Motivic decompositions of projective homogeneous varieties and change of coefficients

Charles De Clercq^{a},

aUniversité Paris VI, Equipe Topologie et géométrie algébriques, 4 place Jussieu 75252 Paris, France

Received *****; accepted after revision +++++ Presented by

Abstract

We prove that under some assumptions on an algebraic group G, indecomposable direct summands of the motive of a projective G-homogeneous variety with coefficients in 𝔽_{p} remain indecomposable if the ring of coefficients is any field of characteristic p. In particular for any projective G-homogeneous variety X, the decomposition of the motive of X in a direct sum of indecomposable motives with coefficients in any finite field of characteristic p corresponds to the decomposition of the motive of X with coefficients in 𝔽_{p}. We also construct a counterexample to this result in the case where G is arbitrary.

To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Résumé

Décompositions motiviques des variétés projectives homogènes et changement des coefficients. Nous prouvons que sous certaines hypothèses sur un groupe algébrique G, tout facteur direct indécomposable du motif associé à une variété projective G-homogène à coefficients dans 𝔽_{p} demeure indécomposable si l’anneau des coefficients est un corps de caractéristique p. En particulier pour toute variété projective G-homogène X, la décomposition du motif de X comme somme directe de motifs indécomposables à coefficients dans tout corps fini de caractéristique p correspond à la décomposition du motif de X à coefficients dans 𝔽_{p}. Nous exhibons de plus un contre-exemple à ce résultat dans le cas où le groupe G est quelconque.

Pour citer cet article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Introduction

Let F be a field, Λ be a commutative ring, CM(F; Λ) be the category of Grothendieck Chow motives with coefficients in Λ, G a semi-simple affine algebraic group and X a projective G-homogeneous F -

Email address: de.clercq.charles@gmail.com (Charles De Clercq).

variety. The purpose of this note is to study the behaviour of the complete motivic decomposition (in a direct sum of indecomposable motives) of X ∈ CM(F; Λ) when changing the ring of coefficients. In the first part we prove some very elementary results in non-commutative algebra and find sufficient conditions for the tensor product of two connected rings to be connected. In the second part we show that under some assumptions on G, indecomposable direct summands of X in CM(F; 𝔽_{p}) remain indecomposable if the ring of coefficients is any field of characteristic p (Theorem 2.1), since these conditions hold for the reduced endomorphism ring of indecomposable direct summands of X. In particular theorem 2.1 implies that the complete decomposition of the motive of X with coefficients in any finite field of characteristic p corresponds to the complete decomposition of the motive of X with coefficients in 𝔽_{p}. Finally we show that theorem 2.1 doesn’t hold for arbitrary G by producing a counterexample.

Let Λ be a commutative ring. Given a field F, an F -variety will be understood as a separated scheme of finite type over F. Given such Λ and an F -variety X, we can consider CH_{i}(X; Λ), the Chow group of idimensional cycles on X modulo rational equivalence with coefficients in Λ, defined as CH_{i}(X)⊗_{ℤ}Λ. These groups are the first step in the construction of the category CM(F; Λ) of Grothendieck Chow motives with coefficients in Λ. This category is constructed as the pseudo-abelian envelope of the category CR(F; Λ) of correspondences with coefficients in Λ. Our main reference for the construction and the main properties of these categories is [2]. For a field extension E/F and any correspondence α ∈ CH(X ×Y; Λ) we denote by α_{E} the pull-back of α along the natural morphism (X × Y)_{E} → X × Y. Considering a morphism of commutative rings φ : Λ ⟶ Λ′ we define the two following functors. The change of base field functor is the additive functor res_{E/F} : CM(F; Λ) ⟶ CM(E; Λ) which maps any summand (X, π)[i] ∈ CM(F; Λ) to (X_{E}, π_{E})[i] and any morphism α : (X, π)[i] → (Y, ρ)[j] to α_{E}. The change of coefficents functor is the additive functor coeff_{Λ′/Λ} : CM(F; Λ) ⟶ CM(F; Λ′) which maps any summand (X, π)[i] to (X, (id ⊗ φ)(π))[i] and any morphism α : (X, π)[i] → (Y, ρ)[j] to (id ⊗ φ)(α).

Acknowledgements I am very grateful to Nikita Karpenko for his suggestions and his support during this work. I also would like to thank François Petit and Maksim Zhykhovich. Finally I am grateful to the referee for the useful remarks.

1. On the tensor product of connected rings

Recall that a ring A is connected if there are no idempotents in A besides 0 and 1. Proposition 1.1 Let A be a finite and connected ring. Then any element a in A is either nilpotent or invertible. The set N of nilpotent elements in A is a two-sided and nilpotent ideal.

In order to prove Proposition 1.1 we will need the following elementary lemma. Lemma 1.2 Let A be a finite ring. An appropriate power of any element a of A is idempotent. Proof. For any a ∈ A, the set {a^{n}, n ∈ ℕ} is finite, hence there is a couple (p, k) ∈ ℕ^{2} (with k non-zero) such that a^{p} = a^{p+k}. The sequence (a^{n})_{n≥p} is k-periodic and for example if s is the lowest integer such that p < sk, a^{sk} is idempotent.

Proof of Proposition 1.1. For any a ∈ A, an appropriate power of a is an idempotent by lemma 1.2. Since A is connected, this power is either 0 or 1, that is to say a is either nilpotent or invertible.

We now show that the set N of nilpotent elements in A is a two-sided ideal. First if a is nilpotent in A, then for any b in A, ab and ba are not invertible, hence ab and ba belong to N.

It remains to show that the sum of two nilpotent elements in A is nilpotent. Setting ν for the number of nilpotent elements in A, we claim that for any sequence a_{1},…, a_{ν} in N, a_{1}...a_{ν} = 0. Indeed if a_{ν+1} is any nilpotent in A the finite sequence Π_{1} = a_{1}, Π_{2} = a_{1}a_{2},…, Π_{ν+1} = a_{1}a_{2}...a_{ν+1} consists of nilpotents and by the pigeon-hole principle Π_{k} = Π_{s}, for some k and s satisfying 1 ≤ k < s ≤ ν + 1. Therefore Π_{s} = Π_{k}a_{k+1}...a_{s} = Π_{k} which implies that Π_{k}(1 − a_{k+1}...a_{s}) = 0 and Π_{k} = 0 since 1 − a_{k+1}...a_{s} is invertible. With this in hand it is clear that for any a and b in N, (a + b)^{ν} = 0. Furthermore N^{ν} = 0 and N is nilpotent. Corollary 1.3 Let A be a finite and connected 𝔽_{p}-algebra endowed with a ring morphism φ : A ⟶ 𝔽_{p}. Then the set N of nilpotent elements in A is precisely ker(φ). Furthermore for any connected 𝔽_{p}-algebra E, A ⊗_{𝔽_{p}} E is connected. Proof. For any a ∈ N and n ∈ ℕ^{∗} such that a^{n} = 0, 0 = φ(a^{n}) = φ(a)^{n}, hence a lies in the kernel of φ. On the other hand if φ(a) = 0, a is not invertible thus a is nilpotent and N = ker(φ). Since N is nilpotent, N ⊗ E is also nilpotent. The sequence

\xymatrix{0 \ar[r] & N ⊗ E \ar[r] & A ⊗ E \ar[r]^{ψ} & E \ar[r] & 0}

is exact and we want to show that any idempotent P in A ⊗_{𝔽_{p}} E is either 0 or 1. Since E is connected, ψ(P) is either 0 or 1. We may replace P by 1 − P and so assume that P lies in the kernel of ψ, which implies that the idempotent P is nilpotent, hence P = 0.

2. Application to motivic decompositions of projective homogeneous varieties

For any semi-simple affine algebraic group G, the full subcategory of CM(F; Λ) whose objects are finite direct sums of twists of direct summands of the motives of projective G-homogeneous F -varieties will be denoted CM_{G}(F; Λ). We now use corollary 1.3 to study how motivic decompositions in CM_{G}(F; Λ) behave when extending the ring of coefficients. A pseudo-abelian category C satisfies the Krull-Schmidt principle if the monoid (ℭ, ⊕) is free, where ℭ denotes the set of the isomorphism classes of objects of C.

In the sequel Λ will be a connected ring and X an F -variety. A field extension E/F is a splitting field of X if the E-motive X_{E} is isomorphic to a finite direct sum of twists of Tate motives. The F -variety X is geometrically split if X splits over an extension of F, and X satisfies the nilpotence principle, if for any field extension E/F the kernel of the morphism res_{E/F} : End(M(X)) ⟶ End(M(X_{E})) consists of nilpotents. Any projective homogeneous variety (under the action of a semi-simple affine algebraic group) is geometrically split and satisfies the nilpotence principle (see [1]), therefore if Λ is finite the Krull-Schmidt principle holds for CM_{G}(F; Λ) by [5, Corollary 3.6], and we can serenely deal with motivic decompositions in CM_{G}(F; Λ).

Let G be a semi-simple affine adjoint algebraic group over F and p a prime. The absolute Galois group Gal(F_{sep}/F) acts on the Dynkin diagram of G and we say that G is of inner type if this action is trivial. By [1] the subfield F_{G} of F_{sep} corresponding to the kernel of this action is a finite Galois extension of F, and we will say that G is p-inner if [F_{G} : F] is a power of p. We now state the main result. Theorem 2.1 Let G be a semi-simple affine adjoint p-inner algebraic group and M ∈ CM_{G}(F; 𝔽_{p}). Then for any field L of characteristic p, M is indecomposable if and only if coeff_{L/𝔽_{p}}(M) is indecomposable. If X is geometrically split the image of any correspondence α ∈ CH_{dim(X)}(X × X; Λ) by the change of base field functor res_{E/F} to a splitting field E/F of X will be denoted \overline{α}. The reduced endomorphism ring of any direct summand (X, π) is defined as res_{E/F} (End_{CM(F;Λ)}((X, π))) and denoted by \overline{End}((X, π)).

Let X be a complete and irreducible F -variety. The pull-back of the natural morphism Spec(F (X)) × X ⟶ X ×X gives rise to mult : CH_{dim(X)}(X ×X; Λ) ⟶ CH_{0}(X_{F (X)}; Λ) ⟶ Λ (where the second map is the usual degree morphism). For any correspondence α ∈ CH_{dim(X)}(X × X; Λ), mult(α) is called the multiplicity of α and we say that a direct summand (X, π) given by a projector π ∈ CH_{dim(X)}(X × X; Λ) is upper if mult(π) = 1. If (X, π) is an upper direct summand of a complete and irreducible F -variety, the multiplicity mult : End_{CM(F;Λ)}((X, π)) ⟶ Λ is a morphism of rings by [4, Corollary 1.7]. Proposition 2.1 Let G be a semi-simple affine algebraic group and M = (X, π) ∈ CM(F; 𝔽_{p}) the upper direct summand of the motive of an irreducible and projective G-homogeneous F -variety. Then for any field L of characteristic p, M is indecomposable if and only if coeff_{L/𝔽_{p}}(M) is indecomposable. Proof. Since the change of coefficients functor is additive and maps any non-zero projector to a non-zero projector, it is clear that if coeff_{L/𝔽_{p}}(M) is indecomposable, M is also indecomposable. Considering a splitting field E of X, the reduced endomorphism ring \overline{End}(M) := \overline{π} ∘\overline{End}(X)∘\overline{π} is connected since M is indecomposable and finite. Corollary 1.3, with A = \overline{End}(M), E = L and φ = mult implies that \overline{End}(M)⊗ L = \overline{End}(coeff_{L/𝔽_{p}}(M)) is connected, therefore by the nilpotence principle End(coeff_{L/𝔽_{p}}(M)) is also connected, that is to say coeff_{L/𝔽_{p}}(M) is indecomposable.

Proof of Theorem 2.1. Recall that G is a semi-simple affine adjoint p-inner algebraic group and consider a projective G-homogeneous F -variety X. By [6, Theorem 1.1], any indecomposable direct summand M of X is a twist of the upper summand of the motive of an irreducible and projective G-homogeneous F -variety Y, thus we can apply proposition 2.1 to each indecomposable direct summand of X.

Remark : If Λ is a finite, commutative and connected ring, complete motivic decompositions in CM(F; Λ) remain complete when the coefficients are extended to the residue field of Λ by [7, Corollary 2.6], hence the study of motivic decompositions in CM_{G}(F; Λ), where Λ is any finite connected ring whose residue field is of characteristic p, is reduced to the study motivic decompositions in CM_{G}(F; 𝔽_{p}).

We now produce a counterexample to Theorem 2.1 in the case where the algebraic group G doesn’t satisfy the needed assumptions. Let L/F be a Galois extension of degree 3. By [1, Section 7], the endomorphism ring End(M(Spec(L))) of the motive associated with the F -variety Spec(L) with coefficients in 𝔽_{2} is the 𝔽_{2}-algebra of Gal(L/F), i.e. \frac{𝔽_{2}[X]}{(X^{3}−1)} ≃ 𝔽_{2}×𝔽_{4}, hence M(Spec(L)) = M ⊕N, with End(N) = 𝔽_{4} and both M and N are indecomposable. Now End(res_{𝔽_{4}/𝔽_{2}}(N)) = 𝔽_{4} ⊗𝔽_{4} is not connected since 1⊗α+α⊗1 is a non-trivial idempotent for any α ∈ 𝔽_{4} \ 𝔽_{2}, hence res_{𝔽_{4}/𝔽_{2}}(N) is decomposable. Consider the (P GL_{2})_{L}-homogeneous L-variety ℙ_{L}^{1}. The Weil restriction R(ℙ_{L}^{1}) is a projective homogeneous F -variety under the action of the Weil restriction of (P GL_{2})_{L}, and the minimal extension such that R((P GL_{2})_{L}) is of inner type is L. By [3, Example 4.8], the motive with coefficients in 𝔽_{2} of R(ℙ_{L}^{1}) contains two twists of Spec(L) as direct summands, therefore at least two indecomposable direct summands of R(ℙ_{L}^{1}) split off over 𝔽_{4}.

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