On the Motive of a Commutative Algebraic Group

Giuseppe Ancona, Stephen Enright-Ward, Annette Huber

Received: December 3, 2014 Revised: June 30, 2015

Communicated by Alexander Merkurjev

Abstract. We prove a canonical Künneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology. We also deduce applications of the decomposition to the existence of a weight filtration, computation of any Weil cohomology theory and study of 1-motives.

2010 Mathematics Subject Classification: 14C15 (primary), 14L10, 19E15, 14F20, 14K99 (secondary) Keywords and Phrases: Voevodsky motives, semiabelian varieties, Künneth decomposition, 1-motives, Kimura finiteness

December 16, 2015

Contents

Introduction  808
1. Notations and Generalities  811
2. The 1-motive M_{1}(G) of G  817
3. Main Theorem  820
4. Special cases  822
5. Properties of M_{1}(G)  827
6. Proof of the Main Theorem  830
7. Consequences  834
Appendix A. Comparison of Chern classes  842
Appendix B. Universal properties of the symmetric (co)algebra  845
Appendix C. Filtrations on the graded symmetric (co)algebra  849
References  856

808 Giuseppe Ancona, Stephen Enright-Ward, Annette Huber

Introduction

The aim of this article is to prove a canonical Künneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology.

Let us start with the concrete example of a connected commutative algebraic group G over ℂ, the field of complex numbers. Then the complex manifold associated with G is homotopy equivalent to a product of unit circles S^{1}. So, by the Künneth formula for singular cohomology, one has

H_{sing}^{i}(G(ℂ), ℚ) = ⋀^{i} H_{sing}^{1}(G(ℂ), ℚ).

This phenomenon exists also for ℓ-adic cohomology. Let G be a connected commutative group scheme over a field k and let ℓ a prime number different from the characteristic of k. Then H^{∗}(Gk̄, ℚ_{ℓ}) is a graded finite-dimensional ℚ_{ℓ}- vector space. Moreover, it has a canonical structure of graded commutative and cocommutative connected Hopf algebra coming from the group structure and the diagonal immersion Δ : G → G × G (the latter induces the cup product). The classification of graded Hopf algebras implies that H^{∗}(Gk̄, ℚ_{ℓ}) has to be the exterior algebra of its primitive part. It turns out to be H^{1}(Gk̄, ℚ_{ℓ}), the dual of the Tate module tensor ℚ_{ℓ}. So one deduces an isomorphism of graded Hopf algebras

H^{∗}(Gk̄, ℚ_{ℓ}) = ⋀^{∗} H^{1}(Gk̄, ℚ_{ℓ}) .

Our main theorem shows that this formula is motivic, up to two minor subtleties. First, in odd degree symmetric powers realize to the exterior powers (Koszul rule of signs), so one has to consider the former. Second, the realization functors from motives to cohomology is contravariant so one has to consider the opposite Hopf algebra structure.

Theorem 1. Let G be a connected commutative algebraic group over a field k. Then there exists a canonical decomposition in DM_{gm}^{eff}(k)_{ℚ}

M(G) = ⨁_{i=0} M_{i}(G),

such that, for any mixed Weil cohomology theory H^{∗}, the motive M_{i}(G) realizes to H^{i}(G). Moreover,

(1) for i big enough the motives Sym^{i}(M_{1}(G)) and M_{i}(G) vanish,

(2) there is a canonical isomorphism

M_{i}(G) ≅ Sym^{i}M_{1}(G) ,

(3) The canonical isomorphism

M(G) ≅ coSym(M_{1}(G)) = ⨁_{i} Sym^{i}M_{1}(G) is an isomorphism of Hopf algebras between the motive of G and the symmetric coalgebra over the motive M_{1}(G).

For more refined statements see Theorems 7.1.1 and 7.2.2 There are a number of useful applications of our motivic decomposition. For instance, we are able to describe the weight filtration on M_{i}(G) explicitly (see Section 7.3). We get a new proof of Kimura finiteness of 1-motives (which implies Kimura finiteness of the motive of a curve). We also show that H^{∗}(M) is concentrated in one degree for any 1-motive M and any mixed Weil cohomology theory. Recently, Sugiyama [Sug14] has applied Theorem 1 to generalize to semiabelian varieties results of Beauville [Bea86] and Bloch [Blo76, Theorem 0.1] on Chow groups of abelian varieties. In a sequel of this paper [AHPL], we will show that all the results in this introduction hold for the relative motive of a general smooth commutative group scheme G over a base scheme S (supposed noetherian and finite dimensional). Note that Theorem 1 was known for abelian schemes over a regular base [DM91, Kün94]. Our generalization will be interesting especially for Néron models of abelian varieties and for the group scheme over compactifications of Shimura varieties.

Theorem 1 was already known for tori and abelian varieties. The straightforward case of tori was discussed partially by Huber-Kahn [HK06]. The abelian case was proven by Künnemann [Kün94] following earlier partial results by Shermenev [Šer74] and Deninger-Murre [DM91]. The main tool in his proof is the Fourier transform for cycles over an abelian variety, introduced by Beauville [Bea86]. This is not available for more general groups. Instead, our starting point is to use the more flexible category motivic complexes. Indeed, we write down the component M_{1}(G) completely explicitly. This is non-trivial, even for abelian varieties. First, following Barbieri-Viale and Kahn [BVK], consider \underline{G} the sheaf on the big étale site on Sm/k represented by G. It inherits from the group structure of G a canonical structure of sheaf of abelian groups. By work of Spieß-Szamuely ([SS03]) \underline{G} has transfers, hence it induces a motive (namely the singular complex \underline{C}_{∗}\underline{G}) that we will note M_{1}(G). Second, consider the map of (pre)sheaves with transfers

Cor(·, G) → \underline{G}

which maps a multivalued map S → G (a correspondence) to the sum of its values in the commutative group G. It induces a canonical map

α_{G} : M(G) → M_{1}(G) .

Using the comultiplication on M(G) this extends to a natural map

φ_{G}^{n} : M(G) → Sym^{n}M_{1}(G) .

Most of the effort of this paper goes into proving the following (Theorem 3.1.4 in the paper):

Theorem 2. Let G be a semiabelian variety over a perfect field k. Then the motive M_{1}(G) is odd (i.e. Sym^{n}M_{1}(G) vanishes for n big enough) and, moreover, the map

φ_{G} = ⨁_{n} φ_{G}^{n} : M(G) ⟶ ⨁_{n=0} Sym^{n}M_{1}(G).

is an isomorphism.

We will easily deduce Theorem 1 from Theorem 2: the construction of the morphism φ_{G} has been done so that it is natural and it respects the Hopf structures (and the unipotent part of a general commutative group will be easy to treat, as well as the non-perfect case). A difficulty in showing this theorem is that we do not have for free natural maps in the opposite direction (the reader can think of the embedding Mor_{Sm/k}(·, G) ⊂ Cor(·, G), but this is just a morphism of sheaves of sets). We prove Theorem 2 by induction on the torus rank with the case of abelian varieties as a starting point. Consider a short exact sequence of semiabelian varieties

1 → 𝔾_{m} → G → H → 1

and suppose to know Theorem 2 for H. On the one hand, the filtration on \underline{G} defined by the above short exact sequence induces triangles

Sym^{n}M_{1}(G) → Sym^{n}M_{1}(H) → Sym^{n−1}M_{1}(H)(1)[2] .

On the other hand, the localization sequence for the 𝔾_{m}-torsor G → H reads

M(G) → M(H) → M(H)(1)[2] .

By comparing these triangles, the hypothesis that φ_{H} is an isomorphism implies that there is a non-canonical isomorphism

ψ : M(G) → ⨁_{n} Sym^{n}(M_{1}(G)) .

This fact has two essential consequences. First, the motive M(G) is Kimura finite. Second, there is a non-trivial morphism from ⨁_{n} Sym^{n}(M_{1}(G)) to M(G). We then show that one can modify this morphism in order to obtain a map which is the inverse of φ_{G} after ℓ-adic realization. Then Kimura finiteness allows us to conclude that φ_{G} is an isomorphism.

We now briefly discuss the structure of the paper. Section 1 settles notations and recalls some facts from the literature. In Section 2 we define the motive M_{1}(G) and the morphism α_{G} and establish basic properties. We will pretty quickly specialize to the case of a semiabelian variety. In Section 3 we construct the morphism φ_{G} and show that it respects the Hopf structures, under the hypothesis that Sym^{n}M_{1}(G) vanishes for n big enough. Section 4 deals with the two special cases: tori and abelian varieties. In Section 5 we show that the motive M_{1}(G) of a semiabelian variety G is odd and geometric. We then compute its ℓ-adic realization. Section 6 finally gives the proof of Theorem 2 for any semiabelian variety G. Section 7 shows how to deduce Theorem 1 from Theorem 2. We also explain how the semiabelian case implies the case of a general commutative group scheme, possibly with a unipotent radical or several connected components. Some other properties are studied, namely the uniqueness of the Künneth decomposition and that the weight filtration in cohomology lifts canonically to a filtration of each Künneth component. Some technical points of the main proof are left to the appendices. Appendix A is the essential input to relate the two triangles above. This is done comparing two obvious definitions of the first Chern class of the line bundle defined by an extension of a semiabelian variety by 𝔾_{m}. Appendix B rewiews the definition of the symmetric coalgebra, its universal properties and a comparison with the standard symmetric algebra in the setting of ℚ-linear pseudo-abelian additive categories. Appendix C constructs a natural filtration on Sym^{n}(V) given a subobject U ⊂ V in the setting of general ℚ-linear abelian categories.

Acknowledgments. Part of these results have already been obtained in the PhD thesis [EW13] of the second author, written under the supervision of the third. The first author would like to thank them for letting him participate to conclude this work and for the warm hospitality in Freiburg. In more detail: the definition of φ_{G} was given in [EW13] and the results on tori and abelian varieties were established there. It also contained the inductive argument of Section 6.1 but not the proof of the main Theorem 2. We would like to thank J. Ayoub, F. Déglise, B. Drew, J. Fresán, M. Huruguen, P. Jossen, B. Kahn, S. Kelly, K. Künnemann, S. Meagher, S. Pepin Lehalleur, J. Scholbach, R. Sugiyama, and C. Vial for useful discussions. R. Sugiyama pointed out a gap in an earlier version of our preprint. The problem is resolved by his [Sug14, Appendix B]. We are most thankful to M. Wendt, who has been involved in the project from the start. He came to act as a coadvisor of the thesis and contributed generously by discussions, advice on references and careful proof reading. The (as it turns out decisive) remark that the very formulation of the main theorem needs a finiteness assumption, was his. Finally, we would like to thank the referee for useful comments.

1. Notations and Generalities

Throughout this paper k denotes a fixed base field of any characteristic. We fix an algebraic closure of k and we call it k̄. We write \overline{(·)} for the pull-back to k̄ on varieties, motives and sheaves.

1.1. Categories of varieties. We denote:

• Sch/k the category of schemes separated and of finite type over k;

• Sm/k the full subcategory of smooth schemes over k; • SmPrVar/k the full subcategory of smooth and projective schemes

over k;

• SmCor/k the category of smooth correspondences. Objects are smooth schemes over k and morphisms are finite correspondences in the sense of Voevodsky [Voe00, Section 2]. The free abelian group generated by primitive finite correspondences from X to Y will be denoted Cor(X, Y);

• cGrp/k the category of commutative group schemes of finite type over k with morphisms of k-group schemes;

• sAb/k the full subcategory of cGrp of semiabelian varieties. By definition an object of sAb/k is a connected smooth commutative group scheme G over k such that its pullback Gk̄ does not have any subgroups which are isomorphic to an additive group.

• Ab/k the full subcategory of cGrp of abelian varieties. We will usually abbreviate Sch instead of Sch/k etc.

Remark 1.1.1. Suppose that the base field is perfect. By a theorem of Barsotti [Bar55] and Chevalley [Che60] (see also [Con02] for a modern presentation) any semiabelian variety can be uniquely decomposed as an extension of an abelian variety by a torus. Moreover, there are no non-constant homomorphism from a torus to an abelian variety, see for example [Con02, Lemma 2.3]. In particular any homomorphism between two semiabelian varieties induces morphisms between the tori and between the abelian varieties of their decompositions.

Remark 1.1.2. Notice that a short exact sequence in cGrp of smooth groups induces a short exact sequence of étale sheaves, as any smooth morphism has a section locally for the étale topology.

1.2. Rational coefficients. When A is an additive category, then we will write A_{ℚ} for the pseudo-abelian hull of the category having the same objects as A and such that for any X, Y ∈ A the set of the homomorphisms from X to Y is the ℚ-vector space

Hom_{A}(X, Y) ⊗_{ℤ} ℚ .

This applies, in particular, to the additive categories SmCor, Ab and sAb.

Remark 1.2.1. (1) Notice that morphisms in SmCor_{ℚ} between two vari-

eties are ℚ-linear combinations of primitive finite correspondences.

(2) The categories Ab_{ℚ} and sAb_{ℚ} are the categories of abelian and semiabelian varieties up to isogeny, respectively. These categories are abelian.

1.3. Symmetric powers, exterior powers and Kimura finiteness. Let A be a pseudo-abelian ℚ-linear symmetric tensor category with unit 𝟙. There are canonical functors

Sym^{n} : A → A .

Indeed, let X ∈ A be an object of A. As the category is symmetric the group of permutations S_{n} acts on X^{⊗n}. The endomorphism

\frac{1}{n!} ∑_{σ∈S_{n}} σ : X^{⊗n} → X^{⊗n}

is a projector. Then one defines Sym^{n}(X) to be its image (notice that it is fonctorial in X ∈ A). We define similarly the functor ⋀^{n} : A → A. By convention, we will write Sym^{0}X = ⋀^{0} X = 𝟙 for all non-zero objects X. Following [Kim05] and [O’S05] we will say that an object X is

• odd of dimension N if Sym^{N}X ≠ 0 and Sym^{N+1}X = 0. In this case we will write det X = Sym^{N}X;

• even of dimension N if ⋀^{N} X ≠ 0 and ⋀^{N+1} X = 0. In this case we will write det X = ⋀^{N} X;

• odd or even finite-dimensional if it is even or odd finite-dimensional for some N;

• Kimura finite if there exists a decomposition (in general not unique) X = X_{+} ⊕ X_{−} such that X_{+} is even finite-dimensional and X_{−} is odd finite-dimensional.

The result we will need about finite-dimensional motives is the following theorem of André and Kahn.

Theorem 1.3.1 (André, Kahn). Let K be a field of characteristic zero, C be a K-linear pseudoabelian symmetric tensor rigid category, with unit object 𝟙 satifsfying End_{C}(𝟙) = K. Let f : X → Y be a map in C between two finite-dimensional objects. Suppose that there exist another K-linear pseudo-abelian symmetric tensor category D, a non-zero K-linear symmetric tensor functor F : C → D (covariant or contravarient) and a map g : Y → X in C such that F (f) and F (g) are inverses of the other. Then f is an isomorphism.

Proof. This is a consequence of the work [AK02]. As this theorem is never stated in this form let us explain how to deduce it from loc. cit. We will suppose F to be covariant (otherwise one can compose with the canonical antiequivalence of D with its opposite category). First of all we can replace C by the K-linear symmetric tensor category generated by finite-dimensional objects, hence suppose that C is a Kimura category. The kernel of the functor F is a tensor ideal of C, which is by hypothesis nontrivial. Then by [AK02, Theorem. 9.2.2] the kernel is contained inside a tensor ideal R called the radical and C is a Wedderburn category. Write π : C → C/R for the canonical functor. Our hypothesis implies that π(f) and π(g) are inverses of each other. We can now conclude because by [AK02, Proposition 1.4.4(b)] the functor π detects the isomorphism, i.e., π(f) is an isomorphism if and only if f is. Notice that the erratum of [AK02] does not concern the statements we are using. □

1.4. Symmetric coalgebra. Let X be an odd object of dimension N in a pseudo-abelian ℚ-linear symmetric tensor category with unit 𝟙. We define the symmetric coalgebra

coSym(X) = ⨁_{n=0}^{N} Sym^{n}(X) .

The coalgebra coSym(X) has a canonical structure of Hopf algebra. Multiplication and comultiplication are defined so that coSym(X) becomes the standard symmetric algebra in the opposite category. For details see Appendix B.

1.5. Graded super-vector spaces. Let F be a field. The category of graded super-F -vector spaces will be denoted by GrVec_{F}^{±}. As an F -linear tensor category it is equivalent to the category of graded F -vector spaces. In particular if V = ⨁_{i∈ℤ} V_{i} and W = ⨁_{i∈ℤ} W_{i}, then

(V ⊗ W)_{n} = ⨁_{p+q=n} V_{p} ⊗ W_{q}.

The difference is in the convention of the symmetry. If σ_{i,j} : V_{i} ⊗W_{j} → W_{j} ⊗V_{i} is the isomorphism of symmetry in the category of F -vector spaces, then the isomorphism of symmetry in GrVec_{F}^{±} is given by (−1)^{i·j}σ_{i,j}.

1.6. Chow and Voevodsky motives. Let k be a perfect field (for the nonperfect case see Section 1.7). Our conventions on motives will follow notations and constructions from Voevodsky [Voe00] and especially from [MVW06, Section 14], where the case of rational coefficients is explicitly treated. We will study the following categories:

ShT_{ét}(k, ℚ) , DM_{−,ét}^{eff}(k, ℚ) , DM_{gm}^{eff}(k)_{ℚ} , DM_{gm}(k)_{ℚ} , Chow(k)_{ℚ}

of étale sheaves with transfers, motivic complexes, effective geometric motives, geometric motives, and Chow motives, respectively. We are going to need the functors

L : Sm_{k} → D^{−}(ShT_{ét}(k, ℚ)) and q : D^{−}(ShT_{ét}(k, ℚ)) → DM_{−,ét}^{eff}(k, ℚ) .

We will write

M = q ∘ L ,

and for any variety X ∈ Sm_{k} we will call M(X) the motive of X. Let us give more details. The category of sheaves of ℚ-vector spaces with transfers on the big étale site on Sm/k is denoted

ShT_{ét}(k, ℚ) .

By [MVW06, Lemma 14.21] a rational presheaf with transfers is a sheaf for the Nisnevich topology if and only if it is a sheaf for étale topology. We decide to work with the latter topology. Tensor product is exact on this category by [Sug14, Proposition B.1]. Moreover, by [MVW06, Theorem 14.30] the categories DM_{−,ét}^{eff}(k, ℚ) and DM_{,Nis}^{eff}(k, ℚ) are equivalent.

By [MVW06, Theorem 14.28] and [MVW06, Remark 14.29] there are two adjoint ℚ-linear exact functors

q : D^{−}ShT_{ét}(k, ℚ) ⇄ DM_{−,ét}^{eff}(k, ℚ) : i .

The functor q is compatible with tensor products. Moreover, the functor i is an embedding and the functor q is a localization. We may identify DM_{−,ét}^{eff}(k, ℚ) with its image in D^{−}ShT_{ét}(k, ℚ) and avoid to write i. The category of effective geometric motives will be denoted by

DM_{gm}^{eff}(k)_{ℚ} .

Several equivalent definitions are possible, here we define it as the full triangulated ℚ-linear pseudo-abelian tensor sub-category of DM_{−,ét}^{eff}(k, ℚ) generated by motives of smooth schemes over k.

Remark 1.6.1. (1) Let DM_{gm}^{eff}(k, ℤ) be the category of effective geometric motives with integer coefficients defined by Voevodsky [Voe00, Definition 2.1.1], then

DM_{gm}^{eff}(k)_{ℚ} = DM_{gm}^{eff}(k, ℤ)_{ℚ} ,

where the tensor product on the right hand side is in the sense of Section 1.2. This fact can be deduced from [CD09, Theorem 11.1.13], [CD09, Remark 9.1.3(3)] and [CD09, Corollary 16.1.6].

(2) The reader should be warned that ShT_{ét}(k, ℚ) and DM_{−,ét}^{eff}(k, ℚ) are not ShT_{ét}(k, ℤ)_{ℚ} and DM_{−,ét}^{eff}(k, ℤ)_{ℚ}. We will always work with ShT_{ét}(k, ℚ) and DM_{−,ét}^{eff}(k, ℚ).

After tensor-inverting the Lefschetz motive inside DM_{gm}^{eff}(k)_{ℚ} we obtain a ℚlinear pseudo-abelian triangulated rigid tensor category, denoted by DM_{gm}(k)_{ℚ} and called the category of geometric motives. The category DM_{gm}^{eff}(k)_{ℚ} is canonically a full subcategory of DM_{gm}(k)_{ℚ} by Voevodsky’s Cancellation Theorem [Voe10]. The ℚ-linear pseudo-abelian rigid tensor sub-category of DM_{gm}(k)_{ℚ} generated by motives of smooth and projective schemes over k will be denoted by Chow(k)_{ℚ} and called the category of Chow (or pure) motives (with ℚ coefficients). Note that by [Voe00, Proposition 2.1.4] together with [Voe02], this category is equivalent to the opposite of the classical category of Chow motives. Although we will use properties of classical Chow motives transposed in this context, we will always keep the notations coming from the conventions of Voevodsky, for instance the motive of the projective line decomposes as M(ℙ_{k}^{1}) = 𝟙 ⊕ 𝟙(1)[2]. We are going to need the following facts.

Proposition 1.6.2. Let f be a morphism in DM_{−,ét}^{eff}(k, ℚ). Then f is an isomorphism if and only if its pull-back \overline{f} to k̄ is. In particular a motive M vanishes if and only if \overline{M} does.

Proof. Let f be a morphism in DM_{−,ét}^{eff}(k, ℚ). Then f is an isomorphism if and only if it is an isomorphism in the derived category of étale sheaves, i.e., if it induces isomorphism on cohomology sheaves without transfers. A morphism of étale sheaves is an isomorphism if it is an isomorphism on geometric stalks. It suffices to consider geometric points over closed points of objects in Sm/k. Hence we can check it after pull back to the algebraic closure. □

Remark 1.6.3. Voevodsky’s tensor product on ShT_{ét}(k, ℚ) is exact because by [SV96, §5 and §6] the category can be seen as a full tensor subcategory of the category of qfh-sheaves with the standard tensor product of sheaves. This argument is due to Sugiyama, see [Sug14, Appendix B] for full details.

1.7. Motives over non-perfect fields. To avoid the hypothesis of perfectness one can consider Beilinson motives with rational coefficients following Cisinski and Déglise [CD09]. These motives are defined over general bases and when the base is a perfect field their definition is equivalent to the category of geometric Voevodsky motives with ℚ-coefficients DM_{gm}^{eff}(k)_{ℚ}, see [CD09, Corollary 16.1.6]. Suppose now that k is not perfect, and let k^{i} be its perfect closure. Then by [CD09, Proposition 2.1.9] and [CD09, Theorem 14.3.3] the pull-back from DM_{gm}^{eff}(k)_{ℚ} to DM_{gm}^{eff}(k^{i})_{ℚ} is an equivalence of category.

1.8. Realization functor. Fix a prime number ℓ different from the characteristic of k. We denote

H^{∗} = ⨁_{i∈ℤ} H^{i} : Sm_{k} → Sm\overline{k} ⟶ GrVec_{ℚ_{ℓ}}^{±}

the contravariant functor of ℓ-adic cohomology. By work of Ivorra [Ivo07, Theorem 4.3] it extends uniquely to a contravariant functor called realization

H^{∗} : DM_{gm}(k) ⟶ GrVec_{ℚ_{ℓ}}^{±}

which is a ℚ-linear symmetric tensor functor, sending the unit to the unit triangles to long exact sequences and which moreover verifies H^{∗}(M(n)) = H^{∗}(M)(−n) and H^{m}(M[n]) = H^{m−n}(M) for all integers m and n and all motives M.

The structure of ℓ-adic cohomology of semiabelian varieties is well known.

Lemma 1.8.1. Let G be a semiabelian variety. Then

H^{∗}(G) = Sym(H^{1}(G)) .

Moreover, let V_{ℓ}(G) be the ℓ-adic Tate module tensored by ℚ_{ℓ}, then we have

H^{1}(G) = V_{ℓ}(G)^{∗} .

Proof. The abelian or torus case are classical. For the semiabelian case see e.g. [BS13, Lemma 4.1 and 4.2]. The rational case follows from their stronger assertion with torsion coefficients. □

Lemma 1.8.2. The functor H^{1} is exact and faithful on sAb_{ℚ}.

Proof. From the explicit formula for H^{1} we get both the exact sequence and a formula for the dimension of H^{1}(G). In particular, it does not vanish, if G is non-trivial. □

2. The 1-motive M_{1}(G) of G

Let k be a perfect field and G be a smooth commutative group scheme over k (often a semiabelian variety). In this section we are going to construct a natural morphism

α_{G} : M(G) ⟶ M_{1}(G)

in the category DM_{−,ét}^{eff}(k, ℚ) of triangulated motives, where M_{1}(G) is the the 1-motive of G to be defined below. This is based on the work of Barbieri-Viale/Kahn [BVK], Spieß/Szamuely [SS03], and Suslin/Voevodsky [SV96]. There is no need to work with rational coefficients in this section, but we prefer to put them in the setting (they will become essential later).

2.1. Adding transfers.

Definition 2.1.1. Let \underline{G} be the presheaf of abelian groups Sm/k defined by G, i.e.,

\underline{G}(S) = Mor_{Sm/k}(S, G)

for S ∈ Sm/k. We denote \underline{G}_{ℚ} = \underline{G} ⊗_{ℤ} ℚ the presheaf tensor product, i.e.,

\underline{G}_{ℚ}(S) = Mor_{Sm/k}(S, G) ⊗_{ℤ} ℚ

for S ∈ Sm/k.

Lemma 2.1.2 (Spieß-Szamuely, Orgogozo). Let G be a smooth commutative group scheme over k, then \underline{G}_{ℚ} is an étale sheaf with transfers of ℚ-vector spaces. When G is a semiabelian variety, \underline{G}_{ℚ} is homotopy invariant.

Proof. Spieß and Szamuely showed in [SS03, Proof of Lemma 3.2] that \underline{G} is a presheaf with transfers (for more details we refer to second author’s thesis [EW13, Section 2.3]). Homotopy invariance follows as in the abelian or torus case, see [Org04, Lemma 3.3.1]. We have to check the sheaf condition for \underline{G}_{ℚ}. It is enough to check this for covers with finitely many objects, as a general cover can be refined to such a cover. On the other hand the sheaf condition in this case is implied by the sheaf condition of S ↦ Mor_{Sm/k}(S, G), which is true by étale descent. □

Let us recall the relation between finite correspondences Cor(S, X) and multivalued maps. Let S be a connected smooth k-scheme and W ⊂ S × X be an irreducible subvariety finite and surjective over S, i.e., a primitive finite correspondence. Let d be the degree of W/S. By work of Suslin and Voevodsky [SV96, Preamble to Theorem 6.8] there is an associated morphism

S → S^{d}(X) ,

where S^{d}(X) = X^{d}/S_{d} is the symmetric power of X. From this description, the following is straightforward.

Proposition 2.1.3. Let G be a commutative group scheme over k. Then there is a canonical map of sheaves with transfers

γ_{G} : Cor(·, G) ⊗_{ℤ} ℚ → \underline{G}_{ℚ} ,

characterized by the fact that it maps a morphism S →^{f} S^{d}(G) to

S →^{f} S^{d}(G) →^{μ} G ,

where μ : S^{d}(G) → G is the summation map. Moreover, γ_{G} is natural in G ∈ cGrp_{k}.

Definition 2.1.4. Let G be a semiabelian variety over k. Recall the functor q : D^{−}ShT_{ét}(k, ℚ) → DM_{−,ét}^{eff}(k, ℚ) (see Section 1.6).

(1) We denote

M_{1}(G) ∈ DM_{−,ét}^{eff}(k, ℚ)

the image under the functor q of the complex given by \underline{G}_{ℚ} concentrated in degree in 0. We call M_{1}(G) the 1-motive defined by G.

(2) We denote

α_{G} : M(G) → M_{1}(G)

the image under the functor q of the morphism

γ_{G} : Cor(·, G) ⊗_{ℤ} ℚ → \underline{G}_{ℚ}

of complexes concentrated in degree in 0.

Remark 2.1.5. The definition of M_{1}(G) is a special case of the embedding of the category of 1-motives into triangulated motives constructed by Barbieri-Viale/Kahn [BVK]. The notation M_{1}(G) should suggest the first Künneth component of G. This intuition is justified by Proposition 2.2.1 and Lemma 2.2.2 (and of course by the main Theorem 3.1.4).

Remark 2.1.6. The adjunction q : D^{−}ShT_{ét}(k, ℚ) ⇄ DM_{−,ét}^{eff}(k, ℚ) : i (Section 1.6) gives a canonical identification

Hom_{DM_{−,ét}^{eff}(k,ℚ)}(M(G), M_{1}(G)) = Hom_{D^{−}ShT_{ét}(k,ℚ)}(L(G), iM_{1}(G)) .

On the other hand \underline{G}_{ℚ} is homotopy invariant (Lemma 2.1.2) so by [MVW06, Corollary 14.9] iM_{1}(G) is the complex \underline{G}_{ℚ} concentrated in degree zero. Altogether we have a canonical identification

Hom_{DM_{−,ét}^{eff}(k,ℚ)}(M(G), M_{1}(G)) = Hom_{ShT_{ét}(k,ℚ)}(L(G), \underline{G}_{ℚ}) .

Under this identification α_{G} corresponds to γ_{G}.

2.2. Elementary properties.

Proposition 2.2.1 (Orgogozo). The assignment

M_{1} : sAb → DM_{−,ét}^{eff}(k, ℚ)

is an exact functor, i.e., it maps short exact sequences to exact triangles. It is isogeny invariant and factors via an exact functor

M_{1} : sAb_{ℚ} → DM_{−,ét}^{eff}(k, ℚ) .

In particular, it maps direct products in sAb_{ℚ} to direct sums and multiplication by n on a semiabelian variety G to multiplication by n on M_{1}(G). Moreover, M_{1} is a full embedding, i.e., for any two semiabelian varieties G and H we have a natural isomorphism

Hom_{sAb_{ℚ}}(G, H) = Hom_{DM_{−,ét}^{eff}(k,ℚ)}(M_{1}(G), M_{1}(H)) .

Proof. The fact that M_{1} is a full embedding is shown more generally for the derived category of 1-motives in [Org04, Proposition 3.3.3]. Exactness is explained in the preamble of loc. cit. □

Lemma 2.2.2. The morphism α_{G} : M(G) → M_{1}(G) is natural in G ∈ sAb/k and it is always non-zero for G ≠ 0 ∈ sAb_{ℚ}.

Proof. By Remark 2.1.6 we can replace in the statement α_{G} by the map γ_{G} (Proposition 2.1.3). Naturality is part of Proposition 2.1.3 and we have γ_{G} ≠ 0 because it does not vanish on id_{G} ∈ Cor(G, G) ⊗_{ℤ} ℚ. □

Corollary 2.2.3. Let G be a semiabelian variety. Let μ be the multiplication, −1 the inverse, 0 the unit and ϵ the structural map. Then

M_{1}(μ) = + :M_{1}(G) ⊕ M_{1}(G) → M_{1}(G) ,
M_{1}(−1) = −1 :M_{1}(G) → M_{1}(G) ,
M_{1}(0) = 0 :M_{1}(Speck) → M_{1}(G) ,
M_{1}(ϵ) = 0 :M_{1}(G) → M_{1}(Speck) .

Proof. The statement on μ holds by definition. The second follows from Proposition 2.2.1 with n = −1. The last two are trivial from M_{1}(Speck) = 0. □

Recall that ¯(·) denotes the pull-back to k̄.

Lemma 2.2.4. Let G be a semiabelian variety over k.

\overline{M(G)} = M(\overline{G}) ,
\overline{M_{1}(G)} = M_{1}(\overline{G}) ,
\overline{α_{G}} = α\overline{G} .

Proof. The analogous statement in ShT_{ét}(k̄, ℚ) for Cor(·, G) ⊗_{ℤ} ℚ, \underline{G}_{ℚ} and γ_{G} can be proved just by checking the definitions. Then one deduces the statement by applying the functor q. □

3. Main Theorem

In this section we state the main theorem. The point is to construct a morphisms φ_{G} and the theorem will essentially state that it is an isomorphism. We also establish some basic properties of φ_{G}.

3.1. The morphism φ_{G}.

Definition 3.1.1. Let G be a semiabelian variety over k, n ≥ 0 be an integer and Δ_{G}^{n} be the n-fold diagonal. We define φ_{G}^{n} to be the morphism

φ_{G}^{n} : M(G) ⟶^{M(Δ_{G}^{n})} M(G)^{⊗n} ⟶^{α_{G}^{⊗n}} M_{1}(G)^{⊗n}.

As Δ_{G}^{n} is invariant under permutations, this factors uniquely

\xymatrix{M(G) \ar[rr]^{α_{G}^{⊗n}M(Δ_{G}^{n})} \ar[dr]_{φ_{G}^{n}} &  & M_{1}(G)^{⊗n} \\  & Sym^{n}(M_{1}(G)) \ar[ur]}

Remark 3.1.2. Equivalently, we have

φ_{G}^{n} : M(G) ⟶^{M(Δ_{G}^{n})} M(G)^{⊗n} ⟶^{α_{G}^{⊗n}} M_{1}(G)^{⊗n} ⟶ Sym^{n}(M_{1}(G)),

which was the original definition in [EW13].

Definition 3.1.3. Let G be a semiabelian variety over k which is an extension of an abelian variety of dimension g by a torus of rank r. Define the morphism φ_{G} as

φ_{G} = ⨁_{n=0}^{2g+r} φ_{G}^{n} : M(G) ⟶ ⨁_{n=0}^{2g+r} Sym^{n}(M_{1}(G)).

Our main theorem is the following; it will be proven in Section 6.

Theorem 3.1.4. Let k be a perfect field and G be a semiabelian variety which is an extension of an abelian variety of dimension g by a torus of rank r. Then the motive M_{1}(G) is odd of dimension 2g + r and the map

φ_{G} : M(G) ⟶ ⨁_{n=0}^{2g+r} Sym^{n}(M_{1}(G)).

is an isomorphism of motives.

More refined statements will be deduced in Section 7.

3.2. First properties and reductions. Let G be a semiabelian variety. The motive M(G) has a canonical Hopf algebra structure induced by morphisms of varieties:

• multiplication by multiplication on G;

• comultiplication by the diagonal Δ : G → G × G; • the antipodal map by the inverse on G;

• the unit by the neutral element;

• the counit by the structure map to the base field. The aim here is to show that the morphism φ_{G} of Definition 3.1.3 is a natural morphism of Hopf algebras from M(G) to the symmetric coalgebra

coSym(M_{1}(G)) = ∏_{n≥0} Sym^{n}(M_{1}(G)) .

To consider such an object one needs to work under the following assumption:

Finiteness Assumption 3.2.1. The motive M_{1}(G) is odd of dimension 2g +r (here G is a semiabelian variety which is an extension of an abelian variety of dimension g by a torus rank r).

Remark 3.2.2. The finiteness assumption is needed to make coSym(M_{1}(G)) an object of DM_{−,ét}^{eff}(k, ℚ). We are going to establish later (see Proposition 5.1.1) that this assumption is always satisfied. The reader who does not want to work under this assumption can simply work unconditionally in the procategory Pro-DM_{−,ét}^{eff}(k, ℚ).

Lemma 3.2.3. Under the finiteness assumption 3.2.1, the map φ_{G} is the unique morphism of commutative coalgebras extending α_{G} (Definition 2.1.4).

Proof. From the definitions it is clear that it is a morphism of coalgebras. Uniqueness then comes by universal property of coSym(M_{1}(G)). □

Lemma 3.2.4. The maps φ_{G}^{n} are natural in G ∈ sAb.

Proof. Clear by construction and naturality of α_{G} (Lemma 2.2.2). □

Lemma 3.2.5. Let k̄ be an algebraic closure of k and denote ¯(·) the base change to k̄. Then

\overline{φ_{G}^{n}} = φn ; \overline{G} .

Proof. By construction this comes from the case of α_{G} (Lemma 2.2.4). □

Corollary 3.2.6. Let G be a semiabelian variety over k. Then Theorem 3.1.4 holds for G if it holds for \overline{G}.

Proof. By Lemma 2.2.4, we have M_{1}(\overline{G}) = \overline{M_{1}(G)}. We apply Proposition 1.6.2 to the motives Sym^{n}\overline{M_{1}(G)} and we obtain that M_{1}(G) is odd of the same dimension as M_{1}(\overline{G}). In particular, the finiteness assumption 3.2.1 is verified for G. By Lemma 3.2.5 we have \overline{φ_{G}} = φ\overline{G}, we then conclude applying Proposition 1.6.2 to f = φ_{G}. □

We want to study compatibility of φ_{G} with products. Let G and H be two semiabelian varieties. Recall that we have

M_{1}(G × H) = M_{1}(G) ⊕ M_{1}(H)
M(G × H) = M(G) ⊗ M(H)
coSym(M_{1}(G) ⊕ M_{1}(H)) = coSym(M_{1}(G)) ⊗ coSym(M_{1}(H))

by additivity of M_{1} (Lemma 2.2.1), Künneth formula and Corollary B.3.1.

Proposition 3.2.7. Let G and H be two semiabelian varieties over k satisfying finiteness assumption 3.2.1. Then, under the above identifications, we have

φ_{G×H} = φ_{G} ⊗ φ_{H} ,

i.e., the diagram

\xymatrix{M(G × H) \ar[d]^{≅} \ar[r]^{φ_{G×H}} & coSym(M_{1}(G × H)) \ar[r]^{≅} & coSym(M_{1}(G) ⊕ M_{1}(H)) \\ M(G) ⊗ M(H) \ar[rr]_{φ_{G}⊗φ_{H}} &  & coSym(M_{1}(G)) ⊗ coSym(M_{1}(H)) \ar[u]^{≅}}

commutes.

Proof. First, notice that G × H verifies the finiteness assumption 3.2.1. Now, by the universal property of the coalgebra coSym it is enough to check that the diagram

\xymatrix{M(G × H) \ar[d]_{≅} \ar[r]^{α_{G×H}} & M_{1}(G × H) \ar[r]^{≅} & M_{1}(G) ⊕ M_{1}(H) \\ M(G) ⊗ M(H) \ar[rr]_{(α_{G}⊗ϵ_{G})⊕(ϵ_{H}⊗α_{H})} &  & M_{1}(G) ⊗ 𝟙 ⊕ 𝟙 ⊗ M_{1}(H) \ar[u]^{≅}}

commutes, which is the case by Corollary 2.2.3. □

Corollary 3.2.8. Let G and H be connected semiabelian varieties satisfying finiteness assumption 3.2.1. Then φ_{G×H} is an isomorphism if and only if φ_{G} and φ_{H} are isomorphisms.

Proof. If φ_{G} and φ_{H} are isomorphisms, so is φ_{G} ⊗ φ_{H}. For the converse note that M(G) is a direct factor of M(G × H) via G → G × H. □

Proposition 3.2.9. Under the finiteness assumption 3.2.1, the morphism φ_{G} is a morphism of Hopf algebras.

Proof. Comultiplication is part of the Lemma 3.2.3. Antipode, unit and counit are special cases of naturality Lemma 3.2.4. Multiplication is also reduced to naturality using Proposition 3.2.7. □

4. Special cases

Before giving a full proof we need to address the cases of tori and abelian varieties, the two building blocks of the category of semiabelian varieties. The case of tori is simple from the properties that we have established so far. In the case of abelian varieties, there are two key ingredients: some properties of the Chow motive of A (after Deninger, Murre, Künnemann and Kings) and a computation of Voevodsky of the motive of a curve. We also draw some consequences from these partial results.

4.1. The case of tori. Recall that (cGrp/k)_{ℚ} is the category of commutative groups schemes of finite type over k up to isogeny.

Definition 4.1.1. Let T be a torus over k. The cocharacter sheaf Ξ(T) of T is defined as the sheaf of ℚ-vector spaces on the small étale site of k given by

K ↦ Hom_{(cGrp/K)_{ℚ}}((𝔾_{m})_{K}, T_{K})

for all finite separable field extensions K/k By abuse of notation, we also write Ξ(T) for the pull-back to the category ShT_{ét}(k, ℚ) of étale sheaves with transfers and for its image in DM_{−,ét}^{eff}(k)_{ℚ}.

Remark 4.1.2. The motive Ξ(T) is an Artin motive.

Proposition 4.1.3. Let T be a torus over k of rank r. Then the main Theorem 3.1.4 holds for G = T, i.e., M_{1}(T) is odd of dimension r and

φ_{T} : M(T) → coSym(M_{1}(T))

is an isomorphism. Moreover, the natural pairing

Hom_{cGrp_{ℚ}}(𝔾_{m}, T) × \underline{𝔾_{m}}_{ℚ} → \underline{T}_{ℚ}

defines a map

Ξ(T) ⊗ 𝟙(1)[1] → M_{1}(T)

which is an isomorphism.

Proof. Let us start with the first part of the statement. Let first T = 𝔾_{m}. In this case it is well-known that

M(𝔾_{m}) = 𝟙 ⊕ 𝟙(1)[1]

with 𝟙(1)[1] = \underline{𝔾_{m}}_{ℚ} and the factor 𝟙 is the image of the projector induced by the constant endomorphism of 𝔾_{m}. We claim that α induces an isomorphism

𝟙(1)[1] → M_{1}(𝔾_{m}) .

The proper way of showing this would be to analyze the constructions carefully. However, as Hom_{DM_{−,ét}^{eff}(k,ℚ)}(𝟙, 𝟙) = ℚ we can use a quicker argument instead. By the naturality of Lemma 2.2.2, the morphism α_{𝔾_{m}} vanishes when restricted to the unit motive 𝟙, hence it suffices to check that

α_{𝔾_{m}} : M(𝔾_{m}) → M_{1}(𝔾_{m})

is non-zero. This was pointed out in Lemma 2.2.2. Note that M_{1}(𝔾_{m}) is odd of dimension 1 because 𝟙 is even of dimension 1. Now let T be general. By Corollary 3.2.6 it suffices to consider the case k algebraically closed. Hence T ≅ 𝔾_{m}^{r}. The assertion now follows from Corollary 3.2.8 and T = 𝔾_{m}. We now turn to the identification of M_{1}(T). In order to check that it is an isomorphism, we use the same reduction steps as before. Without loss of generality, k is algebraically closed (Proposition 1.6.2) and hence T ≅ 𝔾_{m}^{r}. By compatibility with products it suffices to consider the case r = 1 which is tautological. □

Remark 4.1.4. The analogous computation for the associated graded of the slice filtration is shown in [HK06, Proposition 7.2], even with integral coefficients. Note that by loc. cit. Corollary 7.9 we do not expect the integral version of Proposition 4.1.3.

Remark 4.1.5. Let T be a torus of rank r. Then

det(M_{1}(T)) = Sym^{r}(M_{1}(T)) = (⋀^{r} Ξ(T)) (r)[r] = det Ξ(T)(r)[r]

is a finite-dimensional motive. It is odd of dimension 1 when r is odd and it is even of dimension 1 when r is even. However, it is not always isomorphic to 𝟙(r)[r] as the example of a non-split torus of rank 1 shows. The Artin motive det Ξ(T) is even of dimension 1. It is given by a onedimensional continuous representation of the absolute Galois group of k in the category of ℚ-vector spaces. It is either trivial or of order 2.

4.2. The Chow motive of an abelian variety. We recall here some classical results on the Chow motive of an abelian variety. Let us recall some notations and conventions: Chow(k)_{ℚ} is the pseudo-abelian ℚ-linear rigid symmetric tensor category of Chow motives over k with rational coefficients, endowed with the covariant ℚ-linear symmetric tensor functor called motive

M : SmPrVar/k → Chow(k)_{ℚ} .

For a detailed description of this category see [DM91] or [Kün94].

Remark 4.2.1. Recall that the standard convention in the classical literature on Chow motives is to take the functor to Chow motives to be contravariant. By replacing a cycle by its transpose we can pass to the covariant version. Note that this operation interchanges the notions of symmetric algebra and symmetric coalgebra. Note also that [DM91] and [Kün94] use the notation ⋀^{i} instead of Sym^{i}.

Theorem 4.2.2 ([DM91, Thm. 3.1]). For any abelian variety over k of dimension g there is a unique decomposition in Chow(k)_{ℚ}

M(A) = ⨁_{i=0}^{i=2g} 𝔥_{i}(A),

which is natural in A ∈ Ab/k, and such that for all n ∈ ℤ

M(n_{A}) = ⨁_{i=0}^{i=2g} n^{i} · id_{𝔥_{i}(A)}.

Moreover, one has 𝔥_{0}(A) = 𝟙.

Theorem 4.2.3 ([Kün94, Thm. 3.1.1 et 3.3.1]). For any abelian variety A over k of dimension g the following holds:

(1) The Poincaré duality holds

𝔥_{2g−i}(A)^{∨} = 𝔥_{i}(A)(−g)[−2g] .

In particular one has 𝔥_{2g}(A) = 𝟙(g)[2g].

(2) For i > 2g, the motive Sym^{i}𝔥_{1}(A) vanishes.

(3) The canonical morphism of coalgebras

M(A) ⟶ coSym(𝔥_{1}(A))

induced by the projection M(A) → 𝔥_{1}(A) is an isomorphism. It respects the grading, i.e. it induces isomorphisms

𝔥_{i}(A) ⟶ Sym^{i}(𝔥_{1}(A)) .

Proposition 4.2.4 ([Kin98, Prop. 2.2.1]). For all pairs of abelian varieties A and B, the functor 𝔥_{1} induces an isomorphism of ℚ-vector spaces

Hom_{Ab_{ℚ}}(A, B) ⟶^{∼} Hom_{Chow(k)_{ℚ}}(𝔥_{1}(A), 𝔥_{1}(B)).

Proposition 4.2.5. Let C be a smooth and projective curve over k and J(C) its Jacobian. Suppose that C(k) ≠ ∅ and let x_{0} : C → C be a constant map. Then the motive of the curve can be decomposed as

M(C) = 𝟙 ⊕ 𝔥_{1}(J(C)) ⊕ 𝟙(1)[2]

such that the projector to 𝟙 is given by M(x_{0}) and the projector to 𝟙(1)[2] by its Poincaré dual.

Proof. This is classical, see for example [Sch94, Proposition 4.5]. Notice that by Theorem 5.3 of loc. cit. the different notions of motivic 𝔥_{1} for an abelian variety coincide. □

4.3. The Voevodsky motive of an abelian variety. We consider A an abelian variety of dimension g over the base field k. In order to prove the main Theorem 3.1.4 in this special case, the key point is to show that α_{A} induces an isomorphism between 𝔥_{1}(A) and M_{1}(A). We will reduce this to the case of Jacobians and then use Proposition 4.2.5 and a parallel result of Voevodsy for geometric motives.

Lemma 4.3.1. Consider the decomposition M(A) = ⨁_{i=0}^{i=2g} 𝔥_{i}(A) of Theorem 4.2.2 and the map α_{A} : M(A) → M_{1}(A) of definition 3.1.3. Then the restriction of the map to each factor 𝔥_{i}(A) is zero for all i ≠ 1. Moreover, the induced map

α_{A} : 𝔥_{1}(A) → M_{1}(A)

is non-zero.

Proof. By Lemma, 2.2.2 the map α_{A} is natural in sAb_{ℚ}. On the other hand the action of the multiplication n_{A} is equal to n^{i} · id on 𝔥_{i}(A) (Theorem 4.2.2) and to n · id on M_{1}(A) (Proposition 2.2.1). This implies that α_{A} is zero on 𝔥_{i}(A) for i ≠ 1. To conclude, notice that the restriction of α_{A} to 𝔥_{1}(A) has to be non-zero, otherwise the whole α_{A} would be zero, which contradicts Lemma 2.2.2. □

Lemma 4.3.2. Let C be a smooth and projective curve over k with a rational point and J(C) its Jacobian. Then, the motives 𝔥_{1}(J(C)) and M_{1}(J(C)) are isomorphic.

Proof. Consider M(C) ∈ D^{−}(ShT_{ét}(k, ℚ)). It is cohomologically concentrated in degrees 0 and −1. The cohomology in degree 0 is by [Voe00, Theorem 3.4.2] given by \underline{Pic_{C/k}}_{ℚ}. In degree −1 it is equal to \underline{𝔾_{m}}_{ℚ}. We use the projector given by the rational point x and its Poincaré dual to cut off 𝟙 ⊕ 𝟙(1)[2] = ℚ ⊕ \underline{𝔾_{m}}_{ℚ}[1]. The remaining object is cohomologically concentrated in degree 0 and given by the kernel of the degree map \underline{Pic_{C/k}}_{ℚ} → ℚ,

hence isomorphic to M_{1}(J(C)). By comparing with the decomposition in Proposition 4.2.5 we get the result. □

Proposition 4.3.3. For any abelian variety A, the map

α_{A}|_{𝔥_{1}(A)} : 𝔥_{1}(A) → M_{1}(A)

is an isomorphism.

Remark 4.3.4. In [EW13, Section 4.3] this is established by going through the definitions carefully. The proof given here is different.

Proof. We can assume that k is algebraically closed by Proposition 1.6.2 and Lemma 2.2.4. We may decompose A up to isogeny into simple factors. The map α_{A} is natural in Ab_{ℚ} and compatible with direct products. Hence it suffices to consider the case of a simple abelian variety. We can choose a curve C such that A is a factor of J(C) up to isogeny. As k is algebraically closed, we can apply Lemma 4.3.2 and deduce that there is some isomorphism

𝔥_{1}(J(C)) → M_{1}(J(C)) .

Hence we may view 𝔥_{1}(A) and M_{1}(A) as direct factors of the same object X and α|_{𝔥_{1}(A)} as an endomorphism of X. They are both simple factors by Proposition 2.2.1 and Proposition 4.2.4. Hence any non-zero map between them is an isomorphism. By Lemma 4.3.1 this is the case for α_{A}. □

Proposition 4.3.5. Let A be an abelian variety over k of dimension g. Then the main Theorem 3.1.4 holds for G = A, i.e., M_{1}(A) is odd of dimension 2g and

φ_{A} : M(A) → coSym(M_{1}(A))

is an isomorphism.

Proof. First, 𝔥_{1}(A) is odd of dimension 2g by Theorem 4.2.3. So by Proposition 4.3.3 the same holds for M_{1}(A). Let us consider the following commutative diagram:

\xymatrix{M(A) \ar[d] \ar[dr]^{α_{A}} \\ 𝔥_{1}(A) \ar[r]_{_{α_{A}_{|𝔥_{1}(A)}}} & M_{1}(A)}

where the vertical arrow is the projection. By the universal property B.2.1 it induces a unique commutative diagram

\xymatrix{M(A) \ar[d] \ar[dr] \\ coSym(𝔥_{1}(A)) \ar[r] & coSym(M_{1}(A))}

of morphisms of coalgebras. The diagonal morphism is φ_{A} by Lemma 3.2.3. By Proposition 4.3.3 α_{A} : 𝔥_{1}(A) → M_{1}(A) is an isomorphism, so the horizontal arrow coSym(𝔥_{1}(A)) → coSym(M_{1}(A)) is an isomorphism. The vertical arrow is an isomorphism by Theorem 4.2.3. We deduce that φ_{A} is an isomorphism.

□

5. Properties of M_{1}(G)

In all the section, G is a semiabelian variety over k. We consider the basic exact sequence

1 → T → G → A → 1 ,

with T a torus of rank r and A an abelian variety of dimension g. We establish properties for M_{1}(G) which we already know for M_{1}(T) and M_{1}(A).

5.1. The motive M_{1}(G) is Kimura finite.

Proposition 5.1.1. Let G be a semiabelian variety over k which is an extension of an abelian variety of dimension g by a torus T of rank r. Then the motive M_{1}(G) is odd of dimension 2g + r, i.e., Sym^{n}(M_{1}(G)) vanishes for n > 2g + r, and the motive

det(G) := det(M_{1}(G)) = Sym^{2g+r}M_{1}(G)

is of the form

Λ(g + r)[2g + r]

where Λ is the tensor-invertible Artin motive det Ξ(T) of Remark 4.1.5. In particular, if the torus part of G is split, then Λ = 𝟙.

Proof. Let 1 → T → G → A → 1 be the basic sequence and consider the associated filtration of Appendix C.2

Fil_{i}^{M_{1}(T)}Sym^{n}M_{1}(G) .

By Proposition C.3.4 its associated graded pieces have the form

Sym^{i}M_{1}(T) ⊗ Sym^{n−i}M_{1}(A) .

Hence, they vanish in DM_{−,ét}^{eff}(k, ℚ) for i > r (Proposition 4.1.3) or n − i > 2g (Theorem 4.2.3). This implies vanishing for n > 2g + r. For n = 2g + r we get a canonical isomorphism

det M_{1}(G) = det M_{1}(T) ⊗ det M_{1}(A) .

Hence the formula follows from the Proposition 4.1.3 for tori and Theorem 4.2.3 for abelian varieties. Note that Λ is indeed invertible. □

The following is not needed in the sequel.

Corollary 5.1.2. (1) Let C be a curve (not necessarily smooth and pro-

jective). Then M(C) is Kimura finite.

(2) Fix the embedding of the category of 1-motives [Del74] in DM_{−,ét}^{eff}(k, ℚ) to be the one constructed in [BVK, Org04]. Then all 1-motives are even objects in DM_{−,ét}^{eff}(k, ℚ).

Proof. By [BVK, Theorem 11.2.1] the motive M(C) decomposes into the sum of an Artin-motive and a 1-motive (shifted by 1). Then, it is enough to show (2). Consider a 1-motive [F → G] as a complex where F (in degree 0) is a k-group scheme such that Fk̄ is a free abelian group of finite rank and G (in degree 1) is a semiabelian variety. By Proposition C.3.4, the filtration (in the abelian category C^{b}(ShT_{ét}(k, ℚ)))

0 → [0 → \underline{G}_{ℚ}] → [\underline{F}_{ℚ} → \underline{G}_{ℚ}] → [\underline{F}_{ℚ} → 0] → 0

induces a filtration on Sym^{n}[\underline{F}_{ℚ} → \underline{G}_{ℚ}] with associated graded pieces isomorphic to

Sym^{a}[0 → \underline{G}_{ℚ}] ⊗ Sym^{n−a}[\underline{F}_{ℚ} → 0] .

These exact sequences induce triangles in DM_{−,ét}^{eff}(k, ℚ) which reduce to the case of [F → 0] and [0 → G]. The first is just an Artin motive, hence even. The second, by definition, equals to M_{1}(G)[−1], which is even by Proposition 5.1.1. □

Remark 5.1.3. (1) Kimura finiteness of motives of curves was known by the

work of Guletskii [Gul06] and Mazza [Maz04] (using different methods).

(2) The fact that 1-motives are Kimura finite is pointed out in [Maz04, Remark 5.11] (attributed to O’Sullivan) as a consequence of Kimura finiteness of motives of curves. The above is more precise and also more direct.

5.2. The motive M_{1}(G) is geometric.

Proposition 5.2.1. The motive M_{1}(G) ∈ DM_{−,ét}^{eff}(k, ℚ) belongs to the full subcategory DM_{gm}^{eff}(k)_{ℚ}.

Remark 5.2.2. The fact that all 1-motives are geometric is already shown in [Org04]. We give a straight-forward argument in our case.

Proof. If G is a torus or an abelian variety, we have established the isomorphism

φ_{G} : M(G) → coSym(M_{1}(G))

in Proposition 4.1.3 and Proposition 4.3.5. In particular, M_{1}(G) is a direct summand of a geometric motive, hence geometric. (Alternatively, we have given an explicit description of M_{1}(G) in Proposition 4.1.3 and Proposition 4.3.3, which is in both cases geometric.)

In general, consider a basic exact sequence fixed in the beginning of the section

1 → T → G → A → 1 .

It induces an exact triangle

M_{1}(T) → M_{1}(G) → M_{1}(A)

in DM_{−,ét}^{eff}(k, ℚ). The claim follows because the category of geometric motives is triangulated. □

5.3. Computation of realization. Fix a prime ℓ different from the characteristic of the base field k. Let

H^{∗} : DM_{gm}(k) ⟶ GrVec_{ℚ_{ℓ}}^{±}

be the realization functor (see Section 1.8).

Proposition 5.3.1. The realization of the map α_{G} : M(G) → M_{1}(G) of Definition 2.1.4 is zero in all degrees except in degree one where it induces an isomorphism

H^{∗}(α_{G}) : H^{∗}(M_{1}(G)) → H^{1}(G).

Proof. Let us start by showing that the statement holds for all G which satisfy the main Theorem 3.1.4. Indeed, applying H^{∗} one has an isomorphism of Hopf algbras

H^{∗}(G) ≅ Sym(H^{∗}(M_{1}(G))) .

Hence, their primitive parts are isomorphic. On the other hand, by the structure theory of connected graded Hopf algebras (see for example [Lod92, Appendix A]) the primitive part of on the right hand side is H^{∗}(M_{1}(G)) and the primitive part of H^{∗}(G) is H^{1}(G) (Lemma 1.8.1). Note that, in particular, we have shown the statement in the toric case and in the abelian case, by Propositions 4.1.3 and 4.3.5. In the general case, write

1 ⟶ T ⟶^{f} G ⟶^{g} A ⟶ 1 .

By Lemma 2.2.1 one has a triangle

M_{1}(T) → M_{1}(G) → M_{1}(A) →^{+1} .

By applying the functor H^{∗} one obtains a long exact sequence. For i ≠ 1 we have seen that H^{i}(M_{1}(T)) and H^{i}(M_{1}(A)) vanish hence H^{i}(M_{1}(G)) vanishes as well. Then one has the following commutative diagram of short exact sequences

\xymatrix{H^{1}(M_{1}(A)) \ar[d]^{H^{1}(α_{A})} \ar[r]^{H^{1}(M_{1}(g))} & H^{1}(M_{1}(G)) \ar[d]^{H^{1}(α_{G})} \ar[r]^{H^{1}(M_{1}(f))} & H^{1}(M_{1}(T)) \ar[d]^{H^{1}(α_{T})} \\ H^{1}(A) \ar[r]^{H^{1}(g)} & H^{1}(G) \ar[r]^{H^{1}(f)} & H^{1}(T).}

(The two squares are commutative by Lemma 2.2.2 and the second line is also a short exact sequence by Lemma 1.8.2.) As we have shown that the first and third vertical arrows are isomorphisms so is the second.

□

6. Proof of the Main Theorem

The proof is by induction on the torus rank. By comparing two triangles, we establish that there is some isomorphism between M(G) and coSym(M_{1}(G)) and deduce that these two motives are finite-dimensional. In the next section 6.2 we show that φ_{G} is an isomorphism studying its behaviour in the realization and using Kimura finiteness.

6.1. Comparing exact triangles. Throughout this section, we consider a short exact sequence of semiabelian varieties

1 → 𝔾_{m} → G → H → 1 .

Lemma 6.1.1. (1) Let n ≥ 0. We denote

[M_{1}(G)] : M_{1}(H) → 𝟙(1)[2]

the connecting morphism of the exact triangle

𝟙(1)[1] → M_{1}(G) → M_{1}(H) .

Then there is an exact triangle

Sym^{n}(M_{1}(G)) → Sym^{n}(M_{1}(H)) ⟶^{∪[M_{1}(G)]} Sym^{n−1}(M_{1}(H))(1)[2] .

(2) Let

c_{1}([G]) ∈ H_{ét}^{1}(H, 𝔾_{m}) = Mor_{DM_{−,ét}^{eff}(k,ℚ)}(M(H), 𝟙(1)[2])

be the first Chern class of G viewed as a 𝔾_{m}-torsor over H. Then there is an exact triangle

M(G) → M(H) ⟶^{·∪c_{1}(G)} M(H)(1)[2] .

(3) The diagram

\xymatrix{M(H) \ar[r]^{·∪c_{1}([G])} \ar[d]^{φ_{H}} & M(H)(1)[2] \ar[d]^{φ_{H}(1)[2]} \\ coSym(M_{1}(H)) \ar[r]_{·∪[M_{1}(G)]} & coSym(M_{1}(H))(1)[2]}

commutes.

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Proof. For (1), we apply Theorem C.4.4 to the exact sequence of sheaves with transfers

0 → \underline{𝔾_{m}}_{ℚ} → \underline{G}_{ℚ} → \underline{H}_{ℚ} → 0

and the localization functor

q : ShT_{ét}(k, ℚ) → DM_{−,ét}^{eff}(k, ℚ) .

Note that ShT_{ét}(k, ℚ) is ℚ-linear abelian symmetric tensor category with an exact tensor product, see Remark 1.6.3, and that M_{1}(𝔾_{m}) = 𝟙(1)[1] and Sym^{2}(𝟙(1)[1]) = 0. For the second triangle consider 𝔸(G) → H, the line bundle associated to the 𝔾_{m}-torsor G. The zero section of 𝔸(G) identifies H with 0(H), a smooth subvariety of 𝔸(G) of codimension 1. Its complement is G. Hence the localization sequence reads

M(G) → M(𝔸(G)) → M(0(H))(1)[2] .

By homotopy invariance M(𝔸(G)) = M(H). The identification of the boundary map with the first Chern class is carried out in [HK06, Proposition C1]. By compatibility with comultiplication (which holds by definition of the maps), it suffices to check commutativity in degree 1. This is precisely the comparison of Chern classes in Proposition A.1.1. □

Corollary 6.1.2. Assume that there exists a short exact sequence of semiabelian varieties

1 → 𝔾_{m} → G → H → 1,

such that the main Theorem 3.1.4 holds for H. Then there exists an isomorphism

ψ : coSym(M_{1}(G)) → M(G) .

In particular, M(G) is Kimura finite.

Proof. We consider

\xymatrix{M(G) \ar[r] \ar@{-->}[d]_{ψ} & M(H) \ar[r] \ar[d]^{φ_{H}}_{≃} & M(H)(1)[2] \ar[d]^{φ_{H} (1)[2]}_{≃} \ar[r] & M(G)[1] \ar@{-->}[d]^{ψ[1]} \\ coSym(M_{1}(G)) \ar[r] & coSym(M_{1}(H)) \ar[r] & coSym(M_{1}(H))(1)[2] \ar[r] & coSym(M_{1}(G))[1].}

Both triangles are constructed in Lemma 6.1.1. The central square commutes also by Lemma 6.1.1. By assumption φ_{H} is an isomorphism. By the axioms of a triangulated category we obtain an isomorphism ψ as indicated. M(G) is Kimura finite because M_{1}(G) is Kimura finite by Proposition 5.1.1 and the notion is stable under tensor products and direct summands. □

Remark 6.1.3. The above corollary was the main result of [EW13]. We expect φ_{?} to define a morphism of triangles, i.e., ψ = φ_{G}. This would immediately show that φ_{G} is an isomorphism. We were not able to establish this morphism of triangles and use a completely different argument instead.

6.2. φ_{G} is an isomorphism. We modify the non-canonical isomorphism ψ (Corollary 6.1.2) such that its realization is the inverse of the realization of φ_{G}. To conclude we use conservativity of the realization functor on finitedimensional motives.

Isomorphism Assumption 6.2.1. We assume the existence of an isomorphism ψ : coSym(M_{1}(G)) → M(G). We fix such ψ and write ψ_{1} : M_{1}(G) → M(G) for its restriction to M_{1}(G).

Lemma 6.2.2. Under the above isomorphism assumption 6.2.1, the realization of ψ_{1} induces an isomorphism

H^{1}(ψ_{1}) : H^{1}(G) ⟶^{∼} H^{1}(M_{1}(G))

Proof. Recall from Proposition 5.3.1 that H^{∗}(M_{1}(G)) is concentrated in degree one. Hence the realization of ψ gives isomorphisms

H^{n}(G) ⟶^{H^{n}(ψ)} Sym^{n}(H^{1}(M_{1})) .

Moreover, H^{1}(ψ_{1}) = H^{1}(ψ). □

Lemma 6.2.3. Under the isomorphism assumption 6.2.1, the endomorphism α_{G} ∘ ψ_{1} of the motive M_{1}(G) is an isomorphism. In particular, there exists a morphism β_{1} : M_{1}(G) → M(G) such that

α_{G} ∘ β_{1} = id_{M_{1}(G)} .

Proof. We write

α_{G} ∘ ψ_{1} = M_{1}(f_{0}) .

This is possible because by Proposition 2.2.1 any endomorphism of M_{1}(G) is of the form M_{1}(f) where f is in End_{sAb_{ℚ}}(G). It is enough to show that f_{0} is an automorphism of G ∈ sAb_{ℚ}. As H^{1} is exact and faithful on sAb_{ℚ} (see Lemma 1.8.2), we can test this after applying H^{1}. Moreover,

H^{1}(f_{0}) = H^{∗}(M_{1}(f_{0})) = H^{∗}(ψ_{1}) ∘ H^{∗}(α_{G}) .

By Lemma 5.3.1, the map H^{∗}(α_{G}) is an isomorphism onto its image H^{1}(G). By Lemma 6.2.2, the map H^{∗}(ψ_{1}) is an isomorphism when restricted to H^{1}(G). Hence the composition is an isomorphism. □

Lemma 6.2.4. Under the isomorphism assumption 6.2.1, let us fix a morphism β_{1} : M_{1}(G) → M(G) such that α_{G} ∘ β_{1} = id_{M_{1}(G)} as in the previous lemma. Let

β : coSymM_{1}(G) → M(G)

be the induced morphism of algebras (Corollary B.2.5). Then H^{∗}(φ_{G}) and H^{∗}(β) are inverse to each other.

Proof. By Assumption 6.2.1 the vector spaces H^{∗}(coSym(M_{1}(G))) and H^{∗}(M(G)) have the same dimension, in particular it is enough to check that the composition in one direction is the identity. By Proposition 3.2.9, φ_{G} is not only a morphism of coalgebras but also a morphism of algebras. Hence

φ_{G} ∘ β : coSym(M_{1}(G)) → coSym(M_{1}(G))

is also a morphism of algebras and so

H^{∗}(φ_{G} ∘ β) = H^{∗}(β) ∘ H^{∗}(φ_{G}) : Sym(H^{∗}(M_{1}(G))) → Sym(H^{∗}(M_{1}(G)))

is a morphism of coalgebras. By Corollary B.2.5, the bialgebra Sym(H^{1}(M_{1}(G)) also has the universal property with respect to comultiplication. We are going to exploit it in order to establish that H^{∗}(φ_{G} ∘ β) is the identity. By Proposition 5.3.1, H^{∗}(M_{1}(G)) = H^{1}(M_{1}(G)) is concentrated in degree one. In degree one our morphism is equal to H^{∗}(α_{G} ∘ β_{1}) so it is the identity by assumption. □

Proposition 6.2.5. Under the isomorphism assumption 6.2.1, the morphism φ_{G} is an isomorphism.

Proof. By Proposition 5.1.1 and Corollary 6.1.2 the objects coSym(M_{1}(G)) and M(G) are finite-dimensional. Moreover, by Cancellation Theorem [Voe10], effective motives are embedded in non-effective ones, hence we can suppose that φ_{G} is a morphism defined in a rigid category. So we can use Theorem 1.3.1 (applied to the realization functor) and conclude by Lemma 6.2.4. □

6.3. Conclusion.

Proof of the Main Theorem 3.1.4. Let k be a perfect field and G be a semiabelian variety over k which is an extension of an abelian variety A of dimension g and a torus T of rank r. By Proposition 5.1.1, the motive M_{1}(G) is odd of dimension 2g + r. It remains to establish that φ_{G} is an isomorphism. By Corollary 3.2.6 we can suppose that k is algebraically closed. We now argue by induction on r. When r = 0 (and hence G = A) the theorem is proved by Proposition 4.3.5. Let us now consider the case r ≥ 1. As the ground field k is algebraically closed we have T ≅ 𝔾_{m}^{r}. We fix such a splitting and let 𝔾_{m} → 𝔾_{m}^{r} be the inclusion as the first coordinate. This defines a short exact sequence

1 → 𝔾_{m} → G → H → 1,

with H a semiabelian variety of torus rank r − 1. By inductive hypothesis the theorem holds for H. By Corollary 6.1.2, this implies the existence of some isomorphism

ψ : M(G) → coSym(M_{1}(G)) .

This is the isomorphism assumption 6.2.1 for G. Then Proposition 6.2.5 shows that the morphism φ_{G} is an isomorphism. □

7. Consequences

Let k be a field (not necessarily perfect). We deduce from our main Theorem 3.1.4 a Künneth decomposition for the motive of a semiabelian variety, the behaviour under mixed Weil cohomology theories and the existence of a weight filtration. Finally, we also compute the motives of arbitrary commutative group schemes.

7.1. Künneth components. In this section we fix a prime number ℓ and write H^{∗} : DM_{gm}^{eff}(k)_{ℚ} ⟶ GrVec_{ℚ_{ℓ}}^{±} for the ℓ-adic realization, see Section 1.8.

Theorem 7.1.1. Let G be a semiabelian variety over k which is an extension of an abelian variety of dimension g and a torus of rank r. Then there exists a unique decomposition in DM_{gm}^{eff}(k)_{ℚ}

M(G) = ⨁_{i=0}^{2g+r} M_{i}(G)

which is natural in G ∈ sAb/k and such that

H^{∗}(M_{i}(G)) = H^{i}(Gk̄, ℚ_{ℓ}).

Moreover:

(1) The multiplication by n_{G} acts as

M(n_{G}) = ⨁_{i=0}^{i=2g+r} n^{i} · id_{M_{i}(G)}.

In particular, for any non-zero integer n, the morphism M(n_{G}) is an isomorphism and hence the Künneth decomposition is natural in G ∈ sAb_{ℚ}.

(2) If L is a field extension of k, then we have

M_{i}(G)_{L} = M_{i}(G_{L})

in DM_{gm}^{eff}(L)_{ℚ}.

(3) The image of the motive M_{1}(G) in DM_{−,ét}^{eff}(k, ℚ) is given by the homotopy invariant sheaf with transfers

S ↦ Mor_{Sch/k}(S, G) ⊗ ℚ .

(4) For all pairs G_{1}, G_{2} ∈ sAb/k_{ℚ}, the functor M_{1} induces an isomorphism of ℚ-vector spaces

Hom_{sAb_{ℚ}}(G_{1}, G_{2}) ⟶^{∼} Hom_{DM_{gm}^{eff}(k)_{ℚ}}(M_{1}(G_{1}), M_{1}(G_{2})).

(5) Any exact sequence 1 → G_{1} → G_{2} → G_{3} → 1 in sAb_{ℚ} induces an exact triangle

M_{1}(G_{1}) → M_{1}(G_{2}) → M_{1}(G_{3})

in DM_{gm}^{eff}(k)_{ℚ}

(6) For i > 2g + r, the motive Sym^{i}(M_{1}(G)) vanishes.

(7) The canonical morphism of coalgebras

M(G) = ⨁_{i} M_{i}(G) ⟶ coSym(M_{1}(G)) = ⨁_{i} Sym^{i}(M_{1}(G))

induced by the projection M(G) → M_{1}(G) is a graded isomorphism of Hopf algebras.

(8) The motive

det(G) := M_{2g+r}(G) = det(M_{1}(G))

is of the form

Λ(g + r)[2g + r]

with a tensor-invertible Artin motive Λ. If the torus part of G is split, then Λ = 𝟙.

(9) There is an isomorphism

M_{i}(G)^{∨} ≅ M_{2g+r−i}(G) ⊗ det(G)^{−1}
= M_{2g+r−i}(G) ⊗ Λ^{−1}(−g − r)[−2g − r]

natural in G ∈ sAb/k_{ℚ}. In particular there are isomorphisms

M(G)^{∨} ≅ M(G) ⊗ det(G)^{−1} ,
M_{c}(G) ≅ M(G) ⊗ det(G)^{−1}(g + r)[2g + 2r]
= M(G) ⊗ Λ^{−1}[r]

natural in G ∈ sAb/k_{ℚ} (where M_{c}(G) is the motive with compact support of G).

Proof. By Section 1.7, we may assume that k is perfect. We use the main Theorem 3.1.4 and choose

M_{i}(G) = Sym^{i}(M_{1}(G)) .

By Proposition 5.3.1 it has the correct behaviour for the realization of M_{1}(G) and by Lemma 1.8.1 also for all M_{i}(G). By Proposition 2.2.1 it also satisfies (1). Suppose there is another natural decomposition

M(G) = ⨁_{i=0}^{2g+r} M′_{i}(G) .

By naturality, this decomposition is stable under the action of the ℚ-algebra generated by M(n_{G}). Notice now that, for all i, the projector p_{i} defining M_{i}(G) is in this algebra, indeed

p_{i} = \frac{∏_{i≠j} M(n_{G}) − n^{j}id}{∏_{i≠j}(n^{i} − n^{j})} .

So we get a decomposition

M′_{i}(G) = ⨁_{j=0}^{2g+r} M_{i}^{j}(G)

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where the motive M_{i}^{j}(G) is a direct factors of M′_{i}(G) and of M_{j}(G). Hence by hypothesis, its realization is concentrated, on one hand in degree i and on the other hand in degree j, which implies that its realization is zero when i ≠ j. Moreover, M_{i}^{j}(G) is a finite-dimensional motive (as M(G) is by Theorem 3.1.4) so we can apply Theorem 1.3.1 to deduce that M_{i}^{j}(G) vanishes for i ≠ j. This gives the uniqueness. For (2) one can argue in two different ways: using uniqueness of the decomposition or using the multiplication n_{G} of (1). Properties (3) and (7) hold by definition and main Theorem 3.1.4. The properties (4) and (5) come from Proposition 2.2.1. The properties (6) and (8) were established in Proposition 5.1.1. Part (9) comes from a more general statement: we claim that if X is an odd object of dimension d in a ℚ-linear pseudo-abelian symmetric tensor category, then there is a canonical isomorphism

(Sym^{i}X)^{∨} ≅ Sym^{d−i}X ⊗ (Sym^{d}X)^{∨} .

Let us prove the claim. In [O’S05, Lemma 3.2] it is proven that for any even object X of dimension d there is a canonical isomorphism (∧^{i}X)^{∨} ≅ ∧^{d−i}X ⊗ (∧^{d}X)^{∨}. One can change the sign of all symmetries so that one gets a new category which is equivalent as ℚ-linear tensor category to the previous one (but not as ℚ-linear symmetric tensor category). This transformation sends even objects of dimension d to odd objects of dimension d so that [O’S05, Lemma 3.2] implies our claim. □

7.2. Mixed Weil cohomology of semiabelian varieties. Let H^{∗} be any mixed Weil cohomology with coefficients in F, in the sense of [CD12] (we do not ask anymore this to be the ℓ-adic cohomology). Recall that, by definition, this means that F is a field of characteristic zero, the Künneth formula and excision hold, and that, moreover, the cohomology of the point, the affine line and the multiplicative group have the standard dimensions. We also require that the cohomology of any scheme is concentrated in nonnegative degrees. In [CD12, Remark 2.7.15] it is conjectured that this should be deduced from the other axioms (see the comments in loc. cit. after Theorem 2.7.14). We also write

H^{∗} : DM_{gm}^{eff}(k) ⟶ GrVec_{F}^{±}

for the realization functor induced by the mixed Weil cohomology [CD12, Thm 3].

Lemma 7.2.1. Let M be an Artin motive. Then H^{∗}(M) is concentrated in degree 0.

Proof. By [CD12, Thm 1(4)], the theory H^{∗} satisfies Poincaré duality. By assumption it is concentrated in non-negative degrees. Hence H^{∗}(M(SpecL)) is concentrated in degree 0. □

Proposition 7.2.2. Let G be a semiabelian variety over k and M_{i}(G) be the Künneth components constructed in Theorem 7.1.1. Then

H^{∗}(M_{i}(G)) = H^{i}(G)

for any mixed Weil cohomology H^{∗}.

Proof. We apply H^{∗} to the isomorphism of Hopf algebras

φ_{G} : M(G) → coSym(M_{1}(G)) .

Hence H^{∗}(M_{1}(G)) is the primitive part of H^{∗}(G) and it suffices to show that it is concentrated in degree 1. We know that M_{1}(G) is odd of dimension 2g + r where r is the torus rank of G and g the dimension of the abelian part of A. Hence H^{∗}(M_{1}(G)) is of dimension 2g + r. Moreover,

det(H^{∗}(M_{1}(G)) = H^{∗}(det(G)) = H^{∗}(Λ(g + r)[2g + r])

with Λ as in Theorem 7.1.1 an invertible Artin motive. By Lemma 7.2.1 the cohomology of Λ is concentrated in degree 0. Hence the cohomology of det(G) is concentrated in degree 2g + r. By assumption H^{∗}(M_{1}(G)) is odd and H^{∗} is concentrated in non-negative degrees. This is only possible if all of H^{∗}(M_{1}(G)) is concentrated in degree 1. □

Consider now the category of 1-motives over k (introduced by Deligne [Del74]) and its embedding in DM_{−,ét}^{eff}(k, ℚ) [BVK, Org04].

Proposition 7.2.3. Let [F → G] ∈ DM_{−,ét}^{eff}(k, ℚ) be a 1-motive, with F (in degree 0) a k-group scheme such that Fk̄ is a free abelian group of rank d and G (in degree 1) an extension of an abelian variety of dimension g by a torus of rank r. Let H^{∗} be a mixed Weil cohomology theory such that H^{∗}(X) is concentrated in non-negative degrees. Then H^{∗}([F → G]) is concentrated in degree 0 and of dimension 2g + d + r.

Proof. This holds for [F → 0] by Lemma 7.2.1 and for [0 → G] by Proposition 7.2.2 (compare with the Definition 2.1.4 and note the shift on the degree here). By considering the long exact cohomology sequence, it follows for [F → G]. □

Remark 7.2.4. To our knowledge this result is new. There is a discussion of realizations of 1-motives in [BVK, Section 15.4] which is going into a different direction.

7.3. Weight filtration. Recall that Bondarko defines in [Bon10, Definition 1.1.1, Proposition 6.5.3] categories _{≤a}DM_{gm} of motives of weight at most a and _{≥a}DM_{gm} of motives of weight at least a such that

≤aDM_{gm} ∩_{≥a} DM_{gm} = Chow(k)_{ℚ}[−a] .

Our structural knowledge also gives us a weight filtration in the sense of Bondarko on M(G). Let G be semiabelian with torus part T of rank r and abelian part A of dimension g. Recall that there is a natural exact triangle

M_{1}(T) → M_{1}(G) → M_{1}(A)

with M_{1}(A) a Chow motive and M_{1}(T) = Ξ(1)[1] (with Ξ the Artin motive of Proposition 4.1.3) a Chow motive shifted by [−1]. This means that M_{1}(A) is pure of weight 0 and M_{1}(T) is pure of weight 1. Hence M_{1}(G) has weights between 0 and 1 and the above sequence is a weight decomposition. Using the filtration of Appendix C we extend this to all Künneth components.

Proposition 7.3.1. Fix an integer i. For every choice of −∞ ≤ a ≤ b ≤ ∞ in ℤ ∪ {−∞, ∞} there is a functor

a≤w≤bM_{i} : sAb_{ℚ} →_{≤b} DM_{gm} ∩_{≥a} DM_{gm}

together with an exact triangle of functors for every choice of a ≤ b < c

b+1≤w≤cM_{i} →_{a≤w≤c} M_{i} →_{a≤w≤b} M_{i}

such that

a≤w≤bM_{i} = M_{i} for a ≤ i and b ≥ 0.

Moreover, for every semiabelian variety G with torus part T of rank r and abelian part A of dimension g, we have naturally

a≤w≤aM_{i}(G) = M_{a}(T) ⊗ M_{i−a}(A) = Sym^{a}M_{1}(T) ⊗ Sym^{i−a}M_{1}(A) .

The weights of M_{i}(G) are concentrated in the range

M_{i}(G) =_{max(0,i−2g)≤w≤min(i,r)} M_{i}(G) .

Remark 7.3.2. Suppose that G = T × A, then

M_{i}(G) = Sym^{i}(M_{1}(T) ⊕ M_{1}(A)) = ⨁_{j=0} M_{j}(T) ⊗ M_{i−j}(A)

and the definitions

a≤w≤bM_{i}(G) = ⨁_{j=a}^{b} M_{j}(T) ⊗ M_{i−j}(A)

verify all the properties of the statement. In general G will just be an extension of A by T and one needs to replace this description by a filtered one, using Proposition C.3.4.

Proof of Proposition 7.3.1. By Section 1.7, we may assume that k is perfect. Recall that M_{1}(G) = q\underline{G} where q : D^{−}(ShT_{ét}(k, ℚ)) → DM_{−,ét}^{eff}(k, ℚ) is the localization functor and \underline{G} ∈ ShT_{ét}(k, ℚ) is a sheaf. We put

a≤w≤bM_{i}(G) = q (Fil\underline{T} ; a Sym^{i}\underline{G}/Fil\underline{T} ; b+1Sym^{i}\underline{G})

with Fil\underline{T} as defined in Definition C.3.1. The exact triangles are immediate from this construction. The computation of _{a≤w≤a}M_{i}(G) follows from Proposition C.3.4. In particular, it is pure of weight a and geometric. All other statements follow from this case by induction on |b − a|. □

Corollary 7.3.3. Suppose we are in one of the following two cases:

(1) k is embedded into ℂ and H^{∗} is the Betti realization with its natural mixed Hodge structure; (2) k is a field of finite type over its prime field, ℓ a prime different from

the characteristic of k and H^{∗} the ℓ-adic cohomology; In both cases let (W_{n}H^{∗})_{n∈ℤ} be the weight filtration. Then

H^{∗} (_{−∞≤w≤a}M_{i}(G)) = H^{i} (_{−∞≤w≤a}M_{i}(G)) = W_{i+a}H^{i}(G) .

Proof. Let again be T the torus part of G and A the abelian part. Note that by Proposition 7.2.2

H^{∗}(M_{a}(T) ⊗ M_{i−a}(A)) = H^{a}(T) ⊗ H^{i−a}(A)

is concentrated in degree i. As A is smooth projective, H^{i−a}(A) is pure of weight i − a. On the other hand, H^{a}(T) is pure of weight 2a. Hence H^{∗}(_{a≤w≤a}M_{i}(G)) is concentrated in degree i and pure of weight i + a. The corollary follows by induction. □

Remark 7.3.4. In general one does not have to expect that the weight filtration of a cohomology and its graded components lift canonically in DM_{gm}^{eff}(k)_{ℚ}. This has been studied in [Wil09] and a sufficient criterion, called of avoiding weights, has been given. This criterion does not apply here, but our situation allows anyway to define such a canonical lifting.

7.4. General Commutative Group Schemes. Let k be a field and G/k an arbitrary commutative group scheme of finite type over k. Our aim is to extend the previous results to this case. Let us initially consider k to be perfect. Let G^{0} be the connected component of the neutral element and π_{0}(G) the group of connected components of G. The latter is finite. We have a natural short exact sequence

1 → G^{0} → G → π_{0}(G) → 1 .

We are going to express the motive of G in terms of G^{0} and π_{0}(G). Note that we always have

M(G) = M(G^{red})

where G^{red} is the maximal reduced subscheme of G. As k is perfect, this scheme is a group and it is, moreover, smooth. Recall that by the Barsotti-Chevalley structure theorem ([Bar55] and [Che60]), there is a short exact sequence

1 → G^{u} → G^{red} → G^{sa} → 1

with G^{u} unipotent and G^{sa} semiabelian.

Remark 7.4.1. Let F be a finite group scheme over k. Then the Artin motive M(F) has the explicit description as representation of the absolute Galois group:

M(F) ≅ ℚ[F (k̄)]

with operation of Gal(k̄/k) given by the operation on k̄-valued points. The Hopf algebra structure is the one of the group ring. Note that we have naturally F (k̄) = F^{red}(k̄).

Definition 7.4.2 (cf. Definition 2.1.1). Let G be a commutative group scheme over a perfect field k. Define \underline{G} to be the étale sheaf on Sm/k given by

\underline{G}(S) = Mor_{Sch/k}(S, G)

and \underline{G}_{ℚ} = \underline{G} ⊗_{ℤ} ℚ.

Lemma 7.4.3. Let G be a commutative group scheme. Then \underline{G} is an étale sheaf with transfers and we have

\underline{G}_{ℚ} = \underline{G}_{ℚ}^{red} = \underline{G}_{ℚ}^{0} = (\underline{G}^{0})_{ℚ}^{red} .

Proof. First, G^{red} is smooth, hence the sheaf has transfers by Lemma 2.1.2. The equality \underline{G}_{ℚ} = \underline{G}_{ℚ}^{red} holds actually integrally because, for any smooth scheme S, we have

Mor_{Sch/k}(S, G) = Mor_{Sch/k}(S, G^{red})

(note that we consider \underline{G} as a sheaf only on Sm). To conclude note that the finite group of connected components does not contribute after tensoring with ℚ. □

Definition 7.4.4 (cf. Definition 2.1.4). Let G be a commutative group scheme over k. Consider \underline{G}_{ℚ} ∈ D^{−}ShT_{ét}(k, ℚ) as an object concentrated in degree zero and define

M_{1}(G) ∈ DM_{−,ét}^{eff}(k, ℚ)

as the image of \underline{G}_{ℚ} by the functor q (see Section 1.6). Similarly, consider

γ_{G^{red}} : Cor(·, G^{red}) ⊗_{ℤ} ℚ → \underline{G^{red}}_{ℚ} ,

of Proposition 2.1.3, as a map in D^{−}ShT_{ét}(k, ℚ), and define

α_{G} : M(G) = M(G^{red}) → M_{1}(G^{red}) = M_{1}(G)

as the image of γ_{G^{red}} by the functor q. Moreover, let

φ_{G} : M(G) → coSym(M_{1}(G))

be the unique extension of α_{G} compatible with comultiplication.

If G is semiabelian, then this definition agrees with the old one.

Lemma 7.4.5. Let k be perfect, G a smooth connected commutative groups scheme over k. Let G^{sa} be its semi-abelian part. Then

M(G) → M(G^{sa}), M_{1}(G) → M_{1}(G^{sa})

are isomorphisms. In particular, M(G) and M_{1}(G) vanish if G is unipotent.

Proof. It suffices to consider the case where k is algebraically closed. We consider the sequence

1 → G^{u} → G → G^{sa} → 1

with G^{u} unipotent and G^{sa} semiabelian. As M_{1} is exact, we need to show that M_{1}(G^{u}) vanishes. The unipotent group G has a filtration with associated graded components isomorphic to 𝔾_{a}. As M_{1} is exact, it suffices to consider G = 𝔾_{a}. By definition

\underline{𝔾_{a}} = O

is the structure sheaf. We take its image under

ShT_{ét}(k, ℚ) ⟶^{q} DM_{−,ét}^{eff}(k, ℚ) ⟶^{i} D^{−}(ShT_{ét}(k, ℚ))

and get the Suslin complex

iqO = \underline{C}_{∗}(O) = O ⊗_{k} O(Δ^{∗})

where Δ^{∗} is the standard cosimplicial object with Δ^{n} the algebraic n-simplex. It suffices to show that O(Δ^{∗}) is exact. This is the case q = 0 of [BG76, Prop. 1.1] Note that they assume char(k) = 0, but the assumption is not used at this point. We now turn to M(G). We view G → G^{sa} as a G^{u}-bundle. Note that G^{u} is 𝔸^{1}-homotopy equivalent to a point. The surjectivity of the map of étale sheaves \underline{G} → \underline{G}^{sa} means that G^{sa} → G étale locally has a section. This implies that the bundle is étale locally trivial. Hence the M(G) → M(G^{sa}) is an isomorphism. □

Theorem 7.4.6. Let G be a commutative group scheme of finite type over a field k, which is not necessarily perfect. Let π : G → π_{0}(G) be the projection to its group of components. Then the natural map

ψ_{G} : M(G) → coSym(M_{1}(G)) ⊗ M(π_{0}(G))

given by the composition

ψ_{G} : M(G) ⟶^{Δ} M(G) ⊗ M(G) ⟶^{φ_{G}⊗π} coSym(M_{1}(G)) ⊗ M(π_{0}(G))

is an isomorphism of Hopf algebras and it is natural in G ∈ cGrp.

Remark 7.4.7. All other results, e.g., Künneth components, computation of mixed Weil cohomology and weights immediately extend to this case.

Proof. By Section 1.7 we may assume that k is perfect and the above constructions apply. By Lemma 7.4.3 we may assume that G is reduced and hence smooth. By Proposition 1.6.2, we may assume that k is algebraically closed. By Lemma 7.4.8 we may assume that G = G^{0} × F with F finite. The morphism ψ_{G} is a morphism of coalgebras by construction. It is compatible with direct products as in Proposition 3.2.7. Hence it suffices to consider the cases G = G^{0} and G = F separately. The latter case is trivial. From now on let G = G^{0} and assume also that G is reduced, i.e., smooth. By Lemma 7.4.5 we are reduced to the semiabelian case. This is main Theorem 3.1.4. Compatiblity with the Hopf object structure follows by the same reductions from the semiabelian case and the finite case because (G × G)^{0} = G^{0} × G^{0}. □

The following statement must be well-known to experts but we have not found a reference.

Lemma 7.4.8. Let k be algebraically closed and G a smooth commutative group scheme over k. Then

G ≅ G^{0} × F

with G^{0} connected and F finite.

Proof. We claim that

ExtcGrp^{1}(F, G^{0})

vanishes. We compute in the category of étale sheaves on Sch. The finite group scheme F is constant because k is algebraically closed. Hence there is a finite resolution

0 → ℤ^{n} ⟶^{M} ℤ^{n} → F → 0

where M is multiplication by a diagonal matrix. The interesting part of the associated long exact sequence reads

G^{0}(k)^{n} ⟶^{M} G^{0}(k)^{n} → Ext^{1}(F, G^{0}) → Ext^{1}(ℤ^{n}, G^{0})

As k is algebraically closed the functor Hom(ℤ, ·) = Γ(k, ·) is exact, hence the last group vanishes. The first map is surjective because the multiplication by m is surjective on the connected group G^{0}. □

Appendix A. Comparison of Chern classes

In this section we establish a comparison of two possible ways of attaching a cohomology class to a semiabelian variety. This technical result is one of the key inputs into the proof of our main Theorem 3.1.4. We keep the notation of the main text, in particular Section 2.

A.1. The comparison result. Let

1 → 𝔾_{m} → G → H → 1

be a short exact sequence of semiabelian varieties. Note that G is a 𝔾_{m}-torsor on H, hence it has a cohomology class

(1)  c_{1}([G]) ∈H_{ét}^{1}(H, \underline{𝔾_{m}}_{ℚ})
(2)  ≅ Mor_{D^{−}(ShT_{ét}(k,ℚ))}(L(H), i(𝟙(1)[2]))

(3) = Mor_{DM_{−,ét}^{eff}(k,ℚ)}(M(H), 𝟙(1)[2]) ,

where the last equality comes by adjunction. On the other hand, the induced exact sequence

0 → M_{1}(𝔾_{m}) → M_{1}(G) → M_{1}(H) → 0

is an element

[M_{1}(G)] ∈ Ext_{ShT_{ét}(k,ℚ)}^{1}(M_{1}(H), 𝟙(1)[1]) =

= Mor_{D^{−}(ShT_{ét}(k,ℚ))}(M_{1}(H), 𝟙(1)[2]) .

By composition with the summation map (see Definition 2.1.4)

α_{H} : M(H) → M_{1}(H)

we obtain

(4)  α_{H}^{∗}[M_{1}(G)] ∈ Mor_{DM_{−,ét}^{eff}(k,ℚ)}(M(H), 𝟙(1)[2]) .

Proposition A.1.1. Let

1 → 𝔾_{m} → G → H → 1

be a short exact sequence of semiabelian varieties. Then the elements c_{1}([G]) and α_{H}^{∗}[M_{1}(G)] agree in Mor_{DM_{−,ét}^{eff}(k,ℚ)}(M(G), 𝟙(1)[2]). In other words, the diagram

\xymatrix{M(H) \ar[dr]^{c_{1}([G])]} \ar[d]^{α} \\ M_{1}(H) \ar[r]_{[M_{1}(G)]} & 𝟙(1)[2]}

commutes.

Remark A.1.2. We formulate everything with rational coefficients because this is the way we want to apply the statement. However, the result is already true with integral coefficients.

A.2. Proof. The proof will be given at the end of this appendix. We introduce some notation.

Notation A.2.1. Let H be a commutative group scheme. As before we write \underline{H} for the corresponding étale sheaf of abelian groups. Moreover, we write \underline{H}^{tr} for the corresponding étale sheaf with transfers.

Lemma A.2.2. Let H be a commutative group scheme. Then there is a commutative diagram of δ-functors on ShT_{ét}(k, ℚ):

Ext_{ShT_{ét}(k,ℚ)}^{i}(\underline{H}_{ℚ}^{tr}, ·)  ⟶ Ext_{Sh_{ét}(k,ℚ)}^{i}(\underline{H}_{ℚ}, ·)
γ^{∗}↓  ↓^{Y_{Sh_{ét}(k,ℚ)}}
Ext_{ShT_{ét}(k,ℚ)}^{i}(L(H), ·) ⟶_{Y_{ShT_{ét}(k,ℚ)}}  H_{ét}^{i}(H, ·)

with γ : L(H) → \underline{H}_{ℚ}^{tr} the summation map of Proposition 2.1.3 and Y_{?} the Yoneda map in the categories ShT_{ét}(k, ℚ) and Sh_{ét}(k, ℚ), respectively.

Proof. By universality in ShT_{ét}(k, ℚ) and Sh_{ét}(k, ℚ), it suffices to check the case i = 0. Let A be an étale sheaf with transfers and f : \underline{H}_{ℚ}^{tr} → A a morphism in ShT_{ét}(k, ℚ). We describe its image in A(H) via the right-hand side map of the diagram. By forgetting transfers, f gives rise to a map

f : \underline{H}_{ℚ} → A .

Evaluating on H, we get

f(H) : \underline{H}_{ℚ}(H) → A(H) .

844 Giuseppe Ancona, Stephen Enright-Ward, Annette Huber Then Y_{Sh_{ét}(k,ℚ)}(f) is defined as f(H)(id). Going via the left hand side, we have to evaluate the map in ShT_{ét}(k, ℚ)

L(H) ⟶^{γ} \underline{H}_{ℚ}^{tr} ⟶^{f} A

on H and get

Hom_{SmCor/k}(H, H) = L(H)(H) ⟶^{γ(H)} \underline{H}_{ℚ}^{tr}(H) ⟶^{f(H)} A(H) .

Then Y_{ShT_{ét}(k,ℚ)}(γ^{∗}(f)) is given by f(H)γ(H)(id). The summation map γ(H) maps the identity in SmCor/k to the identity in \underline{H}_{ℚ}^{tr}(H) = \underline{H}_{ℚ}(H) hence

Y_{Sh_{ét}(k,ℚ)}(f) = Y_{ShT_{ét}(k,ℚ)}(γ^{∗}(f)) .

□

Corollary A.2.3. Let

1 → 𝔾_{m} → G → H → 1

be a short exact sequence of commutative group schemes. Then

Y_{ShT_{ét}(k,ℚ)}γ^{∗}[\underline{G}_{ℚ}^{tr}] = ∂(id_{H})

where ∂ is the connecting homomorphism

Γ(H, \underline{H}_{ℚ}) → H_{ét}^{1}(H, \underline{𝔾_{m}}_{ℚ}) .

Proof. Note that [\underline{G}_{ℚ}^{tr}] = ∂(id) where ∂ is the boundary map

Hom_{ShT_{ét}(k,ℚ)}(\underline{H}_{ℚ}^{tr}, \underline{H}_{ℚ}^{tr}) → Ext_{ShT_{ét}(k,ℚ)}^{1}(\underline{H}_{ℚ}^{tr}, \underline{𝔾_{m}}_{ℚ}^{tr})

for the short exact sequence

0 → \underline{𝔾_{m}}_{ℚ}^{tr} → \underline{G}_{ℚ}^{tr} → \underline{H}_{ℚ}^{tr} → 0 .

The statement follows from evaluating the δ-homomorphism of Lemma A.2.2

Ext_{ShT_{ét}(k,ℚ)}^{i}(\underline{H}_{ℚ}^{tr}, ·) → H_{ét}^{i}(H, ·)

on this short exact sequence. □ Proof of Proposition A.1.1. Let 1 → 𝔾_{m} → G → H → 1 be an exact sequence of commutative group schemes. It suffices to show that

c_{1}([G]) = Y_{ShT_{ét}(k,ℚ)}γ^{∗}[\underline{𝔾_{m}}_{ℚ}^{tr}] ∈ H_{ét}^{1}(H, \underline{𝔾_{m}}_{ℚ}) .

By Corollary A.2.3 this is reduced to showing

c_{1}([G]) = ∂(id_{H})

where ∂ is the connecting homomorphism

Γ(H, \underline{H}_{ℚ}) → H_{ét}^{1}(H, \underline{𝔾_{m}}_{ℚ})

for the short exact sequence

0 → \underline{𝔾_{m}}_{ℚ} → \underline{G}_{ℚ} → \underline{H}_{ℚ} → 0 .

Documenta Mathematica 20 (2015) 807–858

We use the point of view of Čech cohomology in order to compute explicitly. Choose a local trivialization of the 𝔾_{m}-torsor G, i.e., a covering {U_{i} → H}_{i∈I} and sections s_{i} : U_{i} → G|_{U_{i}} inducing isomorphisms

ϕ_{i} : 𝔾_{m} × U_{i} → G|_{U_{i}} .

By definition c_{1}[G] is given by the cocycle g_{ij} ∈ 𝔾_{m}(U_{i} ∩ U_{j}) where

s_{j}|_{U_{i}∩U_{j}} = s_{i}|_{U_{i}∩U_{j}}g_{ij} .

On the other hand, in order to define ∂(id_{H}) we have to choose preimages of id_{H}|_{U_{i}} in G(U_{i}). We choose s_{i}. We then have to apply the boundary map of the Čech complex of \underline{G}_{ℚ} and get the same cocycle g_{ij}. □

Appendix B. Universal properties of the symmetric (co)algebra

We review the Hopf algebra structure on the symmetric algebra and its opposite. Throughout the appendix let A be a ℚ-linear pseudo-abelian symmetric tensor category with unit object 𝟙. These considerations are going to be applied to the triangulated category DM_{−,ét}^{eff}(k, ℚ).

B.1. The symmetric (co)algebra. Let V be an object of A. We denote

T (V) = ⨁_{i=0}^{∞} V^{⊗i}

the tensor algebra with multiplication

μ : T (V) ⊗ T (V) → T (V)

given by the tensor product. We denote by

Sym^{n}V

the image of the projector \frac{1}{n!} ∑_{σ∈S_{n}} σ. Let

ι^{n} : Sym^{n}V → V^{⊗n}, π^{n} : V^{⊗n} → Sym^{n}V

be the natural inclusion and projection. For n = 0 we put Sym^{0}(V) = 𝟙. The projection

π^{n+m} : V^{⊗n+m} → Sym^{n+m}(V)

factors uniquely via π^{n} ⊗ π^{m} and induces

π_{n+m}^{n,m} : Sym^{n}(V) ⊗ Sym^{m}(V) → Sym^{n+m}(V).

The inclusion

ι^{n+m} : Sym^{n+m}(V) → V^{⊗n} ⊗ V^{⊗m}

factors uniquely via ι^{n} ⊗ ι^{m} and induces

ι_{n+m}^{n,m} : Sym^{n+m}(V) → Sym^{n}(V) ⊗ Sym^{m}(V) .

Definition B.1.1. Assume that V is odd finite-dimensional in the sense of Section 1.3, i.e., there is N such that Sym^{N}(V) = 0. The symmetric algebra on V is given by

Sym(V) = ⨁_{n≥0} Sym^{n}V

with multiplication

μ : Sym^{n}(V) ⊗ Sym^{m}(V) → Sym^{n+m}(V)

given by π_{n,m}^{n+m}. The symmetric coalgebra on V is given by

coSym(V) = ∏_{n≥0} Sym^{n}V

with comultiplication

Δ : Sym^{n+m}V → Sym^{n}(V) ⊗ Sym^{m}(V)

given by ι_{n+m}^{n,m}.

Remark B.1.2. The finiteness assumption ensures that all direct sums and products are finite. If we drop the assumption, the definition of the algebra Sym(V) needs existence of the direct sum and that infinite direct sums commute with ⊗. The definition of the coalgebra coSym(V) needs existence of infinite products and that they commute with ⊗. The latter is not satisfied in standard abelian categories like modules. Alternatively, one may work in the ind-category and the pro-category, respectively. For simplicity, we make the finiteness assumption. This is enough for our application.

B.2. Universal properties. Note that T (V) → Sym(V) is a morphism of algebras.

Lemma B.2.1. Let V be odd finite-dimensional. Let A be a unital algebra object in A and let α : V → A a morphism. Then there is a unique morphism of unital algebras

T (V) → A

extending α. If A is commutative, the map factors through a unique map of algebras

Sym(V) → A .

Let B be an augmented cocommutative coalgebra object in A and α : B → V a morphism. Then there is a unique morphism of augmented coalgebras

B → coSym(V) .

Proof. The argument is the same as for vector spaces, where it is well-known.

□

We apply this principle to the diagonal map

Δ : V → V ⊗ 𝟙 ⊕ 𝟙 ⊗ V ⊂ T (V) ⊗ T (V)

and obtain a comultiplication

Δ : T (V) → T (V) ⊗ T (V) .

It turns T (V) into a bialgebra. The same argument also turns Sym(V) into a bialgebra and

T (V) → Sym(V)

is a morphism of bialgebras. Dually, the summation map

coSym(V) ⊗ coSym(V) → V ⊗ 𝟙 ⊕ 𝟙 ⊗ V ⟶^{+} V

gives rise to a multiplication on coSym(V) making it a bialgebra. Finally, multiplication by −1 defines a map

V ⟶^{−1} V

which induces an antipodal map on Sym(V) and on coSym(V). It turns Sym(V) and coSym(V) into Hopf algebras.

Remark B.2.2. If A is the category of ℚ-vector spaces and V a finitedimensional vector space, then Sym(V) is a polynomial ring, whereas coSym(V) is the algebra of distributions. The two are isomorphic but not equal.

Lemma B.2.3. Let V be odd finite-dimensional.

(1) Let A be a unital commutative bialgebra in A and α : V → A a mor-

phism such that

V  ⟶ A
Δ↓  ↓^{Δ}
V ⊗ 𝟙 ⊕ 𝟙 ⊗ V ⟶ A ⊗ A

commutes. Then the universal morphism Sym(V) → A is a morphism of bialgebras.

(2) Let B be an augmented cocommutative bialgebra in A and β : B → V

a morphism such that

B ⊗ B ⟶ V ⊗ 𝟙 ⊕ 𝟙 ⊗ V
μ↓  ↓^{+}
B ⟶  V

commutes. Then the universal morphism coSym(V) → B is a morphism of bialgebras.

Proof. This is an assertion about algebra morphisms. It follows from the universal property of Sym(V). The second part follows by the analogous argument. □

Lemma B.2.4. (1) The component

Δ_{n+m}^{n,m} : Sym^{n+m}V → Sym^{n}V ⊗ Sym^{m}V

of comultiplication on Sym(V) is equal to

Δ_{n+m}^{n,m} = \binom{n + m}{n}ι_{n+m}^{n,m} .

(2) The component of multiplication on coSym(V)

μ_{n,m}^{n+m} : Sym^{n}(V) ⊗ Sym^{m}(V) → Sym^{n+m}(V)

is equal to

μ_{n,m}^{n+m} = \binom{n + m}{n}π_{n,m}^{n+m} .

(3) Let V be odd finite-dimensional. The universal map of bialgebras

Sym(V) → coSym(V)

induced from Sym(V) → V is given by multiplication by n! in degree n. It is an isomorphism.

In particular, the two bialgebras are not identical.

Proof. The statement on Δ is elementary from the definitions. We now consider the map

Sym(V) → coSym(V)

given by multiplication by n! in degree n. We see from the explicit formula that it is compatible with comultiplication, i.e., it is the canonical one. It satisfies the criterion Lemma B.2.3, hence it is an isomorphism of bialgebras. The formula for multiplication on coSym(V) follows from this isomorphism. □

Corollary B.2.5. Assume that V is odd finite-dimensional. Then the bialgebra Sym(V) has the universal properties of Lemma B.2.1 and Lemma B.2.3 with respect to comultiplication. The bialgebra coSym(V) has the universal properties of Lemma B.2.1 and Lemma B.2.3 with respect to multiplication.

B.3. Direct sums.

Lemma B.3.1. Let V = U ⊕ W in A with U and W odd finite-dimensional. The natural map

coSym(U) ⊗ coSym(W) → U ⊗ 𝟙 ⊕ 𝟙 ⊗ W ≅ U ⊕ W

gives rise to an isomorphism of bialgebras

coSym(U) ⊗ coSym(W) → coSym(U ⊕ W)

with inverse given by

coSym(U ⊕W) ⟶^{Δ} coSym(U ⊕W)⊗coSym(U ⊕W) → coSym(U)⊗coSym(W)

with the second map induced by functoriality of coSym in each factor. The analogous statement for Sym(U ⊕ W) holds as well.

Proof. The isomorphism is well-known for vector spaces. The case of an additive category is a special case of [Del02, Propostion 1.8]. By construction, the isomorphism is the one compatible with the inclusion into the tensor algebra, i.e., the one for the symmetric coalgebra. The case of the symmetric algebra follows because the map is a rational multiple of the one for the coalgebra. □

Remark B.3.2. The isomorphism

Sym^{N}(U ⊕ W) → ⨁_{n+m=N} Sym^{n}(U) ⊗ Sym^{m}(V)

via the symmetric coalgebra is not the same as the one defined on the symmetric algebra!

Appendix C. Filtrations on the graded symmetric (co)algebra

The aim of this appendix is to establish a certain exact triangle for the symmetric coalgebra, see Theorem C.4.4. The basic construction behind it already appears in a paper by Deligne [Del02], more precisely in the Proof of Proposition 1.19 of loc. cit. We wanted to understand the details of the argument and, in particular, keep precise control of the morphisms and the Hopf algebra structure. Hence we decided to give the argument in full detail. Throughout the appendix, let A be a ℚ-linear abelian symmetric tensor category with unit object 𝟙. We assume that ⊗ is exact. These considerations are going to be applied to the abelian category ShT_{ét}(k, ℚ) of etale sheaves of ℚ-vector spaces with transfers. This is possible by Remark 1.6.3.

C.1. The graded symmetric (co)algebra.

Definition C.1.1. Let V be an object of A. We denote by

Sym^{∗}(V) = ⨁_{n≥0} Sym^{n}(V)

the graded symmetric algebra with Sym^{n}(V) in degree n. It is a graded Hopf algebra with structure morphisms as in Appendix B. We denote

coSym^{∗}(V) = ⨁_{n≥0} Sym^{n}(V)

the graded symmetric coalgebra with Sym^{n}(V) in degree n. It is a graded Hopf algebra with structure morphisms as in Appendix B.

The explicit formulae are given in Lemma B.2.4.

Remark C.1.2. Note that we are considering graded objects. In contrast to Appendix B we do not assume that V is odd finite-dimensional. Neither do we need existence of infinite direct sums. The notation ⨁_{n≥0} is just convention. A graded object really is given by a sequence of objects. Switching to graded objects and graded morphisms allows all considerations of Appendix B without finiteness assumptions of any kind.

Lemma C.1.3. The universal properties of Lemma B.2.1 and Lemma B.2.3 are satisfied for Sym^{∗}(V) and coSym^{∗}(V) with respect to graded morphisms. The identity on V induces an isomorphism of graded Hopf algebras

Sym^{∗}(V) → coSym^{∗}(V) .

In degree n it is given by multiplication by n!.

Proof. Same arguments as in Appendix B. Note that the finiteness assumptions are not needed because all morphisms respect the grading. □

C.2. The filtration on the graded tensor algebra.

Definition C.2.1. Let U ⊂ V be a subobject in A. We define a descending filtration on V by

Fil_{i}^{U}V = \begin{cases}V & i = 0, \\ U & i = 1, \\ 0 & i > 1. \end{cases}

For n ≥ 0 let

Fil_{i}^{U}V^{⊗n} ⊂ V^{⊗n}

the product filtration on V^{⊗n} and

Gr_{i}^{U}V^{⊗n} = Fil_{i}^{U}V^{⊗n}/Fil_{i+1}^{U}V^{⊗n}

the associated graded objec object.

This means that an elementary tensor is in ith step of the filtration step, if at least i factors are in U.

Lemma C.2.2. Let U ⊂ V be a subobject in A. The filtration Fil_{i}^{U}T (V) turns the graded tensor algebra into a filtered graded bialgebra, i.e.,

μ : Fil_{i}^{U}V^{⊗n} ⊗ Fil_{j}^{U}V^{⊗m} → Fil_{i+j}^{U}V^{⊗n+m} ,

Δ : Fil_{I}^{U}V^{⊗N} → ∑_{i+i′=I,n+n′=N} Fil_{i}^{U}V^{⊗n} ⊗ Fil_{i′}^{U}V^{⊗n′} .

In particular, Gr_{•}^{U}T^{∗}(V) is a bigraded bialgebra.

Proof. The statement on multiplication is obvious. The statement on comultiplication is reduced by the universal property to the basic case N = 1. □

The aim of this section is to compute the associated graded bialgebra.

Proposition C.2.3. Let

0 → U → V → W → 0

be a short exact sequence in A. Then there is a canonical isomorphism of bigraded bialgebras

Gr_{•}^{U}T^{∗}(V) ≅ Gr_{•}^{U}T^{∗}(U ⊕ W) = T^{•}(U) ⊗ T^{∗−•}(W) .

The proof will be given at the end of the section.

Remark C.2.4. Whenever the sequence has a splitting (e.g. in the case when A is a category of vector spaces), this formula is easy to check. The purpose of the following arguments is to verify that they work for more general A.

Lemma C.2.5. Let A′ ⊂ A and B′ ⊂ B be subobjects in A. Then

(A′ ⊗ B) ∩ (A ⊗ B′) = A′ ⊗ B′

where the intersection is taken in A ⊗ B.

Proof. Consider the monomorphism of short exact sequences

\xymatrix{0 \ar[r] & A ⊗ B′ \ar[r] & A ⊗ B \ar[r] & A ⊗ (B/B′) \ar[r] & 0 \\ 0 \ar[r] & A′ ⊗ B′ \ar[r] \ar@{^{(}->}[u] & A′ ⊗ B \ar@{^{(}->}[u] \ar[r] & A′ ⊗ (B/B′) \ar[r] \ar@{^{(}->}[u] & 0}

The assertion follows from diagram chasing. □

Lemma C.2.6. Let i ≥ 0 and n ≥ 1 be integers, and let U ⊂ V be a subobject in A. Then the sequence

0 → U ⊗ Fil_{i}^{U}V^{⊗(n−1)} →

→ (U ⊗ Fil_{i−1}^{U}V^{⊗(n−1)}) ⊕ (V ⊗ Fil_{i}^{U}V^{⊗(n−1)}) → Fil_{i}^{U}V^{⊗n} → 0

is exact.

Proof. Obviously,

(U ⊗ Fil_{i−1}^{U}V^{⊗n−1}) + (V ⊗ Fil_{i}^{U}V^{⊗n−1}) = Fil_{i}^{U}V^{⊗n} .

It remains to check that

(U ⊗ Fil_{i−1}^{U}V^{⊗n−1}) ∩ (V ⊗ Fil_{i}^{U}V^{⊗n−1}) = U ⊗ Fil_{i}^{U}V^{⊗n−1} .

This is true by Lemma C.2.5. □

Lemma C.2.7. Let n ≥ 1, i ≥ 0 and

0 → U → V → W → 0

be a short exact sequence in A. Then there is a natural isomorphism

Gr_{i}^{U}V^{⊗n} → Gr_{i}^{U}(U ⊕ W)^{⊗n} .

Remark C.2.8. The object Gr_{i}^{U}(U ⊕ W) is a direct sum of tensor products of i copies of U and n − i copies of W, running through all possible choices. E.g.

Gr_{1}^{U}(U ⊕ W)^{⊗3} = (U ⊗ W ⊗ W) ⊕ (W ⊗ U ⊗ W) ⊕ (W ⊗ W ⊗ U) .

Proof of Lemma C.2.7. We argue by induction on n. The case n = 1 holds by definition. For n > 0 we consider the commutative diagram of short exact sequences

\xymatrix{0 \ar[r] & U ⊗ Fil_{i+1}^{U}V^{⊗n} \ar[r] \ar@{^{(}->}[d] & (U ⊗ Fil_{i}^{U}V^{⊗n}) ⊕ (V ⊗ Fil_{i+1}^{U}V^{⊗n}) \ar[r] \ar@{^{(}->}[d] & Fil_{i+1}^{U}V^{⊗n+1} \ar[r] \ar@{^{(}->}[d] & 0 \\ 0 \ar[r] & U ⊗ Fil_{i}^{U}V^{⊗n} \ar[r] & (U ⊗ Fil_{i−1}^{U}V^{⊗n}) ⊕ (V ⊗ Fil_{i}^{U}V^{⊗n}) \ar[r] & Fil_{i}^{U}V^{⊗n+1} \ar[r] & 0}

By the snake lemma we get a short exact sequence of cokernels. By induction it reads

0 → U ⊗ Gr_{i}^{U}(U ⊕ W)^{⊗n} →

→ U ⊗ Gr_{i−1}^{U}(U ⊕ W)^{⊗n} ⊕ V ⊗ Gr_{i}^{U}(U ⊕ W)^{⊗n} → Gr_{i}^{U}V^{⊗n+1} → 0 .

Note that the map

U ⊗ Gr_{i}^{U}(U ⊕ W)^{⊗n} → U ⊗ Gr_{i−1}^{U}(U ⊕ W)^{⊗n}

vanishes, hence

Gr_{i}^{U}V^{⊗n+1} ≅ U⊗Gr_{i−1}^{U}(U⊕W)^{⊗n}⊕W ⊗Gr_{i−1}^{U}(U⊕W)^{⊗n} = Gr_{i}^{U}(U⊕W)^{⊗n+1} .

□

Proof of Proposition C.2.3. We apply Lemma C.2.7 for all i and n. The compatibility with multiplication and comultiplication follows from the construction or more abstractly from naturality. □

C.3. The filtration on the graded symmetric (co)algebra.

Definition C.3.1. Let U ⊂ V be a subobject in A and n ≥ 0. We define a descending filtration on Sym^{n}(V) by

Fil_{i}^{U}Sym^{n}(V) = π^{n} (Fil_{i}^{U}V^{⊗n})

and

Gr_{i}^{U}Sym^{n}(V) = Fil_{i}^{U}Sym^{n}(V)/Fil_{i+1}^{U}Sym^{n}(V)

the associated graded object.

Remark C.3.2. We have the simpler presentation

Fil_{i}^{U}Sym^{n}(V) = π^{n} (U^{⊗i} ⊗ V^{⊗n−i}) .

Lemma C.3.3. Let U ⊂ V be a subobject in A. The filtration

Fil_{i}^{U}coSym^{∗}(V) = ⨁_{n≥0} Fil_{i}^{U}Sym^{n}(V)

turns the graded symmetric coalgebra coSym^{∗}(V) into a filtered graded bialgebra. In particular, Gr_{•}^{U}coSym^{∗}(V) is a bigraded bialgebra. The analogous statement for the symmetric algebra is also true.

Proof. The assertion for the symmetric algebra follows from the case of the tensor algebra. The assertion for the symmetric coalgebra follows because multiplication and comultiplication on coSym^{∗}(V) are degreewise rational multiples of multiplication and comultiplication on Sym^{∗}(V). □

Proposition C.3.4. Let

0 → U → V → W → 0

be a short exact sequence in A. Then there is a natural isomorphism of bigraded bialgebras

Gr_{•}^{U}coSym^{∗}(V) ≅ coSym^{•}(U) ⊗ coSym^{∗−•}(W) .

In particular for 0 ≤ i ≤ n, there are natural short exact sequences

0 → Fil_{i+1}^{U}Sym^{n}(V) → Fil_{i}^{U}Sym^{n}(V) ⟶^{g^{n,i}} Sym^{i}(U) ⊗ Sym^{n−i}W → 0 .

The map g^{n,i} is obtained by factoring

U^{⊗i} ⊗ V^{⊗n−i} → U^{⊗i} ⊗ W^{⊗n−i} ⟶^{μ_{coSym}} Sym^{i}(U) ⊗ Sym^{n−i}(W)

uniquely through Fil_{i}^{U}Sym^{n}(V). The analogous statement for Sym^{∗}(V) is also true.

Proof. We first consider the case of the symmetric algebra. As a projector, the symmetrization map π^{n} preserves short exact sequences. Hence Proposition C.2.3 implies

Gr_{•}^{U}Sym^{∗}(V) ≅ Gr_{•}^{U}Sym^{∗}(U ⊕ W) ≅ Sym^{•}(U) ⊗ Sym^{∗−•}(W) .

Specializing to Gr_{i}^{U}Sym^{n}(V) provides the required short exact sequence. It remains to verify the explicit description of g^{n,i}. Everything is determined on the level of the graded tensor algebra, where the map to the associated graded object is induced from the projection V → W in the appropriate factors. As Fil_{i}^{U}Sym^{n}(V) is the image of U^{⊗i} ⊗ V^{n−i} ⊂ Fil_{i}^{U}V^{⊗n}, it suffices to describe the map on this object. In the case of the symmetric coalgebra everything agrees up to rational factors. Exactness of the sequence follows from the first case. □

Remark C.3.5. This agrees with the map denoted g^{n,i} in second author’s thesis [EW13] (see loc. cit. Notation 5.3.10).

C.4. Cup-product. Let W be an object of A. The coalgebra structure on coSym^{∗}(W) allows to define cup-products.

Definition C.4.1. Let W be an object of A, c : W → K a morphism in the derived category D^{b}(A). Then we define

· ∪ c : coSym^{∗}(W) → coSym^{∗−1}(W) ⊗ K

as the composition

coSym^{∗}(W) ⟶^{Δ} coSym^{∗}(W) ⊗ coSym^{∗}(W)

→ coSym^{∗}(W) ⊗ W

⟶^{id⊗c} coSym^{∗}(W) ⊗ K .

We apply this to the morphism [V] : W → U[1] in D^{n}(A) represented by a short exact sequence

0 → U → V → W → 0

in A.

Proposition C.4.2. Let

0 → U → V → W → 0

be a short exact sequence in A. Then

· ∪ [V] : coSym^{∗}(W) → coSym^{∗−1}(W) ⊗ U[1]

is equal to the extension class

[coSym^{∗}(V)/Fil_{2}^{U}coSym^{∗}(V)]

under the identifications of Proposition C.3.4 Gr_{0}^{U}coSym^{∗}(V) = coSym^{∗}(W) and Gr_{1}^{U}(V) ≅ coSym^{∗}(W) ⊗ U.

Remark C.4.3. This corresponds to the crucial computation [EW13, Lemma 5.3.13].

Proof of Proposition C.4.2. We view the morphisms in D(A) as Yoneda extensions. We need to identify the extension class [Sym^{n}(V)/Fil_{2}^{U}Sym^{n}(V)], i.e.,

0 → Sym^{n−1}(V) ⊗ U → Sym^{n}(V)/Fil_{2}^{U}Sym^{n}(V) → Sym^{n}W → 0

with the pull-back of

0 → Sym^{n−1}W ⊗ U → Sym^{n−1}W ⊗ V → Sym^{n−1}W ⊗ W → 0

via the component

Δ^{n−1,1} : Sym^{n}W → Sym^{n−1}W ⊗ W

of the comultiplication. The same component of the comultiplication but for coSym^{∗}(V) gives rise to a map

Δ̄_{n}^{n−1,1} : Sym^{n}V ⟶^{Δ^{n−1,1}} Sym^{n−1}V ⊗ V → Sym^{n−1}W ⊗ V .

This gives rise to a morphism of short exact sequences

0 ⟶ Gr_{1}^{U}Sym^{n}V ⟶ Sym^{n}V/Fil_{2}^{U}Sym^{n}V ⟶  Sym^{n}W  ⟶ 0
↓  ↓Δ̄_{n}^{n−1,1}  ↓^{Δ_{n}^{n−1,1}}
0 ⟶ Sym^{n−1}W ⊗ U ⟶ Sym^{n−1}W ⊗ V ⟶ Sym^{n−1}W ⊗ W ⟶ 0

It remains to check that the induced map on kernels equals g^{n,1}, i.e., it is induced by multiplication. By the explicit description, it suffices to check that the diagram

\xymatrix{V^{⊗n−1} ⊗ U \ar[r]^{μ^{n}} \ar[d]_{μ^{n−1}⊗id} & Sym^{n}V \ar[d]^{Δ̄_{n}^{n,1}} \\ Sym^{n−1}V ⊗ V \ar[r] & Sym^{n−1}W ⊗ V}

commutes. Recall that multiplication and comultiplication are the ones of the symmetric coalgebra, and hence

μ^{n} = n!π^{n} , μ^{n−1} = (n − 1)!π^{n−1}

and

Δ^{n−1,1} = ι^{n−1,1} = \frac{1}{n} ∑_{i=1}^{n} σ_{i}

where

σ_{i} : V^{⊗n} → V^{⊗n−1} ⊗ V

is the permutation that swaps the i-th factor into the last place and leaves the order otherwise intact. Hence we can equivalently check the commutativity of the following diagram

V^{⊗n−1} ⊗ U ⟶^{π^{n}}  Sym^{n}V
π^{n−1}⊗id↓  ↓^{∑_{i=1}^{n} σ_{i}}
Sym^{n−1}V ⊗ V ⟶ Sym^{n−1}W ⊗ V

It suffices to check the same identity on the level of tensor algebras. By abuse of notation

σ_{i}(g_{1} ⊗ g_{2} ⊗ … ⊗ g_{n}) = (g_{1} ⊗ … ⊗ ĝ_{i} ⊗ … ⊗ g_{n}) ⊗ g_{i}

where ĝ_{i} means that the factor is omitted. The composition

V^{⊗n−1} ⊗ U ⊂ V^{⊗n} ⟶^{σ_{i}} V^{⊗n−1} ⊗ V → W^{⊗n−1} ⊗ V

vanishes for i ≠ n because it involves a factor U → W. Hence only σ_{n} = id contributes to

V^{⊗n−1} ⊗ U ⊂ V^{⊗n} ⟶^{Δ^{n−1,1}} V^{⊗n−1} ⊗ V → W^{⊗n−1} ⊗ V .

This finishes the proof. □

Theorem C.4.4. Let A be a ℚ-linear abelian symmetric tensor category. Moreover, let T be a ℚ-linear, tensor, symmetric, triangulated category and q : D^{b}(A) → T be a ℚ-linear, tensor, symmetric and triangulated functor. Let

0 → U → V → W → 0

be a short exact sequence in A. Suppose that Sym^{2}(q(U)) = 0. Then there is a canonical triangle in T

coSym^{n}(q(V)) → coSym^{n}(q(W)) ⟶^{·∪[V]} coSym^{n−1}(q(W)) ⊗ q(U)[1] .

Proof. By Proposition C.4.2 the short exact sequence in A

0 → coSym^{∗−1}(W) ⊗ U → coSym^{∗}(V)/Fil_{2}^{U}coSym^{∗}(V) → coSym^{∗}(W) → 0

gives rise to the exact triangle

coSym^{∗}(V)/Fil_{2}^{U}coSym^{∗}(V) → coSym^{∗}(W) ⟶^{·∪[V]} coSym^{∗−1}(W) ⊗ U[1] .

We apply q. It remains to show that

q(Fil_{2}^{U}coSym^{∗}(V)) = 0 .

This follows by descending induction from the system of triangles of Proposition C.3.4

q(Fil_{i+1}^{U}Sym^{n}(V)) → q(Fil_{i}^{U}Sym^{n}(V)) → Sym^{i}(q(U)) ⊗ Sym^{n−i}q(W)

and the vanishing of Sym^{i}(q(U)) for i ≥ 2. □

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Giuseppe Ancona Universität Duisburg-Essen Thea-Leymann-Straße 9 45127 Essen Germany monsieur.beppe@gmail.com Stephen Enright-Ward Mathematisches Institut Albert-Ludwigs-Universität Freiburg Eckerstraße 1 79104 Freiburg im Breisgau Germany steveleward@gmail.com

Annette Huber

Mathematisches Institut Albert-Ludwigs-Universität Freiburg Eckerstraße 1 79104 Freiburg im Breisgau Germany annette.huber@math.unifreiburg.de