MOTIVIC DECOMPOSITION OF PROJECTIVE HOMOGENEOUS VARIETIES AND THE KRULL-SCHMIDT

THEOREM

VLADIMIR CHERNOUSOV AND ALEXANDER MERKURJEV

Abstract. Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory M(G, R) of the category of Chow motives with coefficients in R that is the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that M(G, R) is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in M(G, R). We prove that the Krull-Schmidt theorem holds in many cases.

1. Introduction

Let X and X′ be two projective homogeneous varieties of a semisimple group G defined over a field F. In general, the diagonal action of G on the product X × X′ is not transitive, i.e., X × X′ is not a homogeneous G-variety. Nevertheless we prove that the Chow motive of X × X′ is a direct sum of some Tate twists of projective homogeneous G-varieties (Theorem 6.3). The main ingredient of the proof is Proposition 5.1 stating that for every orbit of the G-action on X × X′ there is a flat morphism of that orbit to a projective homogeneous G-variety with all fibers affine spaces. Description of all the orbits and the associated projective homogeneous G-varieties is given in the combinatorial terms of the root system of G.

In the second part of the paper we study the problem of uniqueness of a direct sum decomposition of the motive of a projective homogeneous variety into indecomposable objects in the category of Chow motives (the Krull-Schmidt theorem). Example 9.4 shows that the Krull-Schmidt theorem does not hold in general for integral motives of projective homogeneous varieties. On the other hand, when dealing with a particular motive (such as a projective quadric or a Severi-Brauer variety) the coefficient ring ℤ can be replaced by a localization when all “insufficient” integers are inverted. For example, in the case of a quadric one can invert all odd integers, i.e., replace the ring ℤ by its localization ℤ_{(p)} for p = 2 not losing essential information. The situation with the motives of projective homogeneous varieties is better over the ring ℤ_{(p)}: we prove the Krull-Schmidt theorem in many cases (Corollary 9.7), for example when G is a simple group or of inner type. If one replaces the coefficient ring ℤ_{(p)} by its completion, the Krull-Schmidt theorem holds without restrictions.

2000 Mathematics Subject Classification. 20G15, 14F42.

The first author was supported in part by Canada Research Chairs Program and NSERC Canada Grant G121210944.

The second author was supported in part by NSF Grant #0355166.

1.1. Notation. Basic reference is [2]. Let G_{qs} be an adjoint semisimple quasisplit algebraic group defined over a field F. Choose a maximal quasi-split torus T of G_{qs} and a Borel subgroup B defined over F such that T ⊂ B. Let Σ be the root system of G_{qs} with respect to T and Π ⊂ Σ the system of simple roots corresponding to B. Let Σ^{+} (resp. Σ^{−}) be the set of positive (resp. negative) roots with respect to the basis Π. We write U for the unipotent radical of B and U_{α} for the root subgroup of α ∈ Σ.

Denote by W the Weyl group of Σ. The group W is generated by the set V of reflections with respect to all roots in Π.

Let F_{sep} be a separable closure of F and set Γ_{F} = Gal(F_{sep}/F). The group Γ_{F} acts on T, B, W, Σ, Π. This action is called the ∗-action.

If S is a subset of Π, we denote by P_{S} the standard parabolic subgroup of G corresponding to S. Recall that P_{S} is generated by B and the root subgroups U_{−α} for all α ∈ S. We will say that P_{S} is a parabolic subgroup of type S. Any parabolic subgroup Q of G is conjugate to P_{S} for a unique S ⊂ Π. We will say that Q has type S. If P is a parabolic subgroup we denote by R_{u}P the unipotent radical of P.

We denote by H_{S} the variety of all parabolic subgroups of G of type S, so that H_{S} ≃ G/P_{S}. The variety H_{S} is defined over F if and only if S is invariant under the ∗-action. For example, H_{Π} = Spec F and H_{∅} is the variety of all Borel subgroups of G. Any projective homogeneous G-variety defined over F is isomorphic to H_{S} for a unique subset S ⊂ Π invariant under the ∗-action. If S ⊂ S′ ⊂ Π are two ∗-invariant subsets then there is a canonical morphism H_{S} → H_{S′} defined over F.

Let G be an arbitrary adjoint semisimple group. It is an inner form of a quasi-split group G_{qs} unique up to isomorphism. That is the groups G(F_{sep}) and G_{qs}(F_{sep}) can be identified and Γ_{F} acts on G(F_{sep}) = G_{qs}(F_{sep}) by the formula σ(g) = g_{σ}σ^{∗}(g)g_{σ}^{−1} where (g_{σ}) is a 1-cocycle of Γ_{F} with values in G_{qs}(F_{sep}).

Let F_{G} be the subfield of F_{sep} corresponding to the kernel of the ∗-action on G. Thus, F_{G}/F is a finite Galois extension and F_{G} is the smallest field extension of F such that the group G_{F_{G}} is of inner type. In particular G is of inner type if and only if F_{G} = F.

For a subgroup H ⊂ G and an element g of G we denote by^{g}H the subgroup gHg^{−1}.

We write CH_{p}(X) for the Chow group of classes of dimension p algebraic cycles on an algebraic variety X.

2. Product of two homogeneous projective varieties

Let S, S′ be two subsets of Π invariant with respect to the ∗-action of Γ_{F}. We denote by P and P′ the standard parabolic subgroups of G corresponding to S and S′. Let G_{P} (resp. G_{P′} ) be the subgroup of G generated by U_{±α}, α ∈ S (resp. S′). The groups G_{P} and G_{P′} are the semisimple parts of Levi subgroups of P and P′. The root systems Σ_{P} and Σ_{P′} of G_{P} and G_{P′} with respect to the tori T_{P} = T ∩ G_{P} and T_{P′} = T ∩ G_{P′} are the root subsystems in Σ generated by S and S′. The Weyl groups W_{P} and W_{P′} of G_{P} and G_{P′} are the subgroups of W generated by the sets V_{S} and V_{S′} of reflections with respect to the roots in S and S′.

Let X = H_{S} and X′ = H_{S′} be the projective homogeneous G-varieties. Since S, S′ are stable with respect to the ∗-action, the varieties X, X′ are defined over F . We consider the diagonal action of G on X × X′. The orbits of this action are defined over F_{sep} but may not be defined over F. The Galois group Γ_{F} acts on the set of orbits. Note that this action coincides with the ∗-action.

Lemma 2.1. The assignment w ↦ (P, ^{w}P′) gives rise to a Γ_{F} -equivariant bijection between the set of double cosets W_{P} \W/W_{P′} and the set of G-orbits in X × X′.

Proof. Since X = G/P and X′ = G/P′ the assignment g ↦ (P, ^{g}P′) induces a Γ_{F} -equivariant bijection between P \G/P′ and the set of G-orbits in X × X′.

Consider the natural map

γ : W_{P} \W/W_{P′} → P \G/P′, γ(W_{P} w W_{P′}) = P w P′.

Since G is a disjoint union of Bruhat cells BwB, it follows that γ is surjective. Assume that P w_{1}P′ = P w_{2}P′ for some w_{1}, w_{2} ∈ W. Then we have

(1)  w_{1} ∈ P w_{2}P′ ⊂ ⋃_{(w_{p},w_{p′})∈W_{P} ×W_{P′}} Bw_{p}Bw_{2}Bw_{p′}B.

Recall that for every reflection w_{α} with respect to a simple root α ∈ Π and every w ∈ W one has

w_{α}Bw ⊂ BwB ∪ Bw_{α}wB and wBw_{α} ⊂ BwB ∪ Bww_{α}B.

Therefore there exist w_{p} ∈ W_{P} , w_{p′} ∈ W_{P′} such that w_{1} ∈ Bw′_{p}w_{2}w′_{p′}B where w′_{p}, w′_{p′} are subwords of w_{p}, w_{p′} respectively. Since any subword of w_{p} (resp. w_{p′}) belongs to W_{P} (resp. W_{P′}), we deduce that w_{1} ∈ W_{P} w_{2}W_{P′}. This implies that γ is injective. □

In the following statement we give necessary and sufficient conditions of a transitive G-action on X × X′.

Corollary 2.2. Let Σ_{1}, …, Σ_{m} be simple components of Σ and let Π_{i} = Π∩Σ_{i} for i = 1, 2, …, m be their bases. Then G acts transitively on X × X′ if and only if each Π_{i} is contained in either S or S′.

Proof. By Lemma 2.1, the number of G-orbits in X × X′ is 1 if and only if W = W_{P} W_{P′}. Assume that Π_{i} is contained in S or S′. Then the corresponding Weyl group W_{Π_{i}} is contained in W_{P} or W_{P′}. Since W is directly generated by its subgroups W_{Π_{1}}, …, W_{Π_{m}} we easily get W = W_{P} W_{P′}.

Conversely, let W = W_{P} W_{P′}. Assume that Π_{i} ∩ S ≠ Π_{i} and Π_{i} ∩ S′ ≠ Π_{i} for some i. Replacing Π by Π_{i}, S and S′ by Π_{i} ∩S and Π_{i} ∩S′ respectively we may assume that Σ is irreducible and that S ≠ Π, S′ ≠ Π. Consider a positive root α such that its decomposition α = n_{1}α_{1} +⋯+n_{s}α_{s}, where α_{1}, …, α_{s} ∈ Π and n_{1}, …, n_{s} are positive integers, contains simple roots from S \ S′ and S′ \ S.

Let w ∈ W be the reflection with respect to α and let w = w_{1}w_{2} where w_{1} ∈ W_{P} , w_{2} ∈ W_{P′}. Then w(α) = −α is a negative root. On the other hand, by [4, Lemma 5.2], w_{2}(Σ^{+} \ Σ_{S′}^{+}) ⊂ Σ^{+}. In particular, since the decomposition of α contains a simple root from S \S′, the root w_{2}(α) is positive. Analogously, w_{1}^{−1}(Σ^{−} \ Σ_{S}^{−}) ⊂ Σ^{−} and hence w_{1}^{−1}(−α) is a negative root. This contradicts to the equality w_{2}(α) = w_{1}^{−1}(−α). □

Corollary 2.3. If G is a simple group and none of X and X′ is a point, then G does not act transitively on X × X′.

3. The type of the parabolic subgroup R_{u}P · (P ∩ ^{w}P′)

We keep the notation of the previous section. In particular, P and P′ denote the standard parabolic subgroups of types S and S′. By [2, Proposition 14.22], for every w ∈ W the subgroup R_{u}P · (P ∩ ^{w}P′) of G is parabolic and we are going to determine its type. If w ∈ W, we let l(w) denote the length of w with respect to the set of generators V of W.

Lemma 3.1. Assume that w is an element of minimal length in the double coset D = W_{P} wW_{P′} ⊂ W. Then any element w_{1} ∈ W_{P} wW_{P′} can be written in the form w_{1} = awa′ with a ∈ W_{P} , a′ ∈ W_{P′} such that l(w_{1}) = l(a)+l(w)+l(a′). In particular, D contains a unique element of minimal length.

Proof. Apply the same argument as in [4, Proposition 3.4]. □

Lemma 3.2. In the notation above we have w(Σ_{S′}^{+}) ⊂ Σ^{+} and w^{−1}(Σ_{S}^{+}) ⊂ Σ^{+}.

Proof. Apply the same argument as in [4, Lemma 5.1]. □

Remark 3.3. In the notation of the previous section, every G-orbit of X ×X′ contains a unique element of the form (P, ^{w}P′) where w ∈ W is the element of minimal length in the double coset W_{P} wW_{P′}.

Let D ∈ W_{P} \W/W_{P′} be a double coset and let w ∈ D be the element of minimal length. Denote by R the parabolic subgroup R_{u}P · (P ∩ ^{w}P′).

Lemma 3.4. One has B ⊂ R, in particular R is a standard parabolic subgroup.

Proof. Since R_{u}P ⊂ R, it suffices to show that for every root α ∈ S one has U_{α} ⊂ ^{w}P′ or equivalently ^{w^{−1}}U_{α} ⊂ P′. Since ^{w^{−1}}U_{α} = U_{w^{−1}(α)}, the required result follows from Lemma 3.2. □

Consider the following subset of the set of simple roots:

R_{D} = {α ∈ S | w^{−1}(α) ∈ Σ_{S′}^{+}} ⊂ Π.

Note that in the case w = 1, i.e., D = W_{S}W_{S′}, we have R_{D} = S ∩ S′.

Proposition 3.5. The parabolic subgroup R of G has type R_{D}.

Proof. Since R is standard, it suffices to show that the set consisting of simple roots α ∈ Π such that U_{±α} ⊂ R coincides with R_{D}.

Clearly, for every α ∈ R_{D} we have U_{±α} ⊂ R. Conversely, let α be a root with this property. Since R_{u}P is contained in the unipotent radical of R we first conclude that U_{±α} ∈ G_{P} ∩ ^{w}P′, in particular α ∈ S. Since

U_{±w^{−1}(α)} = ^{w^{−1}}U_{±α} ⊂ P′,

we have w^{−1}(α) ∈ Σ_{S′}. By Lemma 3.2, w^{−1}(α) ∈ Σ^{+} ∩ Σ_{S′} = Σ_{S′}^{+} implying α ∈ R_{D}. □

4. Structure of G-orbits on X × X′

Since S, S′ are ∗-stable, the ∗-action on W extends to the set of double cosets W_{P} \W/W_{P′}. Let D ∈ W_{P} \W/W_{P′} and let O_{D} be the corresponding G-orbit in X × X′ (Lemma 2.1). Denote by Z_{D} the projective homogeneous G-variety H_{R_{D}} and consider the G-equivariant map

λ_{D} : O_{D} → Z_{D}, λ_{D}(Q, Q′) = R_{u}Q · (Q ∩ Q′).

Clearly, λ_{D} is isomorphic to the canonical morphism G/(P ∩ ^{w}P′) → G/R, where w ∈ D is the element of minimal length, and therefore, λ_{D} is a flat morphism of varieties.

Proposition 4.1. Assume that the double coset D is ∗-stable. Then the varieties O_{D}, Z_{D} and the morphism λ_{D} are defined over F.

Proof. The element ∗(w) belongs to D and has the same length as w. It follows that ∗(w) = w. Hence the set R_{D} is ∗-stable and therefore, Z_{D} is defined over F. By Lemma 2.1, the orbit O_{D} is Γ_{F} -stable, hence O_{D} is defined over F. It follows from the definition that λ_{D} is Γ_{F} -equivariant and hence is also defined over F. □

Remark 4.2. Since R_{D} ⊂ S, there is a canonical morphism Z_{D} → X.

Let D ∈ W_{P} \W/W_{P′} be an arbitrary double coset. Let F_{D} be the finite separable field extension of F corresponding to the stabilizer of D in Γ_{F} (with respect to the ∗-action). By Proposition 4.1, the varieties O_{D} and Z_{D} and the morphism λ_{D} are defined over F_{D}. Let O_{D} → Spec F_{D} and Z_{D} → Spec F_{D} be the corresponding structure morphisms. We consider O_{D} and Z_{D} as schemes over F with respect to the composites O_{D} → Spec F_{D} → Spec F and Z_{D} → Spec F_{D} → Spec F. By Proposition 4.1, we can view λ_{D} : O_{D} → Z_{D} as a morphism of schemes defined over F.

Proposition 4.3. Let z ∈ Z_{D} be a point (not necessary closed). Then λ_{D}^{−1}(z) is isomorphic to the affine space over F (z) of dimension l(w).

Proof. Replacing F by F (z) we may assume that Z_{D} has a point over F. This point corresponds to the parabolic subgroup in G conjugate to R. Replacing T, B, U and etc. by the corresponding conjugate objects we may assume that z = R. Since R is defined over F , we have g_{σ}σ^{∗}(R)g_{σ}^{−1} = g_{σ}Rg_{σ}^{−1} = R, hence g_{σ} ∈ R ⊂ P for all σ ∈ Γ_{F}. It follows that P is defined over F . Since D and w are Γ_{F} -invariant and R ⊂ ^{w}P′, analogously, ^{w}P′ is defined over F.

Since λ_{D} is G-equivariant, by definition of R,

λD^{−1}(z) ≃ R/(P ∩ ^{w}P′) ≃ R_{u}P/(R_{u}P ∩ ^{w}P′).

The next step is to show that

dim R_{u}P/(R_{u}P ∩ ^{w}P′) = l(w).

Set

Σ′ = (Σ^{+} \ Σ_{S}^{+}) ∩ w(Σ^{−} \ Σ_{S′}^{−}), Σ′_{w} = {α ∈ Σ^{+} | w^{−1}(α) ∈ Σ^{−}}.

Note that

|Σ′_{w}| = l(w) and |Σ′| = dim R_{u}P/(R_{u}P ∩ ^{w}P′).

We claim that Σ′ = Σ′_{w}. The inclusion Σ′ ⊂ Σ′_{w} follows from our construction. Conversely, let α ∈ Σ′_{w}. By Lemma 3.2, α ∈ Σ^{+} \ Σ_{S}^{+}. Since w^{−1}(α) is a negative root and since w(−S′) ⊂ Σ^{−} (Lemma 3.2) we also have w^{−1}(α) ∈ Σ^{−} \ Σ_{S′}^{−} that implies α ∈ w(Σ^{−} \ Σ_{S′}^{−}). Thus α ∈ Σ′ as required.

It remains to show that R_{u}P/(R_{u}P ∩ ^{w}P′) is isomorphic to an affine space over F. We set Q′ = ^{w}P′ and Ψ = Σ^{+} \ Σ_{S}^{+}. Let Ψ_{1} be the subset in Ψ consisting of all roots α such that the corresponding root subgroup U_{α} is contained in R_{u}P ∩ Q′. Clearly, Ψ_{1} is closed, i.e., if α, β ∈ Ψ_{1} and α + β is a root, then α + β ∈ Ψ_{1}. Since R_{u}P and Q′ are defined over F, Ψ_{1} is Γ_{F} -stable. Hence the subgroup U_{1} generated by U_{α}, α ∈ Ψ_{1} (it coincides with R_{u}P ∩ Q′) is defined over F.

Let R_{u}^{−}Q′ be the negative unipotent radical of Q′. Let Ψ_{2} be the subset in Ψ consisting of all roots α such that U_{α} is not contained in Q′ or equivalently U_{α} ⊂ R_{u}P ∩R_{u}^{−}Q′. As above, Ψ_{2} is closed and Γ_{F} -stable. Let U_{2} be a subgroup generated by U_{α}, α ∈ Ψ_{2} (it coincides with R_{u}P ∩ R_{u}^{−}Q′).

Since Ψ_{1} ∩ Ψ_{2} = ∅ and Ψ = Ψ_{1} ∪ Ψ_{2} the product morphism U_{1} × U_{2} → R_{u}P is an F -isomorphism. It follows that the variety R_{u}P/(R_{u}P ∩ Q′) = R_{u}P/U_{1} is isomorphic to U_{2}. By [3, Cor. 3.18], U_{2} is an F -split subgroup of G. In view of [2, Rem. 15.13], any F -split subgroup as a variety is isomorphic to an affine space over F. □

Corollary 4.4. There is only one closed orbit O_{D} in X × X′ where D = W_{S}W_{S′}.

Proof. The orbit is closed if and only if the affine fibers of the projection O_{D} → Z_{D} are trivial, or equivalently, l(w) = 0, i.e., w = 1 and D = W_{S}W_{S′}. □

5. Filtration on X × X′

Consider the ∗-action on the set of double cosets W_{P} \W/W_{P′}. Let Δ be the set of all ∗-orbits in W_{P} \W/W_{P′}. For an orbit δ ∈ Δ, the length l(D) does not depend on the choice of a representative D ∈ δ. We denote this number by l_{δ}. The varieties O_{D} and Z_{D} considered as schemes over F also do not depend on D ∈ δ. We denote them by O_{δ} and Z_{δ} respectively.

Proposition 5.1. One can number all elements δ_{1}, δ_{2}, …, δ_{n} of Δ so that there exists a filtration by closed subvarieties

∅ = V_{0} ⊂ V_{1} ⊂ V_{1} ⊂ … ⊂ V_{n} = X × X′

together with flat morphisms f_{i} : V_{i} \V_{i−1} → Z_{δ_{i}} of constant relative dimension l(δ_{i}) for every i = 1, 2, …, n, with the property that the fiber of every f_{i} over any point s ∈ Z_{δ_{i}} is isomorphic to the affine space over F (s) of dimension l_{δ_{i}}.

Proof. Recall that we consider the diagonal G-action in X × X′. For every j ≥ 0 let V′_{j} be the union of all orbits of dimension at most j. Clearly V′_{j} is closed in X × X′ and defined over F. Thus we have a filtration by closed subvarieties

∅ = V′_{0} ⊂ V′_{1} ⊂ … ⊂ V′_{m} = X × X′

such that for every j > 0, the variety V′_{j} \ V′_{j−1} is the union of G-orbits of the dimension j. The Galois group Γ_{F} permutes the orbits in V′_{j} \ V′_{j−1} according to the ∗-action on the set of double cosets W_{P} \W/W_{P′} in view of Lemma 2.1. We can then rearrange the filtration:

∅ = V_{0} ⊂ V_{1} ⊂ … ⊂ V_{n} = X × X′

in such a way that every V_{i} \ V_{i−1} coincides with O_{δ} for an appropriate δ ∈ Δ. By Proposition 4.1, there is a flat morphism λ_{δ} : O_{δ} → Z_{δ} of constant relative dimension l_{δ}. In view of Proposition 4.3, every fiber of λ_{δ} is an affine space. □

6. Category of Chow motives

Let J be a class of objects of an additive category A. The subcategory A′ generated by J is the smallest full additive subcategory A′ ⊂ A containing all objects of J. Objects of A′ are finite direct sums of objects from J.

An additive category A is called pseudo-abelian if every projector in A splits [1, Ch. I, §3]. Let J be a class of objects of a pseudo–abelian category A. The pseudo–abelian subcategory A′ generated by J is the smallest full pseudo– abelian subcategory A′ ⊂ A containing all objects of J. Objects of A′ are direct summands of finite direct sums of objects from J.

We say that an additive category A is a Tate category if an additive endofunctor T : A → A is given (called the Tate twist functor). We write A(n) = T^{n}(A) for every A ∈ A and n ≥ 0. Let A be a Tate category. We say that a full subcategory A′ ⊂ A is a Tate subcategory if A′ is additive and closed under the Tate twist functor, that is, if A ∈ A′ then A(n) ∈ A′ for every n ≥ 0. Let J be a class of objects of A. The Tate subcategory of A generated by J is the smallest Tate subcategory of A containing all objects of J. Objects of this subcategory are direct sums of objects of the form A(n) where A ∈ J. A Tate category A is finitely generated if it is generated by a finite set of objects as a Tate category.

An object A of an additive category is called indecomposable if A is not isomorphic to the direct sum of two nonzero objects. If A is pseudo-abelian, A is indecomposable if and only if the ring End_{A}(A) has no nontrivial idempotents.

We say that the Krull-Schmidt theorem holds for an additive category A if any two finite direct sum decompositions of an object of A into indecomposable objects are isomorphic. Recall that a ring S is called local if whenever a+b = 1 for a, b ∈ S then either a or b is invertible in S. We will be using the following

Theorem 6.1. [1, Ch. I, Th. 3.6] Let A be a pseudo-abelian category. Suppose that the endomorphism ring of every indecomposable object is local. Then the Krull-Schmidt theorem holds for A.

Let F be a field and let Var be the category of smooth complete varieties over F. For any commutative ring R we define the additive category C(R) of correspondences with coefficients in R as follows. The objects of C(R) are the same as in Var. The group of morphisms between X and Y in the case when X is integral of dimension d is equal to

Mor_{C(R)}(X, Y) = CH_{d}(X × Y) ⊗_{ℤ} R.

In general, Mor_{C(R)}(X, Y) is the direct sum of Mor_{C(R)}(X_{i}, Y) over all irreducible (connected) components X_{i} of X.

The direct sum of X and Y in C(R) is the disjoint union X + Y.

We have the functor Var → C(R) that is identical on objects and taking a morphism f to the class of the cycle [Γ_{f}] ⊗ 1_{R} where Γ_{f} ⊂ X × Y is the graph of f.

The category of Chow motives M(R) with coefficients in R is the pseudoabelian closure of C(R). The objects of M(R) (called the motives) are the pairs (X, ρ) where X is a smooth complete variety over F and ρ ∈ End_{C(R)}(X) is an idempotent (projector). The R-module of all morphisms between (X, ρ) and (X′, ρ′) is the submodule

ρ′ ∘ Mor_{C(R)}(X, X′) ∘ ρ ⊂ Mor_{C(R)}(X, X′).

The direct sum operation is given by the disjoint union of varieties:

(X, ρ) ⊕ (X′, ρ′) = (X + X′, ρ + ρ′).

The category M(R) is a symmetric tensor additive category with respect to the tensor product defined by

(X, ρ) ⊗ (X′, ρ′) = (X × X′, ρ × ρ′).

The composition of functors Var → C(R) → M(R) takes any X to the motive M(X) = (X, id_{X}) of X.

The motive (ℙ^{1}, ρ) in M(R), where ρ is given by the cycle x × ℙ^{1} ⊂ ℙ^{1} × ℙ^{1} for a rational point x ∈ ℙ^{1}, is denoted by R(1) and called the Tate motive.

Let R(n) for n ≥ 0 be the n-th tensor power of R(1). In particular, R(0) = M(Spec F). We have

(2)  Hom_{M(R)}(R(i), R(j)) = \{ \begin{matrix}R, & if i = j; \\ 0, & otherwise. \end{matrix}

For a motive M in M(R) we set M(n) = M ⊗ R(n). In particular, M(0) = M. Clearly, M(R) is a Tate category with respect to the Tate twist functor M ↦ M(1).

Example 6.2. (Projective bundle theorem [9, §7]) Let E be a vector bundle of rank r + 1 over X in Var and let ℙ(E) be the associated projective bundle. Then in M(R):

M(P (E)) ≃ M(X) ⊕ M(X)(1) ⊕ ⋯ ⊕ M(X)(r).

For any two motives M and M′ the natural homomorphism

Mor_{M(R)}(M, M′) → Mor_{M(R)}(M(n), M′(n))

is an isomorphism for every n ≥ 0. In particular, the endomorphism rings of M and M(n) are isomorphic for every n ≥ 0. Hence the motive M is indecomposable if and only if M(n) is indecomposable.

We keep the notation of Section 5. The theorem below is a corollary of Proposition 5.1 and results of N. Karpenko [7, Th. 6.5, Cor. 6.11] (see also [4, Th. 7.2, Rem. 7.3]).

Theorem 6.3. Let X and X′ be projective homogeneous varieties of a semisimple group G over F. There is an isomorphism in M(R):

M(X × X′) = ∐_{δ∈Δ} M(Z_{δ})(l_{δ}),

where each Z_{δ} is a projective homogeneous G-variety.

Remark 6.4. The variety Z_{δ} is defined over the field F_{D} for D ∈ δ. Note that F_{D} ⊂ F_{G}.

Remark 6.5. By Corollary 4.4, the direct sum contains exactly one term with the zero Tate twist - the motive of the closed orbit O_{D} = Z_{D} for D = W_{S}W_{S′}. We have O_{D} ≃ Z_{D} = H_{S∩S′}. Moreover, the canonical morphism H_{S∩S′} → H_{S} × H_{S′} = X × X′ induces an isomorphism between H_{S∩S′} and the orbit O_{D}.

Remark 6.6. There are canonical morphisms Z_{δ} → X over F (Remark 4.2). Interchanging the roles of X and X′ we can get another decomposition

M(X × X′) = ∐_{δ∈Δ} M(Z′_{δ})(l_{δ}),

where Z′_{δ} are projective homogeneous G-varieties equipped with the morphisms Z′_{δ} → X′ (see Remark 4.2).

7. Motives of zero-dimensional varieties

Let L/F be a finite Galois field extension with the Galois group Γ. Denote by C(R)^{L} the additive subcategory of C(R) generated by all zero-dimensional varieties X = Spec E where E is a field such that F ⊂ E ⊂ L.

Recall that a RΓ-module P is called a permutation module if there is an R-basis of P permuted by Γ. For a variety X ∈ C(R)^{L}, the R-module

Hom_{C(R)}(Spec L, X) = CH_{0}(X_{L}) ⊗ R

has a natural structure of a permutation RΓ-module. For another variety Y ∈ C(R)^{L}, the composition law yields a homomorphism

Hom_{C(R)}(X, Y) → Hom_{Γ}(CH_{0}(X_{L}) ⊗ R, CH_{0}(Y_{L}) ⊗ R).

This is an isomorphism since it can be obtained by taking the RΓ-invariant elements of the two Γ-modules in the isomorphism (over L)

Hom_{C(R)}(X_{L}, Y_{L}) → Hom_{ℤ}(CH_{0}(X_{L}) ⊗ R, CH_{0}(Y_{L}) ⊗ R).

Thus, the subcategory C(R)^{L} is equivalent to the category of permutation RΓ-lattices.

An RΓ-module I is called invertible if it is a direct summand of a permutation RΓ-module. The category Inv(RΓ) of invertible RΓ-modules is pseudoabelian. It is equivalent to the pseudo-abelian closure M(R)^{L} of the category C(R)^{L}.

If Γ is trivial, the category Inv(RΓ) coincides with the category P(R) of finitely generated projective R-modules. Thus the category M(R)^{F} is equivalent to P(R).

If R = ℤ and Γ is a cyclic group of prime order ≤ 19, the Krull-Schmidt theorem holds for the category Inv(ℤΓ) [5, Th. 34.41].

Consider the case of a local ring. Let R be a discrete valuation ring with the maximal ideal 𝔪 and the quotient field K. Denote by R̂ and K̂ the completions of R and K respectively. Let Γ be a finite group of order relatively prime to char F, so that the group algebra KΓ is semisimple. Let D_{1}, D_{2}, …, D_{s} be the endomorphism division K-algebras of all simple K[Γ]-modules. The group Γ is called split if D_{i} = K for all i.

We will be considering the following condition on Γ:

(∗) For all i, D_{i} ⊗ K̂ is a division K̂-algebra.

The following statement was proved in [5, Th. 30.18]:

Proposition 7.1. Let Γ satisfy the condition (∗). Then the Krull-Schmidt theorem holds for the category Inv(RΓ).

Remark 7.2. The condition (∗) holds in the following cases:

(1) Γ is a split group. For example, Γ = S_{n} the symmetric group;

(2) R = ℤ_{(p)} and Γ is a p-group [5, Th. 36.1];

(3) R is complete.

Remark 7.3. The automorphism group Γ of a connected Dynkin diagram satisfies (∗) over R = ℤ_{(p)}. Indeed, Γ is either a cyclic group of order ≤ 3 or the symmetric group S_{3}.

Remark 7.4. In general, the Krull-Schmidt theorem does not hold for Inv(RΓ) over a d.v.r. R. The following example is due to E. Beneish.

Let R = ℤ_{(p)} for a prime p ≥ 5 and let Γ be the semi-direct product of a cyclic group Λ of order p with its automorphism group Φ. Let τ ∈ Φ be a generator (of order p − 1). Consider the ring L = R[ξ] where ξ is a primitive (p − 1)-th root of unity. We regard L as a RΓ-module of rank φ(p − 1) via the composition RΓ → RΦ → L where the first map is induced by the canonical homomorphism Γ → Φ and the second one takes τ to ξ. Since L_{K} = L⊗_{R}K is a field, the RΓ-module L is indecomposable. Since the homomorphism RΦ → L splits, L is a direct summand of R[Φ] and therefore is invertible.

Let θ ∈ R̂ be a primitive root of unity of degree p − 1. For every k with (k, p − 1) = 1 let L^{k} be the R̂Γ-module of rank 1 so that Λ acts trivially on L^{k} and τ acts by the multiplication by θ^{k}. We have

LR̂ = L ⊗_{R} R̂ = ∐_{(k,p−1)=1^{1≤k<p−1,}} L^{k}.

Consider M = R[Γ/Φ] as a (permutation) RΓ-module of rank p and let V be the kernel of the augmentation Γ-homomorphism M → R. Over the field K̂,

(3)  MK̂ = M ⊗_{R} K̂ = VK̂ ⊕ K̂.

The dimension count shows that VK̂ is the only (up to isomorphism) indecomposable (irreducible) K̂Γ-module of rank p − 1. Therefore, VK̂ ⊗ Lk ; K̂ ≃ VK̂ for every k with (k, p − 1) = 1. The decomposition (3) does not descent to R̂, hence MR̂ is indecomposable over R̂.

Consider the following direct sums of indecomposable R̂Γ-modules:

C = L^{1} ⊕ ∐_{(k,p−1)=1^{1<k<p−1,}}(M ⊗_{R} L^{k}), C′ = (M ⊗_{R} L^{1}) ⊕ ∐_{(k,p−1)=1^{1<k<p−1,}} L^{k}.

Since

CK̂ ≃ V φ(p−1)−1 ; K̂ ⊕ LK̂, C′ ; K̂ ≃ VK̂ ⊕ LK̂,

the modules CK̂ and C′ ; K̂ are defined over K. By [5, Cor. 30.10], there are RΓ-modules D and D′ such that DR̂ ≃ C and D′ ; R̂ ≃ C′. We have

DR̂ ⊕ D′ ; R̂ ≃ C ⊕ C′ ≃ (MR̂ ⊗ LR̂) ⊕ LR̂,

By [5, Prop. 30.17],

D ⊕ D′ ≃ (M ⊗ L) ⊕ L.

Note that since p ≥ 5, we have LR̂ ≠ L^{1}, hence L is not a direct summand of D or D′ even over R̂ since the Krull-Schmidt theorem holds over R̂ by [5, Th. 30.18].

8. Category M(G, R)

Let G be a semisimple algebraic group defined over F. Recall that F_{G} is the subfield of F_{sep} corresponding to the kernel of the ∗-action.

Let J be the class of all projective homogeneous varieties X over L of the group G_{L} for all field L such that F ⊂ L ⊂ F_{G}, where X is considered as a variety over F via the natural morphism Spec L → Spec F. Denote by M(G, R) the Tate pseudo-abelian subcategory of M(R) generated by the motives of varieties from J. An object of M(G, R) is a direct summand of a motive of the form ∐ M(X_{i})(n_{i}) where all X_{i} ∈ J and n_{i} ≥ 0.

Theorem 6.3 and Remark 6.4 imply that the tensor product of two motives M(X) and M(Y) for two projective homogeneous varieties X and Y is an object of M(G, R). In other words the following holds:

Theorem 8.1. The category M(G, R) is a symmetric tensor category.

We will be using the following generalization of the Nilpotence theorem [4, Th. 8.2]:

Theorem 8.2. Let M be an object of M(G, R). Then for every field extension L/F, the kernel of the natural ring homomorphism End(M) → End(M_{L}) is a nil ideal.

Proof. We may assume that M is a direct sum of motives of the form M(Y)(i) for a projective homogeneous G-variety Y. The motive M(Y)(i) is a direct summand of M(Y × ℙ^{i}) (Example 6.2). Hence we are reduced to the case when M = M(X) where X is the disjoint union of homogeneous varieties of the form Y × ℙ^{i}. The statement follows from [4, Th. 8.2] (note that the proof goes through if X is not connected). □

Corollary 8.3. Let M be an object of M(G, R) and let L/F be a field extension such that M_{L} = 0. Then M = 0.

Proof. The identity endomorphism id_{M} is zero over L, hence it is nilpotent, and therefore zero. □

The category M(G, R) contains R(0) = M(Spec F). Let M(G, R)_{s} be the Tate pseudo-abelian subcategory of M(G, R) generated by R(0). By (2), M(G, R)_{s} is equivalent to the direct sum of the full subcategories M(R)^{F} (n) over all n ≥ 0 and hence it is equivalent to the direct sum of countably many copies of the category P(R):

(4)  M(G, R)_{s} ≃ P(R)^{ℕ}.

where ℕ = {0, 1, 2, … }.

For any motive M ∈ M(G, R), the motive M_{sep} = M ⊗_{F} F_{sep} of the split group G_{sep} defines then a sequence of projective modules P_{n} ∈ P(R), n ≥ 0, almost all zero. Suppose that Spec R is connected. Then rank(P_{n}) ∈ ℕ of every P_{n} is well defined. For every n ≥ 0 set

rank_{n}(M) = rank(P_{n}) ∈ ℤ.

We have rank_{n}(M ⊕M′) = rank_{n}(M)+rank_{n}(M′) for any two motives M and M′.

Proposition 8.4. Let Spec(R) be connected. A motive M ∈ M(G, R) is zero if and only if rank_{n}(M) = 0 for all n ≥ 0.

Proof. Let rank_{n}(M) = 0. Since the category M(G_{sep}, R) is equivalent to M(G_{sep}, R)_{s}, in view of the equivalence (4) we get M_{sep} = 0, hence M = 0 by Corollary 8.3. □

Theorem 8.5. Let Spec(R) be connected. Then

(1) Every motive in M(G, R) is isomorphic to a finite direct sum of indecomposable motives.

(2) Suppose the Krull-Schmidt theorem holds for M(G, R). Then the category M(G, R) is finitely generated.

Proof. (1) Let M ∈ M(G, R). We proceed by induction on

r(M) = ∑_{n≥0} rank M_{n}.

If N is a nontrivial direct summand of a motive M then rank_{n}(N) ≤ rank_{n}(M) for all n, and it follows from Proposition 8.4 that the strict inequality holds for at least one n. Hence r(N) < r(M) and the statement holds for N by induction.

(2) Let J be the finite set of indecomposable direct summands of M(X) for all (finitely many up to isomorphism) projective homogeneous varieties of G over (finitely many) fields between F and F_{G}. The set J generates M(G, R). Indeed, an indecomposable object M is a direct summand of a motive of the form ∐_{i}^{m} M(X_{i})(n_{i}) where X_{i} are projective homogeneous G-varieties. By the Krull-Schmidt theorem, M is isomorphic to N(n_{i}) for some N ∈ J and n_{i} ≥ 0. □

9. Krull-Schmidt theorem

Let M(G, R)_{qs} be the Tate subcategory of M(G, R) generated by the motives of zero-dimensional varieties Spec(L) where L is a field such that F ⊂ L ⊂ F_{G}. According to the results in Section 7 we get an equivalence of categories

(5)  M(G, R)_{qs} ≃ Inv(RΓ_{F})^{ℕ}.

Lemma 9.1. Let G be a quasi-split group. Then the inclusion M(G, R)_{qs} ↪ M(G, R) is an equivalence of categories.

Proof. Let L be a field such that F ⊂ L ⊂ F_{G}. By [4, Th. 7.5], the motive of a projective homogeneous G_{L}-variety over L is a direct sum of the twisted motives of anisotropic projective homogeneous varieties defined over fields E between L and F_{G}. Since G_{E} is quasi-split, every anisotropic homogeneous variety over E is the point Spec E. □

Proposition 9.2. Let G become quasi-split over a separable field extension of degree n. If n is invertible in R then the category M(G, R) is equivalent to Inv(RΓ_{F})^{ℕ}.

Proof. Consider first the case when G is quasi-split. By Lemma 9.1, the category M(G, R) is equivalent to M(G, R)_{qs} and therefore to Inv(RΓ_{F})^{ℕ} by (5).

In the general case, let L/F be a separable field extension of degree n such that G_{L} is quasi-split. For any two projective homogeneous G-varieties X and Y, the homology groups of the Čech complex

0 → CH_{∗}(X × Y) ⊗ R → CH_{∗}(X_{L} × Y_{L}) ⊗ R → CH_{∗}(X_{L⊗L} × Y_{L⊗L}) ⊗ R

have exponent n. Since n is invertible in R, the complex is acyclic, hence the group Mor(M(X), M(Y)) is isomorphic to the kernel of

Mor(M(X_{L}), M(Y_{L})) → Mor(M(X_{L⊗L}), M(Y_{L⊗L})).

Let G^{qs} be the quasi-split twisted form of G and let X^{qs} and Y^{qs} be the twisted forms of X and Y respectively. Since the groups G_{L}^{qs} and G_{L} are isomorphic, then the group Mor(M(X^{qs}), M(Y^{qs})) is isomorphic to the same kernel. Thus, the categories M(G, R) and M(G^{qs}, R) are equivalent. The result follows then from the first part of the proof. □

Corollary 9.3. The Krull-Schmidt theorem holds for M(G, R) in the following cases:

(1) G is split and every finitely generated projective R-module is free;

(2) R is a d.v.r. and G is quasi-split such that Γ_{F} satisfies the condition (∗). For example, G is a simple quasi-split group.

Proof. The first statement follows from the last paragraph of Section 6. The second follows from Proposition 7.1. □

The following example shows that the Krull-Schmidt theorem does not hold over R = ℤ even for a group G of inner type.

Example 9.4. Let A and B be two central simple F -algebras generating the same subgroup in the Brauer group Br(F). Let X and Y be the corresponding Severi-Brauer varieties. We can view X and Y as projective homogeneous varieties of the group G = PGL_{1}(A)×PGL_{1}(B) of inner type. By assumption, the projections of X × Y to X and Y are projective vector bundles. By the Projective Bundle theorem (Example 6.2),

M(X × Y) ≃ M(X) ⊕ M(X)(1) ⊕ ⋯ ⊕ M(X)(m),

M(X × Y) ≃ M(Y) ⊕ M(Y)(1) ⊕ ⋯ ⊕ M(Y)(n),

where n = dim X and m = dim Y. If A and B are division algebras then the motives M(X) and M(Y) are indecomposable by [6, Th. 2.2.1]. On the other hand, the integral motives of X and Y are not isomorphic if A is not isomorphic to B or the opposite algebra B^{op} [8, Criterion 7.1].

Remark 9.5. A. Vishik proved in [10] that the Krull-Schmidt theorem holds for the integral motives of projective quadrics.

We use the notation of Section 7.

Theorem 9.6. Let R be a discrete valuation ring and let G be a semisimple group such that the group Γ_{F} satisfies the condition (∗). Then

(1) The Krull-Schmidt theorem holds for M(G, R).

(2) The category M(G, R) is finitely generated.

Proof. The second statement follows from (1) and Theorem 8.5. Let M be an indecomposable motive in M(G, R). By Theorem 6.1, it suffices to show that the ring End(M) is local. Let S be the image of End(M) under the R-algebra homomorphism f : End(M) → End(M_{sep}). Since G is split over F_{sep}, the R-module End(M_{sep}) is finitely generated, hence S is an R-order in S ⊗_{R} K. By Theorem 8.2, the kernel of f is a nil ideal. Therefore, it is sufficient to prove that S is a local ring.

The K-algebra S ⊗_{R} K is the endomorphism ring of M_{K} in M(G, K). By Proposition 9.2 the category M(G, K) is equivalent to Inv(KΓ_{F})^{ℕ}, hence S ⊗_{R} K is isomorphic to a product of rings of the form End_{KΓ_{F}} (N) for some finitely generated KΓ_{F} -module N. It follows that the endomorphism ring of a simple S ⊗_{R} K-module is isomorphic to the K-algebra D_{i} for some i. By the condition (∗), D_{i} ⊗K̂ is a division K̂-algebra. It is proven in [5, Th. 30.18] that for every indecomposable S-lattice T, the Ŝ-lattice T̂ is also indecomposable. Applying this statement to T = S we see that the ring Ŝ has no nontrivial idempotents. The lifting property for the idempotents [1, Ch. III, Prop. 2.10] shows that the factor ring S/𝔪S = Ŝ/𝔪Ŝ has no nontrivial idempotents. The ring S/𝔪S is a finite dimensional algebra over the field R/𝔪R and hence is local.

We are finally ready to prove that S is a local ring. Let a + b = 1 for a, b ∈ S. Since the ring S/𝔪S is local one of the residues ā or b̄ in S/𝔪S, say ā, is invertible. By Nakayama Lemma, the left and right multiplication endomorphisms by a of the finitely generated R-module S are surjective, therefore, a is invertible in S and hence S is a local ring. □

Remarks 7.2 and 7.3 imply

Corollary 9.7. The Krull-Schmidt theorem holds for M(G, R) in the following cases:

(1) R = ℤ_{(p)} and G is a simple group.

(2) G has inner type.

(3) The group Gal(F_{G}/F) is split.

(4) R is a complete discrete valuation ring.

References

[1] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

[2] A. Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991.

[3] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. (1965), no. 27, 55–150.

[4] V. Chernousov, S. Gille, and A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, to appear in Duke Math. J.

[5] C. W. Curtis and I. Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1990, With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Interscience Publication.

[6] N. A. Karpenko, Grothendieck Chow motives of Severi-Brauer varieties, Algebra i Analiz 7 (1995), no. 4, 196–213.

[7] N. A. Karpenko, Cohomology of relative cellular spaces and of isotropic flag varieties, Algebra i Analiz 12 (2000), no. 1, 3–69.

[8] N. A. Karpenko, Criteria of motivic equivalence for quadratic forms and central simple algebras, Math. Ann. 317 (2000), no. 3, 585–611.

[9] Ju. I. Manin, Correspondences, motifs and monoidal transformations, Mat. Sb. (N.S.) 77 (119) (1968), 475–507.

[10] A. Vishik, Motives of quadrics with applications to the theory of quadratic forms, Lect. Notes Math., 1835 (2003) Springer-Verlag, Berlin.

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Department of Mathematics, University of California, Los Angeles, CA 90095-1555

E-mail address: chernous@math.ualberta.ca
E-mail address: merkurev@math.ucla.edu