zainoulline [Reference Vishik and Zainoulline18, Section 3].

Proof (i) Identifying $\operatorname {\mathrm {End}}_{X}([X\times _{k}X\xrightarrow {\pi _{2}} X])_{R}\simeq {\mathrm {CH}}^{\dim X}(X\times _{k}X\times _{k}X)_{R}$ the morphism ${\mathrm {res}}_{X/k}$ is equal to

$$ \begin{align*}\pi_{12}^{\ast}\, :\;{\mathrm{CH}}^{\dim X}(X\times_{k}X)_{R}\,\longrightarrow\,{\mathrm{CH}}^{\dim X}(X\times_{k}X\times_{k}X)_{R}, \end{align*} $$

where $\pi _{12}:X\times _{k}X\times _{k}X\longrightarrow X\times _{k}X$ is the projection onto the first two factors. This morphism is split by the pullback along $\operatorname {\mathrm {id}}_{X}\times \Delta _{X}$ , where $\Delta _{X}:X\longrightarrow X\times _{k}X$ is the diagonal morphism.

Part (ii) is essentially a special case of Vishik and Zainulline [Reference Vishik and Zainoulline18, Lemma 3.2], which does not assert but proves that ${\mathcal I}_{U/X}^{R}([X\times _{k}X\xrightarrow {\pi _{2}} X])$ is a nil ideal of bounded exponent d, where $d-1$ is the largest integer $\leq \frac {\dim X}{{\mathrm {codim}}_{X} (X\setminus U)}$ .

However, the argument in [Reference Vishik and Zainoulline18, Proof of Lemma 3.2] actually shows even more, namely, that we have

$$ \begin{align*}\big[{\mathcal I}_{U/X}^{R}([X\times_{k}X\xrightarrow{\pi_{2}} X])\big]^{d}\, =\, 0. \end{align*} $$

In fact, instead of considering only one element $\phi $ in ${\mathcal I}_{U/X}^{R}([X\times _{k}X\xrightarrow {\pi _{2}} X]),$ one takes d elements $\rho _{1},\ldots ,\rho _{d}$ in this kernel (using the notation of [Reference Vishik and Zainoulline18, Proof of Lemma 3.2]), and sets $\phi _{i}:=\pi _{i,i+1}^{\ast }(\rho _{i})$ , where $\pi _{i,i+1}:X^{\times _{k}\, (d+1)}\longrightarrow X\times _{k}X$ is the projection onto the ith and $(i+1)$ th coordinates. Following then word by word, the rest of the proof gives $\rho _{1}\circ \,\dots \,\circ \rho _{d}=0$ .

3.1

Most proofs of Rost nilpotence rely on a lemma due to Rost [Reference Rost15, Proposition 1], which we recall here in a seemingly more general version. However, this is what is actually proven in [Reference Rost15] as we explain now.

Lemma 3.2 (Rost’s Lemma)

Let X and Y be smooth and projective k-schemes with Y connected, and $\alpha _{0},\ldots ,\alpha _{d}\in \operatorname {\mathrm {End}}_{k}(X)_{R}$ , where $d=\dim Y$ . Assume that

$$ \begin{align*}(\alpha_{i\,k(y)})_{\ast}\big(\,{\mathrm{CH}}_{i}(k(y)\times_{k}X)_{R}\,\big)\; =\, 0 \end{align*} $$

for all $y\in Y$ and $0\leq i\leq \dim X$ . Then,

$$ \begin{align*}(\alpha_{0}\circ\alpha_{1}\circ\,\dots\,\circ\alpha_{d-1}\circ\alpha_{d})_{\ast}\big(\,{\mathrm{CH}}_{j}(Y\times_{k}X)_{R}\big)\, =\, 0 \end{align*} $$

for all $0\leq j\leq \dim X+\dim Y$ . In particular, if $X=Y$ , we have

$$ \begin{align*}\alpha_{0}\circ\alpha_{1}\circ\,\dots\,\circ\alpha_{d-1}\circ\alpha_{d}\, =\, 0. \end{align*} $$

Proof Let $\pi :Y\times _{k}X\longrightarrow Y$ be the projection onto the first factor. Define for $0\leq j\leq \dim X+\dim Y$ a filtration on the Chow group ${\mathrm {CH}}_{j}(Y\times _{k}X)_{R}$ by setting

$$ \begin{align*}F_{-1}{\mathrm{CH}}_{j}(Y\times_{k}X)_{R}\, =\, 0 \end{align*} $$

and letting $F_{p}{\mathrm {CH}}_{j}(Y\times _{k}X)_{R}$ be the subgroup generated by the classes of j-dimensional subvarieties V of $Y\times _{k}X$ with $\dim \pi (V)\leq p$ for $0\leq p\leq d=\dim Y$ .

Now, Rost [Reference Rost15, Proof of Proposition 1], or Brosnan [Reference Brosnan1] for a more geometric argument, show that if $\alpha \in \operatorname {\mathrm {End}}_{k}(X)_{R}$ satisfies

$$ \begin{align*}\alpha_{k(y)\,\ast}\big(\,{\mathrm{CH}}_{i}(k(y)\times_{k}X)_{R}\,\big)\, =\, 0 \end{align*} $$

for all $y\in Y$ and all $0\leq i\leq \dim X,$ then we have

$$ \begin{align*}\alpha_{\ast}\big(\,F_{p}{\mathrm{CH}}_{j}(Y\times_{k}X)_{R}\,\big)\,\subseteq\, F_{p-1}{\mathrm{CH}}_{j}(Y\times_{k}X)_{R} \end{align*} $$

for all $d\geq p\geq 0$ . Hence, the assumption on the $\alpha _{i}$ ’s implies that

$$ \begin{align*}\alpha_{0\,\ast}\Big(\,\alpha_{1\,\ast}\big(\,\ldots\,\alpha_{d\,\ast}({\mathrm{CH}}_{j}(Y\times_{k}X)_{R})\,\ldots\,\big)\,\Big) \;\subseteq\; F_{-1}{\mathrm{CH}}_{j}(Y\times_{k}X)_{R}\, =\, 0 \end{align*} $$

as claimed.

The strong version of Rost nilpotence is “additive.” More precisely, we have the following fact.

Proof By induction, it is enough to show this for $d=2$ . Let for this $M,N\in \mathfrak {Chow} (k,R)$ satisfy strong Rost nilpotence, and $E\supseteq k$ a field extension. We set for brevity of notation ${\mathcal I}_{M}:={\mathcal I}_{E/k}^{R}(M)$ and ${\mathcal I}_{N}:={\mathcal I}_{E/k}^{R}(N)$ . Let further B and C be the kernels of ${\mathrm {res}}_{E/k}:\operatorname {\mathrm {Hom}}_{k}(N,M)_{R}\longrightarrow \operatorname {\mathrm {Hom}}_{E}(N_{E},M_{E})_{R}$ and of ${\mathrm {res}}_{E/k}:\operatorname {\mathrm {Hom}}_{k}(M,N)_{R}\longrightarrow \operatorname {\mathrm {Hom}}_{E}(M_{E},N_{E})_{R}$ , respectively.

Identifying

$$ \begin{align*}\operatorname{\mathrm{End}}_{k}(M\oplus N)_{R}\,\simeq\,\left( \begin{array}{c@{\;\;\;}c} \operatorname{\mathrm{End}}_{k}(M)_{R} & \operatorname{\mathrm{Hom}}_{k}(N,M)_{R} \\[3mm] \operatorname{\mathrm{Hom}}_{k}(M,N)_{R} & \operatorname{\mathrm{End}}_{k}(N)_{R} \end{array}\right), \end{align*} $$

we have

$$ \begin{align*}{\mathcal I}_{E/k}^{R}(M\oplus N)\, =\,\left( \begin{array}{c@{\;\;\;}c} {\mathcal I}_{M} & B \\[3mm] C & {\mathcal I}_{N} \end{array}\right)\, =\, \left( \begin{array}{c@{\;\;\;}c} {\mathcal I}_{M} & 0 \\[3mm] C & 0 \end{array}\right)\, +\, \left( \begin{array}{c@{\;\;\;}c} 0 & B \\[3mm] 0 & {\mathcal I}_{N} \end{array}\right). \end{align*} $$

By assumption, both ${\mathcal I}_{M}$ and ${\mathcal I}_{N}$ are nilpotent ideals, and so

$$ \begin{align*}A_{M}:=\left(\begin{array}{c@{\;\;\;}c} {\mathcal I}_{M} & 0 \\[3mm] C & 0 \end{array}\right)\qquad\mbox{and}\qquad A_{N}:=\left(\begin{array}{c@{\;\;\;}c} 0 & B \\[3mm] 0 & {\mathcal I}_{N} \end{array}\right) \end{align*} $$

are nilpotent subrings (without $1$ ) of the ring ${\mathcal I}_{E/k}^{R}(M\oplus N)$ . Hence, by the main theorem of Kegel [Reference Kegel10], their sum $A_{M}+A_{N}$ , which is the kernel ideal ${\mathcal I}_{E/k}^{R}(M\oplus N)$ , is nilpotent as well.

Proof By the blow-up formula, see Manin [Reference Manin12, Section 9, Corollary] or [Reference Fulton7, Chapter 6], we have

$$ \begin{align*}Z\,\simeq\, X\oplus\;\bigoplus\limits_{i=1}^{r-1}Y(i) \end{align*} $$

in $\mathfrak {Chow} (k,R)$ and so the corollary follows from Theorem 3.3 above.

3.1 The case of bounded exponents

We show now that Rost nilpotence with bounded exponents is also “additive,” i.e., the analog of Theorem 3.3 holds for this slightly weaker version of Rost nilpotence.

Let $M,N$ be motives in $\mathfrak {Chow} (k,R)$ , which satisfy Rost nilpotence with bounded exponent, and $E\supseteq k$ a field extension. Let further $r=r(E)$ and $s=s(E)$ be integers $\geq 1$ , such that $x^{r+1}=0$ and $y^{s+1}=0$ for all $x\in {\mathcal I}_{E/k}^{R}(M)$ and all $y\in {\mathcal I}_{E/k}^{R}(N)$ , respectively. In particular, this means that the R-algebras (without $1$ ) ${\mathcal I}_{E/k}^{R}(M)$ and ${\mathcal I}_{E/k}^{R}(N)$ are PI-algebras.

We aim to show that then ${\mathcal I}_{E/k}^{R}(M\oplus N)$ is also a nil ideal with bounded nilpotence index, i.e., there exists $t=t(E)\geq 1$ , such that $\alpha ^{t}=0$ for all $\alpha $ in ${\mathcal I}_{E/k}^{R}(M\oplus N)$ .

Given $\alpha \in {\mathcal I}_{E/k}^{R}(M\oplus N),$ we can write

(3.1) $$ \begin{align} \alpha\, =\,\left( \begin{array}{c@{\;\;\;}c} x & b \\[2mm] c & y \end{array}\right) \end{align} $$

with $x\in {\mathcal I}_{E/k}^{R}(M)$ , $y\in {\mathcal I}_{E/k}^{R}(N)$ , and $b\in \operatorname {\mathrm {Hom}}_{k}(N,M)_{R}$ and $c\in \operatorname {\mathrm {Hom}}_{k}(M,N)_{R}$ satisfying $b_{E}=c_{E}=0$ .

Let ${\mathcal I}_{M}$ be the $\mathbb {Z}$ -subalgebra (without $1$ !) of ${\mathcal I}_{E/k}^{R}(M)$ generated by the elements $x,x^{2},\ldots ,x^{r}$ , $bc,byc,by^{2}c,\ldots ,by^{s}c$ , and ${\mathcal I}_{N}$ be the $\mathbb {Z}$ -subalgebra (without $1$ !) of ${\mathcal I}_{E/k}^{R}(N)$ generated by the elements $y,y^{2},\ldots ,y^{s}$ , $cb,cxb,cx^{2}b,\ldots ,cx^{r}b$ . Both ${\mathcal I}_{M}$ and ${\mathcal I}_{N}$ are finitely generated subalgebras of the PI-algebras ${\mathcal I}_{E/k}^{R}(M)$ and ${\mathcal I}_{E/k}^{R}(N)$ , respectively, and therefore by [Reference Procesi14, Chapter VI, Theorem 2.13], there exist integers $u,v\geq 0$ , such that ${\mathcal I}_{M}^{u}=0$ and ${\mathcal I}_{N}^{v}=0$ . The integers u and v do not depend on $x,y,b$ , and c but only on the number of generators of ${\mathcal I}_{M}$ and ${\mathcal I}_{N}$ , respectively, and the nilpotence indices r and s, hence on r and s only.

We define now two abelian groups. Let for this $b,c$ be as in (3.1). The first abelian group, denoted B, is the subgroup of $\operatorname {\mathrm {Hom}}_{k}(N,M)_{R}$ generated by the set of morphisms

$$ \begin{align*}\Big\{\, b,\, fb,\, bg,\, fbg\;\big|\; f\in{\mathcal I}_{M}\, ,\; g\in{\mathcal I}_{N}\,\Big\}, \end{align*} $$

and the second one, denoted C, is the subgroup of $\operatorname {\mathrm {Hom}}_{k}(M,N)_{R}$ generated by the set of morphisms

$$ \begin{align*}\Big\{\, c,\, cf,\, gc,\, gcf\;\big|\; f\in{\mathcal I}_{M}\, ,\; g\in{\mathcal I}_{N}\,\Big\}. \end{align*} $$

Given motives $M_{i}$ , $i=1,2,3$ , in $\mathfrak {Chow} (k,R)$ and subsets $X\subseteq \operatorname {\mathrm {Hom}}_{k}(M_{1},M_{2})_{R}$ and $Y\in \operatorname {\mathrm {Hom}}_{k}(M_{2},M_{3})_{R}$ , we write $YX$ for the abelian subgroup of $\operatorname {\mathrm {Hom}}_{k}(M_{1},M_{3})_{R}$ generated by all $y\cdot x$ with $x\in X$ and $y\in Y$ .

Proof Part (i) is by definition of B and C. For (ii) and (iii), we note that an element of ${\mathcal I}_{M}$ is a sum of elements of the form

(3.2) $$ \begin{align} x^{i_{1}}\,\cdot\,(\prod\limits_{l=1}^{m_{1}}by^{i_{1l}}c)\,\cdot\, x^{i_{2}}\,\cdot\, (\prod\limits_{l=1}^{m_{2}}by^{i_{2l}}c)\,\cdot\,\ldots\,\cdot\, (\prod\limits_{l=1}^{m_{m}}by^{i_{ml}}c)\,\cdot\, x^{i_{m+1}}, \end{align} $$

where $m,m_{l}\geq 1$ are integers and $i_{l}$ and $i_{hl}$ are integers $\geq 0$ . Analogously, an element of ${\mathcal I}_{N}$ is a sum of elements of the form

(3.3) $$ \begin{align} y^{j_{1}}\,\cdot\,(\prod\limits_{l=1}^{n_{1}}cx^{j_{1l}}b)\,\cdot\, y^{j_{2}}\,\cdot\, (\prod\limits_{l=1}^{n_{2}}cx^{j_{2l}}b)\,\cdot\,\ldots\,\cdot\, (\prod\limits_{l=1}^{n_{n}}cx^{j_{nl}}b)\,\cdot\, y^{i_{n+1}}, \end{align} $$

where $n,n_{l}\geq 1$ are integers and $j_{l}$ and $j_{hl}$ are integers $\geq 0$ .

If now $a\in {\mathcal I}_{M}$ is of the form (3.2), then $c\cdot a\cdot b$ is an element of the form (3.3) and so in ${\mathcal I}_{N}$ , and analogous if $d\in {\mathcal I}_{N}$ is an element of the form (3.3), then $b\cdot d\cdot c$ is of the form (3.2) and so in ${\mathcal I}_{M}$ . From these two observations, the sublemma follows.

We come back to the element $\alpha $ in ${\mathcal I}_{E/k}^{R}(M\oplus N)$ (see (3.1)). We want to show that there exists an integer $t=t(E)\geq 0$ depending only on the nilpotence indices u and v of ${\mathcal I}_{M}$ and ${\mathcal I}_{N}$ , respectively, such that $\alpha ^{t}=0$ .

We claim that $t:=2^{1+\max \{ u,v\}}$ does the job. To this end, we compute first

$$ \begin{align*}\alpha^{2}\, =\,\left( \begin{array}{c@{\;\;\;}c} x & b \\[2mm] c & y \end{array}\right)^{2}\, =\,\left( \begin{array}{c@{\;\;\;}c} x^{2}+bc & xb+by \\[2mm] cx+yc & cb+y^{2} \end{array}\right), \end{align*} $$

from which we conclude that

$$ \begin{align*}\alpha^{4}\;\,\in\;\left( \begin{array}{c@{\;\;\;}c} {\mathcal I}_{M}^{2}+B{\mathcal I}_{N}^{2}C & {\mathcal I}_{M}^{2}B+B{\mathcal I}_{N}^{2}+{\mathcal I}_{M}B{\mathcal I}_{N}\\[2mm] C{\mathcal I}_{M}^{2}+{\mathcal I}_{N}^{2}C +{\mathcal I}_{N}C{\mathcal I}_{M}& C{\mathcal I}_{M}^{2}B+{\mathcal I}_{N}^{2} \end{array}\right). \end{align*} $$

Now, a straightforward induction using the identities of Sublemma 3.6 shows that

$$ \begin{align*}\alpha^{2^{l}}\;\,\in\left( \begin{array}{c@{\;\;\;}c} {\mathcal I}_{M}^{l}+B{\mathcal I}_{N}^{l}C & {\mathcal I}_{M}^{l}B+B{\mathcal I}_{N}^{l}+{\mathcal I}_{M}^{l-1}B{\mathcal I}_{N}^{l-1}\\[2mm] C{\mathcal I}_{M}^{l}+{\mathcal I}_{N}^{l}C +{\mathcal I}_{N}^{l-1}C{\mathcal I}_{M}^{l-1}& C{\mathcal I}_{M}^{l}B+{\mathcal I}_{N}^{l} \end{array}\right) \end{align*} $$

for all $l\geq 2$ . Therefore, we have $\alpha ^{2^{l}}=0$ for all $l\geq 1+\max \{ u,v\}$ , as claimed.

We have shown that ${\mathcal I}_{E/k}^{R}(M\oplus N)$ is a nil ideal with bounded exponent. By another induction on the number of summands, this gives the next result.

Proof Recall first that an R-algebra A is said to be integral of bounded degree if there exists an integer $u\geq 1$ , such that for every $a\in A$ exists a monic polynomial $f(T)\in R[T]$ of degree $\leq u$ with $f(a)=0$ .

We now prove the corollary. By assumption, there exist geometrically split motives $N_{i}$ , $1\leq i\leq l$ , which satisfy Rost nilpotence with bounded exponent, such that M is direct summand of $N:=\bigoplus \limits _{i=1}^{l}N_{i}$ . Since $N_{\bar {k}}$ is split, its endomorphism ring is isomorphic to a product of matrix rings over R, say $\operatorname {\mathrm {End}}_{\bar {k}}(N_{\bar {k}})_{R}\simeq \prod \limits _{j=1}^{m}{\mathrm {M}}_{s_{j}}(R)$ for some integer $m\geq 1$ and positive integers $s_{j}$ , $1\leq j\leq m$ . Therefore, by the Cayley–Hamilton theorem, there exists, for every $a\in \operatorname {\mathrm {End}}_{\bar {k}}(N_{\bar {k}})_{R}$ , a monic polynomial $f(T)\in R[T]$ (depending on a) with $f(a)=0$ and $\deg f(T)\leq \sum \limits _{j=1}^{m}s_{j}=:v$ .

By assumption and the theorem above, the kernel ${\mathcal I}_{\bar {k}/k}^{R}(N)$ of

$$ \begin{align*}{\mathrm{res}}_{\bar{k}/k}\, :\;\operatorname{\mathrm{End}}_{k}(N)_{R}\longrightarrow\operatorname{\mathrm{End}}_{\bar{k}}(N_{\bar{k}})_{R} \end{align*} $$

is nil of bounded exponent, say u, and so for every $x\in \operatorname {\mathrm {End}}_{k}(N)_{R}$ , there is a monic polynomial $f(T)$ with R-coefficients of degree $\leq uv$ , such that $f(x)=0$ . Hence, $\operatorname {\mathrm {End}}_{k}(N)_{R}$ and so also its R-subalgebra $\operatorname {\mathrm {End}}_{k}(M)_{R}$ are integral R-algebras of bounded degree. Now, [Reference Procesi14, Chapter I, Theorem 3.22 and Chapter VI, Corollary 2.8(1)] implies that these algebras are PI-algebras over R, which are locally finite if R is noetherian.

We close this section with a technical lemma needed in the next one.

Proof By Remark 3.9 (i) above, it is enough to show that ${\mathcal I}_{\bar {k}/k}^{R}(M)$ is a nilpotent ideal.

By assumption, we have ${\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(M)^{t}=0$ for some $t\geq 1$ and there is a short exact sequence

$$ \begin{align*}0\longrightarrow\,{\mathcal I}_{\bar{k}/k}^{\mathbb{Z}}(M)\,\longrightarrow\,\operatorname{\mathrm{End}}_{k}(M)\,\xrightarrow{\;\iota\;}\,\overline{\operatorname{\mathrm{End}}_{k}(M)}\,\longrightarrow 0, \end{align*} $$

where $\overline {\operatorname {\mathrm {End}}_{k}(M)}$ denotes the image of ${\mathrm {res}}_{\bar {k}/k}:\operatorname {\mathrm {End}}_{k}(M)\longrightarrow \operatorname {\mathrm {End}}_{\bar {k}}(M_{\bar {k}})$ . Tensoring above exact sequence with $R,$ we get an exact sequence

(3.4) $$ \begin{align} R\otimes_{\mathbb{Z}}{\mathcal I}_{\bar{k}/k}^{\mathbb{Z}}(M)\,\longrightarrow\,R\otimes_{\mathbb{Z}}\operatorname{\mathrm{End}}_{k}(M)\,\xrightarrow{\;\operatorname{\mathrm{id}}_{R}\otimes\,\iota\;} \, R\otimes_{\mathbb{Z}}\overline{\operatorname{\mathrm{End}}_{k}(M)}\,\longrightarrow 0, \end{align} $$

and consequently the kernel of $\operatorname {\mathrm {id}}_{R}\otimes \,\iota $ is a nilpotent ideal. We are reduced to prove that the kernel of

$$ \begin{align*}\operatorname{\mathrm{id}}_{R}\otimes j\, :\; R\otimes_{\mathbb{Z}}\overline{\operatorname{\mathrm{End}}_{k}(M)}\,\longrightarrow\, R\otimes_{\mathbb{Z}}\operatorname{\mathrm{End}}_{\bar{k}}(M_{\bar{k}}) \end{align*} $$

is a nilpotent ideal, where j denotes the inclusion map $\overline {\operatorname {\mathrm {End}}_{k}(M)}\hookrightarrow \operatorname {\mathrm {End}}_{\bar {k}}(M_{\bar {k}})$ .

Since M is geometrically split, the ring $\operatorname {\mathrm {End}}_{\bar {k}}(M_{\bar {k}})$ is a free abelian group of finite rank and so the subgroup $\overline {\operatorname {\mathrm {End}}_{k}(M)}$ is free abelian of finite rank as well. Hence, $R\otimes _{\mathbb {Z}}\overline {\operatorname {\mathrm {End}}_{k}(M)}$ is a free R-module of finite rank and therefore a noetherian R-algebra. Hence, by Levitzki’s theorem, see, e.g., [Reference Rowen16, Theorem 2.6.23], it is enough to show that the kernel of $\operatorname {\mathrm {id}}_{R}\otimes j$ is a nil ideal. To verify this, let $\alpha \in \operatorname {\mathrm {Ker}} (\operatorname {\mathrm {id}}_{R}\otimes j)$ . We have by the exact sequence (3.4) that $\alpha =(\operatorname {\mathrm {id}}_{R}\otimes \,\iota )(\beta )$ for some $\beta $ in the endomorphism ring $\operatorname {\mathrm {End}}_{k}(M)_{R}=R\otimes _{\mathbb {Z}}\operatorname {\mathrm {End}}_{k}(M)$ .

But $\beta $ is in the kernel of the restriction homomorphism

$$ \begin{align*}R\otimes_{\mathbb{Z}}\operatorname{\mathrm{End}}_{k}(M)\,\longrightarrow\, R\otimes_{\mathbb{Z}}\operatorname{\mathrm{End}}_{\bar{k}}(M_{\bar{k}}), \end{align*} $$

and so nilpotent by our assumption. It follows that $\alpha $ is nilpotent as well.

4.1 Projective homogenous varieties

Let X be a projective homogenous variety over k. In [Reference Chernousov, Gille and Merkurjev3], see also Brosnan [Reference Brosnan2], it is shown that Rost nilpotence with bounded exponent holds for the motive of X in $\mathfrak {Chow} (k,R)$ (any coefficient ring R). But the argument there shows that in fact strong Rost nilpotence holds for X. For the sake of completeness, we recall the details.

Note that since the motive of X is geometrically split, it is enough to show that the ideal ${\mathcal I}_{\bar {k}/k}^{R}(X)$ is nilpotent (see Remark 3.9 (i)).

By the main result of [Reference Chernousov, Gille and Merkurjev3], we have

(4.1) $$ \begin{align} X\,\simeq\,\bigoplus\limits_{i=1}^{l}Y_{i}(n_{i}) \end{align} $$

in $\mathfrak {Chow} (k,R)$ for anisotropic projective homogeneous varieties $Y_{i}$ of dimension strictly less than $\dim X$ if $l\geq 2$ , and integers $n_{i}\geq 0$ .

We claim that given a field k and a projective homogeneous k-variety X for some semisimple algebraic group G over $k,$ then ${\mathcal I}_{\bar {k}/k}^{R}(X)$ is a nilpotent ideal with nilpotence exponent only depending on $\dim X$ and the number l of summands in (4.1). We prove this by descending induction on l.

If l is maximal, then X is split and so a sum of twists of Tate motives. Otherwise, let $1\leq i\leq l$ and $y\in Y_{i}$ . Then, $Y_{i,k(y)}$ is isotropic and therefore splits into a direct sum of at least two twists of motives of projective homogeneous varieties. The induction assumption gives then ${\mathcal I}_{\overline {k(y)}/k(y)}^{R}(X_{k(y)})^{\, t}=0$ , where $\overline {k(y)}$ is an algebraic closure of the residue field $k(y)$ , for some integer $t\geq 0,$ which depends only on $\dim X$ and the number of summands in the motivic decomposition of $X_{k(y)}$ into twists of motives of projective homogeneous varieties. In particular, we can find $t_{i}>0$ which works for all $y\in Y_{i}$ , and which depends only on $\dim X=\dim X_{k(y)}$ and the (minimal) number of summands in the motivic decompositions of $X_{k(y)}$ , $y\in Y_{i}$ . Rost’s Lemma 3.2 then implies

$$ \begin{align*}{\mathcal I}_{\bar{k}/k}^{R}(X)^{t_{i}\cdot (1+\dim Y_{i})}_{\ast}\big({\mathrm{CH}}_{j}(Y_{i}\times X)_{R}\big)\; =\, 0 \end{align*} $$

for all $0\leq j\leq \dim X+\dim Y_{i}$ .

Finally, setting $\ell :=\max \limits _{1\leq i\leq d}\big [ t_{i}\cdot (1+\dim Y_{i})\big ]$ , we get ${\mathcal I}_{\bar {k}/k}^{R}(X)^{\ell }=0$ by (4.1).

4.2 Geometrically rational surfaces

Let S be a geometrically rational surface over k. We claim that strong Rost nilpotence holds for S in $\mathfrak {Chow} (k,R)$ for R either the ring of integers $\mathbb {Z}$ or integers modulo $m\geq 2$ , denoted $\mathbb {Z}/m$ . Note that the motive of S is geometrically split.

We first assume that $R=\mathbb {Z}$ . By the Hochschild–Serre spectral sequence, we know that ${\mathrm {Pic}} (S_{k(s)})\simeq {\mathrm {CH}}_{1}(S_{k(s)})$ is torsion free for all $s\in S$ and the same holds for ${\mathrm {CH}}_{2}(S_{k(s)}),$ since S is geometrically integral. Hence, if $\alpha $ is in ${\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(S),$ then $\alpha _{k(s)}$ acts trivial on both ${\mathrm {CH}}_{1}(S_{k(s)})$ and ${\mathrm {CH}}_{2}(S_{k(s)})$ for all $s\in S$ . Hence, to apply Rost’s lemma, we are left to study the action of $\alpha _{k(s)}$ on ${\mathrm {CH}}_{0}(S_{k(s)})$ for all $\alpha \in {\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(S)$ and all $s\in S$ .

To this end, we denote for a field extension $E\supseteq k$ by ${\mathrm {A}}_{0}(S_{E})$ the kernel of the degree map $\deg \, :\;{\mathrm {CH}}_{0}(S_{E})\longrightarrow \mathbb {Z}$ . By the main result of Coombes [Reference Coombes5], the surface S is rational over the separable closure $k_{\mathrm {{sep}}}$ of k and so its motive is split in $\mathfrak {Chow} (k_{\mathrm {{sep}}})$ . Hence, there exists for a field extension $E\supseteq k$ a Galois extension $L\supseteq E$ , such that $(\alpha _{E})_{L}=0$ . If $S(E)\not =\emptyset ,$ the subgroup ${\mathrm {A}}_{0}(S_{E})$ is the torsion part of ${\mathrm {CH}}_{0}(S_{E})$ .

This implies that given $\alpha \in {\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(S),$ we have $\alpha _{k(s)\,\ast }\big ({\mathrm {CH}}_{0}(S_{k(s)})\big )\subseteq {\mathrm {A}}_{0}(S_{k(s)})$ . and by [Reference Gille8, Corollary 4.9] (see [Reference Gille9, Section 2.4] if the base field is not perfect), we have $\alpha _{k(s)\,\ast }\big ({\mathrm {A}}_{0}(S_{k(s)})\big )=0$ .

Given now $\alpha ,\beta \in {\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(S)$ then $(\alpha \circ \beta )_{k(s)\,\ast }\big ({\mathrm {CH}}_{0}(S_{k(s)})\big )=0$ and so by Rost’s Lemma 3.2, we have

$$ \begin{align*}\alpha_{1}\circ\alpha_{2}\circ\alpha_{3}\circ\alpha_{4}\circ\alpha_{5}\circ\alpha_{6}\, =\, 0 \end{align*} $$

for all $\alpha _{1},\ldots ,\alpha _{6}\in {\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(S)$ . Hence, ${\mathcal I}_{\bar {k}/k}^{\mathbb {Z}}(S)^{6}=0$ .

Let now $R=\mathbb {Z}/m$ for some integer $m\geq 2$ . In [Reference Gille9, Theorem 15], it is shown that Rost nilpotence holds for S in $\mathfrak {Chow} (k,\mathbb {Z}/m)$ . Hence, by Lemma 3.10 above, we get that strong Rost nilpotence holds for S in $\mathfrak {Chow} (k,\mathbb {Z}/m)$ .

4.3 Surfaces

Assume that $\operatorname {\mathrm {char}} k=0$ and $R=\mathbb {Z}$ . Let S be a k-surface, i.e., a smooth projective integral k-scheme of dimension $2$ . We claim that strong Rost nilpotence holds for S in $\mathfrak {Chow} (k)$ . Again the proof is only a slight modification of the verification of Rost nilpotence for surfaces in [Reference Gille9, Section 2.5]. We briefly sketch the details.

Let for this $E\supseteq k$ be a field extension and $\alpha _{i}\in {\mathcal I}_{E/k}^{\mathbb {Z}}(S)$ , $i=1,2,3$ . We have a tower of fields $E\supseteq F\supseteq k$ with F a purely transcendental extension of k and E an algebraic extension of F. Since ${\mathrm {res}}_{F/k}:\operatorname {\mathrm {End}}_{k}(S)\longrightarrow \operatorname {\mathrm {End}}_{F}(S_{F})$ is an isomorphism, see, e.g., [Reference Flenner, O’Carrol and Vogel6, Proposition 2.1.8], replacing k by $F,$ we can assume that $E\supseteq k$ is algebraic.

Let $s\in S$ . Since $\alpha _{i\, E(s)}=0$ and $\operatorname {\mathrm {char}} k=0,$ there is a Galois extension $L\supseteq k(s)$ , such that $\alpha _{i\, L}=0$ for all $1\leq i\leq 3$ . It follows from [Reference Gille9, Theorem 4] that we have then

$$ \begin{align*}(\alpha_{1}\circ\alpha_{2}\circ\alpha_{3})_{k(s)}\, =\, 0. \end{align*} $$

Taking into account Rost’s Lemma 3.2, this implies

$$ \begin{align*}\alpha_{1}\circ\alpha_{2}\circ\,\dots\,\circ\alpha_{8}\circ\alpha_{9}\, =\, 0 \end{align*} $$

for all $\alpha _{1},\ldots ,\alpha _{9}\in {\mathcal I}_{E/k}^{\mathbb {Z}}(S)$ , and so $[{\mathcal I}_{E/k}^{\mathbb {Z}}(S)]^{9}=0$ .

4.4 Varieties whose motives split generically

Let k be again an arbitrary field and X a smooth projective and integral k-scheme. We say that the motive of X is generically split in $\mathfrak {Chow} (k,R)$ if it is split in $\mathfrak {Chow} (k(X),R)$ . Vishik and Zainoulline [Reference Vishik and Zainoulline18] have shown that for such schemes Rost nilpotence holds (for any coefficient ring R). But their argument shows that actually strong Rost nilpotence holds in this case.

Let for this $E\supseteq k$ be a field extension. Since by assumption, the motive of $X_{k(X)}$ is split, ${\mathrm {res}}_{E(X)/k(X)}:\operatorname {\mathrm {End}}_{k(X)}(X_{k(X)})_{R}\longrightarrow \operatorname {\mathrm {End}}_{E(X)}(X_{E(X)})_{R}$ is injective, cf. Remark 3.1 (i), and hence it is enough to show that ${\mathcal I}_{k(X)/k}^{R}(X)$ is nilpotent.

That ${\mathcal I}_{k(X)/k}^{R}(X)$ is nilpotent is true for all integral $X\in {\mathrm {PSm}}_{k}$ and not only for smooth projective and integral schemes whose motive is generically split in $\mathfrak {Chow} (k,R)$ . In fact, given an integral smooth and projective k-scheme $X,$ the morphism ${\mathrm {res}}_{k(X)/k}:\operatorname {\mathrm {End}}_{k}(X)_{R}\longrightarrow \operatorname {\mathrm {End}}_{k(X)}(X_{k(X)})_{R}$ factors

$$ \begin{align*}\operatorname{\mathrm{End}}_{k}(X)_{R}\xrightarrow{{\mathrm{res}}_{X/k}}\operatorname{\mathrm{End}}_{X}([X\times_{k}X\longrightarrow X])_{R}\xrightarrow{{\mathrm{res}}_{U/X}} \operatorname{\mathrm{End}}_{U}([X\times_{k}U\longrightarrow U])_{R}\quad \end{align*} $$

(4.2) $$ \begin{align} \qquad\qquad\qquad\qquad\xrightarrow{{\mathrm{res}}_{k(X)/U}}\operatorname{\mathrm{End}}_{k(X)}(X_{k(X)})_{R} \end{align} $$

for every open $U\subseteq X$ . The map ${\mathrm {res}}_{X/k}$ is injective by Lemma 2.1 (i), and by part (ii) of this lemma, we have $\big [\operatorname {\mathrm {Ker}} ({\mathrm {res}}_{U/X})\big ]^{1+\dim X}=0$ . Now,

$$ \begin{align*}\operatorname{\mathrm{End}}_{k(X)}(X_{k(X)})_{R}\,\simeq\,{\mathrm{CH}}^{\dim X}(X\times_{k}X\times_{k}k(X))_{R} \end{align*} $$

is isomorphic to the direct limit

$$ \begin{align*}\lim\limits_{\begin{array}{c} U\subseteq X \\ \mbox{open}\end{array}}{\mathrm{CH}}^{\dim X}(X\times_{k}X\times_{k}U)_{R}. \end{align*} $$

Since ${\mathrm {CH}}^{\dim X}(X\times _{k}X\times _{k}U)\simeq \operatorname {\mathrm {End}}_{U}([X\times _{k}U\longrightarrow U])_{R}$ , we conclude from (4.2) the following remarkable fact.

Theorem 4.1 (Vishik–Zainoulline)

Let X be an integral scheme in ${\mathrm {PSm}}_{k}$ . Then, ${\mathcal I}_{k(X)/k}^{R}(X)$ is a nilpotent ideal for all coefficient rings R. More precisely, we have

$$ \begin{align*}\big[{\mathcal I}_{k(X)/k}^{R}(X)\big]^{1+\dim X}\, =\, 0. \end{align*} $$

If moreover the motive of X is generically split in $\mathfrak {Chow} (k,R),$ then strong Rost nilpotence holds for X.