Commutative Algebraic Groups up to Isogeny

Michel Brion

Received: February 10, 2016 Revised: June 28, 2016

Communicated by Henning Krause

Abstract. Consider the abelian category C_{k} of commutative group schemes of finite type over a field k. By results of Serre and Oort, C_{k} has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of C_{k} by the full subcategory F_{k} of finite k-group schemes. We show that C_{k}/F_{k} has homological dimension 1, and we determine its projective or injective objects. We also obtain structure results for C_{k}/F_{k}, which take a simpler form in positive characteristics.

2010 Mathematics Subject Classification: 14K02, 14L15, 18E35, 20G07. Keywords and Phrases: commutative algebraic groups, isogeny category, homological dimension.

1 Introduction

There has been much recent progress on the structure of algebraic groups over an arbitrary field; in particular, on the classification of pseudo-reductive groups (see [CGP15, CP15]). Yet commutative algebraic groups over an imperfect field remain somewhat mysterious, e.g., extensions with unipotent quotients are largely unknown; see [To13] for interesting results, examples, and questions. In this article, we explore the category of commutative algebraic groups up to isogeny, in which the problems raised by imperfect fields become tractable; this yields rather simple and uniform structure results. More specifically, denote by C_{k} the category with objects the group schemes of finite type over the ground field k, and with morphisms, the homomorphisms of k-group schemes (all group schemes under consideration will be assumed commutative). By a result of Grothendieck (see [SGA3, VIA, Thm. 5.4.2]), C_{k} is an abelian category. We define the category of ‘algebraic groups up to isogeny’ as the quotient category of C_{k} by the Serre subcategory of finite group schemes; then C_{k}/F_{k} is obtained from C_{k} by inverting all isogenies, i.e., all morphisms with finite kernel and cokernel. It will be easier to deal with the full subcategory \underline{C}_{k} of C_{k}/F_{k} with objects the smooth connected algebraic groups, since these categories turn out to be equivalent, and morphisms in \underline{C}_{k} admit a simpler description. As a motivation for considering the ‘isogeny category’ \underline{C}_{k}, note that some natural constructions involving algebraic groups are only exact up to isogeny; for example, the formations of the maximal torus or of the largest abelian variety quotient, both of which are not exact in C_{k}. Also, some structure theorems for algebraic groups take on a simpler form when reformulated up to isogeny. A classical example is the Poincaré complete reducibility theorem, which is equivalent to the semi-simplicity of the isogeny category of abelian varieties, i.e., the full subcategory \underline{A}_{k} of \underline{C}_{k} with objects abelian varieties. Likewise, the isogeny category of tori, \underline{T}_{k}, is semi-simple. We gather our main results in the following:

Theorem. (i) The category \underline{C}_{k} is artinian and noetherian. Its simple objects are the additive group 𝔾_{a,k}, the simple tori, and the simple abelian varieties.

(ii) The product functor \underline{T}_{k} × \underline{U}_{k} → \underline{L}_{k} yields an equivalence of categories, where \underline{U}_{k} (resp. \underline{L}_{k}) denotes the isogeny category of unipotent (resp. linear) algebraic groups.

(iii) If char(k) > 0, then the product functor \underline{S}_{k} × \underline{U}_{k} → \underline{C}_{k} yields an equivalence of categories, where \underline{S}_{k} denotes the isogeny category of semi-abelian varieties. If in addition k is locally finite, then the product functor \underline{T}_{k} × \underline{A}_{k} → \underline{S}_{k} yields an equivalence of categories as well.

(iv) The base change under any purely inseparable field extension k′ of k yields an equivalence of categories \underline{C}_{k} → \underline{C}_{k′}.

(v) The homological dimension of \underline{C}_{k} is 1.

We also describe the projective objects of the category \underline{C}_{k} (Theorem 5.14) and its injective objects (Theorem 5.16). Moreover, in characteristic 0, we obtain a structure result for that category (Proposition 5.13), which turns out to be more technical than in positive characteristics. Let us now compare the above statements with known results on C_{k} and its full subcategories A_{k} (resp. T_{k}, U_{k}, L_{k}, S_{k}) of abelian varieties (resp. tori, unipotent groups, linear groups, semi-abelian varieties). About (i) (an easy result, mentioned by Serre in [Se60]): C_{k} is artinian and not noetherian. Also, every algebraic group is an iterated extension of ‘elementary’ groups; these are the simple objects of \underline{C}_{k} and the simple finite group schemes.

About (ii): denoting by M_{k} the full subcategory of C_{k} with objects the algebraic groups of multiplicative type, the product functor M_{k} × U_{k} → L_{k} yields an equivalence of categories if k is perfect. But over an imperfect field, there exist non-zero extensions of unipotent groups by tori, and these are only partially understood (see [To13, §9] again; our study of \underline{C}_{k} brings no new insight in these issues). About (iii): the first assertion follows from recent structure results for algebraic groups (see [Br17, §5]), together with a lifting property for extensions of such groups with finite quotients (see [Br15] and [LA15]). The second assertion is a direct consequence of the Weil-Barsotti isomorphism (see e.g. [Oo66, §III.18]). About (iv): this is a weak version of a result of Chow on abelian varieties, which asserts (in categorical language) that base change yields a fully faithful functor A_{k} → A_{k′} for any primary field extension k′ of k (see [Ch55], and [Co06, §3] for a modern proof). About (v), the main result of this article: recall that the homological dimension of an abelian category D is the smallest integer, hd(D), such that Ext_{D}^{n}(A, B) = 0 for all objects A, B of D and all n > hd(D); these Ext groups are defined as equivalence classes of Yoneda extensions. In particular, hd(D) = 0 if and only if D is semi-simple. It follows from work of Serre (see [Se60, 10.1 Thm. 1] together with [Oo64, Thm. 3.5]) that hd(C_{k}) = 1 if k is algebraically closed of characteristic 0. Also, by a result of Oort (see [Oo66, Thm. 14.1]), hd(C_{k}) = 2 if k is algebraically closed of positive characteristic. Building on these results, Milne determined hd(C_{k}) when k is perfect (see [Mi70, Thm. 1]); then the homological dimension can be arbitrarily large. In the approach of Serre and Oort, the desired vanishing of higher extension groups is obtained by constructing projective resolutions of elementary groups, in the category of pro-algebraic groups. The latter category contains C_{k} as a full subcategory, and has enough projectives. In contrast, to show that hd(\underline{C}_{k}) = 1 over an arbitrary field k, we do not need to go to a larger category. We rather observe that tori are projective objects in \underline{C}_{k}, and abelian varieties are injective objects there. This yields the vanishing of all but three extension groups between simple objects of \underline{C}_{k}; two of the three remaining cases are handled directly, and the third one reduces to the known vanishing of Ext_{C_{k}}^{2}(𝔾_{a,k}, 𝔾_{a,k}) when k is perfect. When k has characteristic 0, the fact that hd(C_{k}) ≤ 1 follows from a similar result for the category of Laumon 1-motives up to isogeny (obtained by Mazzari in [Ma10, Thm. 2.5]). Indeed, C_{k} is equivalent to a Serre subcategory of the latter category; moreover, if an abelian category has homological dimension at most 1, then the same holds for any Serre subcategory (as follows e.g. from [Oo64, §3]). Likewise, the fact that the category of Deligne 1-motives up to isogeny has homological dimension at most 1 (due to Orgogozo, see [Or04, Prop. 3.2.4]) implies the corresponding assertion for the isogeny category of semi-abelian varieties over an arbitrary field. Abelian categories of homological dimension 1 are called hereditary. The most studied hereditary categories consist either of finite-dimensional modules over a finite-dimensional hereditary algebra, or of coherent sheaves on a weighted projective line (see e.g. [Ha01]). Such categories are k-linear and Hom-finite, i.e., all groups of morphisms are finite-dimensional vector spaces over the ground field k. But this seldom holds for the above isogeny categories. More specifically, \underline{A}_{k} and \underline{T}_{k} are both ℚ-linear and Hom-finite, but not \underline{C}_{k} unless k is a number field. When k has characteristic 0, we may view \underline{C}_{k} as a mixture of k-linear and ℚ-linear categories. This is already displayed by the full subcategory \underline{V}_{k} with objects the vector extensions of abelian varieties: as shown in §5.1, \underline{V}_{k} has enough projectives, and these are either the unipotent groups (k-linear objects), or the vector extensions of simple abelian varieties (ℚ-linear objects). In positive characteristic, one may also consider the quotient category of C_{k} by the Serre subcategory I_{k} of infinitesimal group schemes. This yields the abelian category of ‘algebraic groups up to purely inseparable isogeny’, which is equivalent to that introduced by Serre in [Se60]; as a consequence, it has homological dimension 1 if k is algebraically closed. For any arbitrary field k, the category C_{k}/I_{k} is again invariant under purely inseparable field extensions; its homological properties may be worth investigating.

Notation and conventions. We will use the book [DG70] as a general reference, especially for affine algebraic groups, and the expository text [Br17] for some further results. Throughout this text, we fix the ground field k and an algebraic closure k̄; the characteristic of k is denoted by char(k). We denote by k_{s} the separable closure of k in k̄, and by Γ_{k} the Galois group of k_{s} over k. We say that k is locally finite, if it is algebraic over 𝔽_{p} for some prime p. By an algebraic k-group, we mean a commutative group scheme G of finite type over k; we denote by G^{0} the neutral component of G. The group law of G will be denoted additively: (x, y) ↦ x + y. By a k-subgroup of G, we mean a closed k-subgroup scheme. Morphisms are understood to be homomorphisms of k-group schemes. The (scheme-theoretic) image of a morphism f : G → H will be denoted by Im(f) or f(G), and the (scheme-theoretic) pull-back of a k-subgroup H′ ⊂ H, by G×_{H} H′ or f^{−1}(H′). Recall that an abelian variety over k is a smooth connected proper algebraic k-group. Also, recall that a k-group scheme G is an affine algebraic k-group if and only if G is isomorphic to a k-subgroup of the general linear group GL_{n,k} for some n. We will thus call affine algebraic k-groups linear. We say that an algebraic k-group G is of multiplicative type if G is isomorphic to a k-subgroup of some k-torus. To simplify the notation, we will suppress the mention of the ground field k whenever this yields no confusion. For example, the category C_{k} will be denoted by C, except when we use base change by a field extension. Given an algebraic group G and two subgroups G_{1}, G_{2}, we denote by G_{1} + G_{2} the subgroup of G generated by G_{1} and G_{2}. Thus, G_{1} + G_{2} is the image of the morphism G_{1} × G_{2} → G, (x_{1}, x_{2}) ↦ x_{1} + x_{2}.

An isogeny is a morphism with finite kernel and cokernel. Two algebraic groups G_{1}, G_{2} are isogenous if they can be connected by a chain of isogenies. We say that two subgroups G_{1}, G_{2} of an algebraic group G are commensurable if both quotients G_{1}/G_{1} ∩ G_{2} and G_{2}/G_{1} ∩ G_{2} are finite; then G_{1} and G_{2} are isogenous. Given an algebraic group G and a non-zero integer n, the multiplication by n yields a morphism

n_{G} : G ⟶ G.

We denote its kernel by G[n], and call it the n-torsion subgroup. We say that G is divisible if n_{G} is an epimorphism for all n ≠ 0; then n_{G} is an isogeny for all such n. When char(k) = 0, the divisible groups are the connected algebraic groups; when char(k) = p > 0, they are just the semi-abelian varieties, that is, the extensions of abelian varieties by tori (see e.g. [Br17, Thm. 5.6.3] for the latter result). Still assuming that char(k) = p > 0, we say that an algebraic group G is a p-group if p_{G}^{n} = 0 for n ≫ 0. Examples of p-groups include the unipotent groups and the connected finite algebraic groups, also called infinitesimal.

2 Structure of algebraic groups

2.1 Preliminary results

We will use repeatedly the following simple observation:

Lemma 2.1. Let G be a smooth connected algebraic group.

(i) If G′ is a subgroup of G such that G/G′ is finite, then G′ = G.

(ii) Any isogeny f : H → G is an epimorphism.

Proof. (i) The quotient G/G′ is smooth, connected and finite, hence zero. (ii) This follows from (i) applied to Im(f) ⊂ H.

The following lifting result for finite quotients will also be frequently used:

Lemma 2.2. Let G be an algebraic group, and H a subgroup such that G/H is finite.

(i) There exists a finite subgroup F ⊂ G such that G = H + F.

(ii) If G/H is infinitesimal (resp. a finite p-group), then F may be chosen infinitesimal (resp. a finite p-group) as well.

Proof. (i) This is a special case of [Br15, Thm. 1.1]. (ii) Assume G/H infinitesimal. Then the quotient G/(H + F^{0}) is infinitesimal (as a quotient of G/H) and étale (as a quotient of F/F^{0}), hence zero. Thus, we may replace F with F^{0}, an infinitesimal subgroup.

Next, assume that G/H is a finite p-group. Denote by F [p^{∞}] the largest psubgroup of F. Then the quotient G/(H + F [p^{∞}]) is a finite p-group and is killed by the order of F/F [p^{∞}]. Since the latter order is prime to p, we obtain G/(H + F [p^{∞}]) = 0. Thus, we may replace F with F [p^{∞}].

Next, we recall a version of a theorem of Chevalley:

Theorem 2.3. (i) Every algebraic group G contains a linear subgroup L such that G/L is an abelian variety. Moreover, L is unique up to commensurability in G, and G/L is unique up to isogeny.

(ii) If G is connected, then there exists a smallest such subgroup, L = L(G), and this subgroup is connected.

(iii) If in addition G is smooth, then every morphism from G to an abelian variety factors uniquely through the quotient map G → G/L(G).

Proof. The assertion (ii) follows from [Ra70, Lem. IX 2.7] (see also [BLR90, 9.2 Thm. 1]). To prove (i), note that G contains a finite subgroup F such that G/F is connected (as follows from Lemma 2.2). Then we may take for L the pull-back of a linear subgroup of G/F with quotient an abelian variety. If L′ is another linear subgroup of G such that G/L′ is an abelian variety, then L + L′ is linear, as a quotient of L × L′. Moreover, the natural map q : G/L → G/(L + L′) is the quotient by (L + L′)/L, a linear subgroup of the abelian variety G/L. It follows that (L + L′)/L is finite; thus, q is an isogeny and L′/L ∩ L′ is finite. Likewise, q′ : G/L′ → G/(L + L′) is an isogeny and L/L ∩ L′ is finite; this completes the proof of (i). Finally, the assertion (iii) is a consequence of [Br17, Thm. 4.3.4].

The structure of linear algebraic groups may be described as follows (see [DG70, Thm. IV.3.1.1]):

Theorem 2.4. Let G be a linear algebraic group. Then G has a largest subgroup of multiplicative type, M; moreover, G/M is unipotent. If k is perfect, then G = M × U, where U denotes the largest unipotent subgroup of G.

Also, note the following orthogonality relations:

Proposition 2.5. (i) Let M be a group of multiplicative type, and U a unipotent group. Then Hom_{C}(M, U) = 0 = Hom_{C}(U, M).

(ii) Let L be a linear algebraic group, and A an abelian variety. Then Hom_{C}(A, L) = 0, and every morphism L → A has finite image. Moreover, Hom_{C}(L, A) is n-torsion for some positive integer n.

Proof. (i) This follows from [DG70, Cor. IV.2.2.4]. (ii) The image of a morphism A → L is proper, smooth, connected and affine, hence zero. Likewise, the image of a morphism L → A is affine and proper, hence finite.

To show the final assertion, we may replace k with any field extension, and hence assume that k is perfect. Then the reduced neutral component L_{red}^{0} is a smooth connected subgroup of L, the quotient L/L_{red}^{0} is finite, and Hom_{C}(L_{red}^{0}, A) = 0 by the above argument. As a consequence, we have Hom_{C}(L, A) = Hom_{C}(L/L_{red}^{0}, A) and this group is n-torsion, where n denotes the order of the finite group scheme L/L_{red}^{0} (indeed, L/L_{red}^{0} is n-torsion in view of [SGA3, VIIA, Prop. 8.5]).

Next, we obtain a key preliminary result. To state it, recall that a unipotent group G is said to be split if it admits a finite increasing sequence of subgroups 0 = G_{0} ⊂ G_{1} ⊂ ⋯ ⊂ G_{n} = G such that G_{i}/G_{i−1} ≅ 𝔾_{a} for i = 1, …, n.

Proposition 2.6. Let G be an algebraic group.

(i) There exists a finite subgroup F ⊂ G such that G/F is smooth and connected.

(ii) If G is unipotent, then we may choose F such that G/F is split.

Proof. (i) By Lemma 2.2, we have G = G^{0} +F for some finite subgroup F ⊂ G. Thus, G/F ≅ G^{0}/F ∩ G^{0} is connected; this yields the assertion if char(k) = 0. If char(k) = p > 0, then we may assume G connected by the above step. Consider the relative Frobenius morphism F_{G/k} : G → G^{(p)} and its iterates F_{G/k}^{n} : G → G^{(p^{n})}, where n ≥ 1. Then Ker(F_{G/k}^{n}) is finite for all n; moreover, G/Ker(F_{G/k}^{n}) is smooth for n ≫ 0 (see [SGA3, VIIA, Prop. 8.3]), and still connected. (ii) We argue by induction on the dimension of G. The statement is obvious if dim(G) = 0. In the case where dim(G) = 1, we may assume that G is smooth and connected in view of Lemma 2.2 again; then G is a k-form of 𝔾_{a}. By [Ru70, Thm. 2.1], there exists an exact sequence

0 ⟶ G ⟶ 𝔾_{a}^{2} ⟶^{f} 𝔾_{a} ⟶ 0,

where f ∈ O(𝔾_{a}^{2}) ≅ k[x, y] satisfies f(x, y) = y^{p^{n}} − a_{0} x − a_{1} x^{p} − ⋯ − a_{m} x^{p^{m}} for some integers m, n ≥ 0 and some a_{0}, …, a_{m} ∈ k with a_{0} ≠ 0. Thus, the projection

p_{1} : G ⟶ 𝔾_{a}, (x, y) ⟼ x

lies in an exact sequence

0 ⟶ α_{p^{n}} ⟶ G ⟶^{p_{1}} 𝔾_{a} ⟶ 0,

where α_{p^{n}} denotes the kernel of the endomorphism x ↦ x^{p^{n}} of 𝔾_{a}. This yields the assertion in this case. If dim(G) ≥ 2, then we may choose a subgroup G_{1} ⊂ G such that 0 < dim(G_{1}) < dim(G) (as follows from [DG70, Prop. IV.2.2.5]). By the induction assumption for G/G_{1}, there exists a subgroup G_{2} ⊂ G such that G_{1} ⊂ G_{2}, G_{2}/G_{1} is finite, and G/G_{2} is split. Next, the induction assumption for G_{2} yields a finite subgroup F ⊂ G_{2} such that G_{2}/F is split. Then G/F is split as well.

Remark 2.7. By Proposition 2.6, every algebraic group G admits an isogeny u : G → H, where H is smooth and connected. If k is perfect, then there also exists an isogeny v : K → G, where K is smooth and connected: just take v to be the inclusion of the reduced neutral component G_{red}^{0}. But this fails over any imperfect field k. Indeed, if such an isogeny v exists, then its image must be G_{red}^{0}. On the other hand, by [SGA3, VIA, Ex. 1.3.2], there exists a connected algebraic group G such that G_{red} is not a subgroup.

By combining Lemma 2.2, Theorems 2.3 and 2.4, and Proposition 2.6, we obtain readily:

Proposition 2.8. Every algebraic group G admits a finite increasing sequence of subgroups 0 = G_{0} ⊂ G_{1} ⊂ ⋯ ⊂ G_{n} = G such that each subquotient G_{i}/G_{i−1}, i = 1, …, n, is finite or isomorphic to 𝔾_{a}, a simple torus, or a simple abelian variety. Moreover, G is linear if and only if no abelian variety occurs.

2.2 Characteristic zero

In this subsection, we assume that char(k) = 0. Recall that every unipotent group is isomorphic to the additive group of its Lie algebra via the exponential map; this yields an equivalence between the category U of unipotent groups and the category of finite-dimensional k-vector spaces (see [DG70, Prop. IV.2.4.2]). In particular, every unipotent group is connected. Next, consider a connected algebraic group G. By Theorem 2.3, there is a unique exact sequence 0 → L → G → A → 0, where A is an abelian variety, and L is connected and linear. Moreover, in view of Theorem 2.4, we have L = T × U, where T is a torus and U is unipotent. We now extend the latter structure results to possibly non-connected groups:

Theorem 2.9. (i) Every algebraic group G lies in an exact sequence

0 ⟶ M × U ⟶ G ⟶ A ⟶ 0,

where M is of multiplicative type, U is unipotent, and A is an abelian variety. Moreover, U is the largest unipotent subgroup of G: the unipotent radical, R_{u}(G). Also, M is unique up to commensurability in G, and A is unique up to isogeny.

(ii) The formation of the unipotent radical commutes with base change under field extensions, and yields an exact functor

R_{u} : C ⟶ U,

right adjoint to the inclusion U → C.

(iii) The projective objects of C are the unipotent groups.

Proof. (i) Theorem 2.3 yields an exact sequence

0 ⟶ L ⟶ G ⟶ A ⟶ 0,

where L is linear (possibly non-connected), and A is an abelian variety. By Theorem 2.4, we have L = M × U, where M is of multiplicative type and U is unipotent. Since M and A have no non-trivial unipotent subgroups, we have U = R_{u}(G). Given another exact sequence

0 ⟶ M′ × U ⟶ G ⟶ A′ ⟶ 0

satisfying the same assumptions, the image of M′ in A ≅ G/(M × U) is finite by Proposition 2.5. In other words, the quotient M′/M′ ∩ (M × U) is finite. Likewise, M/M ∩ (M′ × U) is finite as well. Since we have (M × U) ∩ M′ = M ∩ M′ = M ∩ (M′ × U), we see that M, M′ are commensurable in G. Then A = G/(M × U) and A′ = G/(M′ × U) are both quotients of G/(M ∩ M′) × U by finite subgroups, and hence are isogenous. (ii) In view of (i), G/R_{u}(G) is an extension of an abelian variety by a group of multiplicative type. Since these two classes of algebraic groups are stable under base change by any field extension k′ of k, it follows that (G/R_{u}(G))_{k′} has zero unipotent radical. Thus, R_{u}(G)_{k′} = R_{u}(G_{k′}). Next, note that every morphism f : G → H sends R_{u}(G) to R_{u}(H). Consider an exact sequence

0 ⟶ G_{1} ⟶^{f} G_{2} ⟶^{g} G_{3} ⟶ 0

and the induced complex

0 ⟶ R_{u}(G_{1}) ⟶^{R_{u}(f)} R_{u}(G_{2}) ⟶^{R_{u}(g)} R_{u}(G_{3}) ⟶ 0.

Clearly, R_{u}(f) is a monomorphism. Also, we have

Ker(R_{u}(g)) = R_{u}(G_{2}) ∩ Ker(g) = R_{u}(G_{2}) ∩ Im(f) = Im(R_{u}(f)).

We now show that R_{u}(g) is an epimorphism. For this, we may replace G_{2} with g^{−1}(R_{u}(G_{3})), and hence assume that G_{3} is unipotent. Next, we may replace G_{2} with G_{2}/R_{u}(G_{2}), and hence assume (in view of (i) again) that G_{2} is an extension of an abelian variety by a group of multiplicative type. Then Hom(G_{2}, G_{3}) = 0 by Proposition 2.5; this completes the proof of the exactness assertion. The assertion about adjointness follows from the fact that every morphism U → H, where U is unipotent and H arbitrary, factors through a unique morphism U → R_{u}(H). (iii) Consider an epimorphism φ : G → H, a unipotent group U, and a morphism ψ : U → H. Then ψ factors through R_{u}(H). Also, by (ii), φ restricts to an epimorphism R_{u}(G) → R_{u}(H), which admits a section as unipotent groups are just vector spaces. Thus, ψ lifts to a morphism U → G. This shows that U is projective in C. Conversely, let G be a projective object in C. We claim that the (abstract) group Hom_{C}(G, H) is divisible for any divisible algebraic group H. Indeed, the exact sequence

0 ⟶ H[n] ⟶ H ⟶^{n_{H}} H ⟶ 0

yields an exact sequence

0 ⟶ Hom_{C}(G, H[n]) ⟶ Hom_{C}(G, H) ⟶^{×n} Hom_{C}(G, H) ⟶ 0,

for any positive integer n. Next, the exact sequence 0 → L → G → A → 0 yields an exact sequence

0 ⟶ End_{C}(A) ⟶ Hom_{C}(G, A) ⟶ Hom_{C}(L, A),

where the abelian group End_{C}(A) is free of finite rank (see [Mi86, Thm. 12.5]), and Hom_{C}(L, A) is killed by some positive integer (Proposition 2.5). On the other hand, Hom_{C}(G, A) is divisible by the above claim. It follows that End_{C}(A) is zero, and hence so is A. Thus, G is linear, and hence G = M × U as above. Since U is projective, so is M. Choose a torus T containing M; then again, the group Hom_{C}(M, T) is finitely generated and divisible, hence zero. Thus, T = 0 and G = U.

Remark 2.10. With the notation of the above theorem, we have a natural map

G ⟶ G/M ×_{A} G/U,

which is a morphism of M × U-torsors over A, and hence an isomorphism. Moreover, G/M is an extension of an abelian variety by a unipotent group; such ‘vector extensions’ will be studied in detail in §5.1. Also, G/U is an extension of an abelian variety by a group of multiplicative type, and hence of a semi-abelian variety by a finite group. The semi-abelian varieties will be considered in §5.2.

2.3 Positive characteristics

In this subsection, we assume that char(k) = p > 0. Then the assertions of Theorem 2.9 are no longer valid. For example, the formation of the unipotent radical (the largest smooth connected unipotent subgroup) is not exact, and does not commute with arbitrary field extensions either (see Remark 2.12 (i) for details). Also, C has no non-zero projective objects, as will be shown in Corollary 5.15. Yet Theorem 2.9 has a useful analogue, in which the unipotent radical is replaced by the largest unipotent quotient:

Theorem 2.11. Let G be an algebraic group.

(i) G has a smallest subgroup H such that U := G/H is unipotent. Moreover, H is an extension of an abelian variety A by a group of multiplicative type M. Also, M is unique up to commensurability in G, and A is unique up to isogeny.

(ii) Every morphism H → U is zero; every morphism U → H has finite image.

(iii) The formation of U commutes with base change under field extensions,

and yields a functor

U : C ⟶ U,

which is left adjoint to the inclusion of U in C. Moreover, every exact sequence in C

0 ⟶ G_{1} ⟶^{f} G_{2} ⟶^{g} G_{3} ⟶ 0

yields a right exact sequence

0 ⟶ U(G_{1}) ⟶^{U(f)} U(G_{2}) ⟶^{U(g)} U(G_{3}) ⟶ 0,

where Ker(U(f)) is a finite p-group.

(iv) There exists a subgroup V ⊂ G such that G = H + V and H ∩ V is a finite p-group.

Proof. (i) Since the underlying topological space of G is noetherian, we may choose a subgroup H ⊂ G such that G/H is unipotent, and H is minimal for this property. Let H′ ⊂ G be another subgroup such that G/H′ is unipotent. Then so is G/H ∩ H′ in view of the exact sequence

0 ⟶ G/H ∩ H′ ⟶ G/H × G/H′.

By minimality of H, it follows that H ∩ H′ = H, i.e., H ⊂ H′. Thus, H is the smallest subgroup with unipotent quotient. Since the class of unipotent groups is stable under extensions, every unipotent quotient of H is zero. Also, by the affinization theorem (see [Br17, Thm. 1, Prop. 5.5.1]), H is an extension of a linear algebraic group L by a semi-abelian variety S. Then every unipotent quotient of L is zero, and hence L must be of multiplicative type in view of Theorem 2.4. By [DG70, Cor. IV.1.3.9], the reduced neutral component L_{red}^{0} is its maximal torus, T; the quotient L/T is a finite group of multiplicative type. Denote by S′ the preimage of T in H; then S′ is a semi-abelian variety (extension of T by S) and we have an exact sequence

0 ⟶ S′ ⟶ H ⟶ L/T ⟶ 0.

By Lemma 2.2, there exists a finite subgroup F ⊂ H such that H = S′ + F; equivalently, the quotient morphism H → L/T restricts to an epimorphism F → L/T. Also, by Theorem 2.4 again, F has a largest subgroup of multiplicative type, M_{F}, and the quotient F/M_{F} is unipotent. Since L/T is of multiplicative type, it follows that the composition M_{F} → F → L/T is an epimorphism as well. Thus, we may replace F with M_{F}, and assume that F is of multiplicative type. Let T′ be the maximal torus of the semi-abelian variety S′, and M := T′ + F. Then M is of multiplicative type; moreover, H/M is a quotient of S′/T′, and hence is an abelian variety. The uniqueness assertions may be checked as in the proof of Theorem 2.9. (ii) This follows readily from Proposition 2.5. (iii) The assertion on base change under field extensions follows from the stability of the classes of unipotent groups, abelian varieties, and groups of multiplicative type, under such base changes. The adjointness assertion may be checked as in the proof of Theorem 2.9 (ii). Next, consider an exact sequence as in the statement. Clearly, U(g) is an epimorphism. Also, Ker(U(g))/Im(U(f)) ≅ g^{−1}(H_{3})/(u(G_{1}) + H_{2}), where H_{i} denotes the kernel of the quotient map G_{i} → U(G_{i}). Thus, Ker(U(g))/Im(U(f)) is a quotient of g^{−1}(H_{3})/u(G_{1}) ≅ H_{3}. As Ker(U(g))/Im(U(f)) is unipotent, it is trivial by (ii). Finally, Ker(U(f)) ≅ (G_{1} ∩ u^{−1}(H_{2}))/H_{1} is isomorphic to a subgroup of H_{2}/u(H_{1}). Moreover, H_{2}/u(H_{1}) is an extension of an abelian variety by a group of multiplicative type. Since Ker(U(f)) is unipotent, it is finite by (ii) again. (iv) If G is linear, then H = M is the largest subgroup scheme of multiplicative type of G. By [Br17, Thm. 5.3.1], there exists a subgroup scheme W ⊂ G such that G = H + W and H ∩ W is finite. Equivalently, the composition W → G → U is an epimorphism and its kernel is finite. As a consequence, W is n-torsion for some n ≥ 1, and hence W = V + F where V is a p-group and F is finite of order prime to p. Thus, G/(H + V) is a quotient of U and a quotient of F, hence trivial. So G = H + V, where H ∩ V is a finite p-group. For an arbitrary algebraic group G, Theorem 2.3 yields an exact sequence

0 ⟶ L ⟶ G ⟶ A ⟶ 0,

where L is linear and A is an abelian variety. In view of (iii) and the vanishing of U(A), we obtain an exact sequence

0 ⟶ F ⟶ U(L) ⟶ U(G) ⟶ 0,

where F is a finite p-group. By the preceding step, there exists a subgroup V ⊂ L such that the composition V → L → U(L) is an epimorphism with kernel a finite p-group. Thus, so is the composition V → G → U(G).

Remarks 2.12. (i) The formation of the unipotent radical does not commute with purely inseparable field extensions, in view of [SGA3, XVII.C.5]. This formation is not exact either, as seen e.g. from the exact sequence

0 ⟶ α_{p} ⟶ 𝔾_{a} ⟶^{F} 𝔾_{a} ⟶ 0,

where F denotes the relative Frobenius endomorphism. When k is perfect, one may show that every exact sequence

0 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 0

in C yields a complex

0 ⟶ R_{u}(G_{1}) ⟶ R_{u}(G_{2}) ⟶ R_{u}(G_{3}) ⟶ 0

with finite homology groups. But this fails when k is imperfect; more specifically, choose a finite purely inseparable field extension K of k of degree p, and consider G := R_{K/k}(𝔾_{m,K}), where R_{K/k} denotes the Weil restriction. By [CGP15, Prop. A.5.11], G is smooth, connected, and lies in an exact sequence

0 ⟶ 𝔾_{m} ⟶ G ⟶ U ⟶ 0,

where U is unipotent of dimension p − 1. Moreover, every morphism from a smooth connected unipotent group to G is constant, as follows from the adjointness property of the Weil restriction (see [CGP15, (A.5.1)]). In other terms, R_{u}(G) = 0. (ii) The functor U of Theorem 2.11 is not left exact, as seen from the exact sequence

0 ⟶ α_{p} ⟶ E ⟶^{F} E^{(p)} ⟶ 0,

where E denotes a supersingular elliptic curve, and F its relative Frobenius morphism. Also, note that the torsion subgroups E[p^{n}], where n ≥ 1, form a strictly increasing sequence of infinitesimal unipotent groups; in particular, E has no largest connected unipotent subgroup.

Corollary 2.13. Let G be an algebraic group.

(i) There exists a finite subgroup F ⊂ G such that G/F ≅ S × U, where S is a semi-abelian variety, and U a split unipotent group. Moreover, S and U are unique up to isogeny.

(ii) If k is locally finite, then we may choose F so that S ≅ T × A, where T is a torus, and A an abelian variety. Moreover, T and A are unique up to isogeny.

Proof. (i) With the notation of Theorem 2.11, we have isomorphisms

G/H ∩ V ≅ G/V × G/H ≅ (H/H ∩ V) × U.

Also, H/H ∩ V is an extension of an abelian variety, H/(H ∩ V) + M, by a group of multiplicative type, M/M ∩ V. Moreover, U is an extension of a split unipotent group by a finite group (Proposition 2.6). Thus, we may assume that G = H. Then G_{red}^{0} is a semi-abelian variety, as follows from [Br17, Lem. 5.6.1]. Since G/G_{red}^{0} is finite, applying Lemma 2.2 yields that G is an extension of a semi-abelian variety by a finite group.

(ii) By [Br17, Cor. 5.5.5], there exists an abelian subvariety A ⊂ S such that S = T + A, where T ⊂ S denotes the maximal torus. Then T ∩ A is finite, and S/T ∩ A ≅ T/T ∩ A × A/T ∩ A. This completes the proof of the existence assertions in (i) and (ii). The uniqueness up to isogeny follows from Proposition 2.5.

3 The isogeny category of algebraic groups

3.1 Definition and first properties

Recall that C denotes the category of commutative algebraic groups, and F the full subcategory of finite groups. Since F is stable under taking subobjects, quotients and extensions, we may form the quotient category C/F; it has the same objects as C, and its morphisms are defined by

Hom_{C/F}(G, H) = lim_{→} Hom_{C}(G′, H/H′),

where the direct limit is taken over all subgroups G′ ⊂ G such that G/G′ is finite, and all finite subgroups H′ ⊂ H. The category C/F is abelian, and comes with an exact functor

Q : C ⟶ C/F,

which is the identity on objects and the natural map

Hom_{C}(G, H) ⟶ lim_{→} Hom_{C}(G′, H/H′), f ⟼ \underline{f}

on morphisms. The quotient functor Q satisfies the following universal property: given an exact functor R : C → D, where D is an abelian category, such that R(F) = 0 for any finite group F, there exists a unique exact functor S : C/F → D such that R = S ∘ Q (see [Ga62, Cor. III.1.2, Cor. III.1.3] for these results). Alternatively, C/F may be viewed as the localization of C at the multiplicative system of isogenies (see [GZ67, §I.2] or [SP16, §4.26] for localization of categories); this is easily checked by arguing as in the proof of [SP16, Lem. 12.9.6]. We now show that C/F is equivalent to a category with somewhat simpler objects and morphisms:

Lemma 3.1. Let \underline{C} be the full subcategory of C/F with objects the smooth connected algebraic groups.

(i) The inclusion of \underline{C} in C/F is an equivalence of categories.

(ii) Hom\underline{C}(G, H) = lim Hom_{C}(G, H/H′), where the direct limit is taken over all finite subgroups H′ ⊂ H.

(iii) Let \underline{f} ∈ Hom\underline{C}(G, H) be represented by a morphism f : G → H/H′ in C. Then \underline{f} is zero (resp. a monomorphism, an epimorphism, an isomorphism) if and only if f is zero (resp. has a finite kernel, is an epimorphism, is an isogeny).

Proof. (i) This follows from Proposition 2.6. (ii) This follows from Lemma 2.1. (iii) By [Ga62, Lem. III.1.2], \underline{f} is zero (resp. a monomorphism, an epimorphism) if and only if Im(f) (resp. Ker(f), Coker(f)) is finite. By Lemma 2.1 again, the finiteness of Im(f) is equivalent to f = 0, and the finiteness of Coker(f) is equivalent to f being an epimorphism. As a consequence, \underline{f} is an isomorphism if and only if f is an isogeny.

The abelian category \underline{C} will be called the isogeny category of (commutative) algebraic groups. Every exact functor R : C → D, where D is an abelian category and R(f) is an isomorphism for any isogeny f, factors uniquely through \underline{C} (indeed, R must send any finite group to zero).

We may now prove the assertion (i) of the main theorem:

Proposition 3.2. (i) The category \underline{C} is noetherian and artinian.

(ii) The simple objects of \underline{C} are 𝔾_{a}, the simple tori, and the simple abelian varieties.

Proof. (i) Let G be a smooth connected algebraic group, and (G_{n})_{n≥0} an increasing sequence of subobjects of G in \underline{C}, i.e., each G_{n} is smooth, connected, and equipped with a C-morphism

φ_{n} : G_{n} ⟶ G/G′_{n},

where Ker(φ_{n}) and G′_{n} are finite; moreover, we have C-morphisms

ψ_{n} : G_{n} ⟶ G_{n+1}/G′′_{n+1},

where Ker(ψ_{n}) and G′′_{n+1} are finite. Thus, dim(G_{n}) ≤ dim(G_{n+1}) ≤ dim(G). It follows that dim(G_{n}) = dim(G_{n+1}) for n ≫ 0, and hence ψ_{n} is an isogeny. So G_{n} ≅ G_{n+1} in \underline{C} for n ≫ 0. This shows that \underline{C} is noetherian. One may check likewise that \underline{C} is artinian. (ii) This follows from Proposition 2.8.

Next, we relate the short exact sequences in C with those in \underline{C}:

Lemma 3.3. Consider a short exact sequence in C,

ξ : 0 ⟶ G_{1} ⟶^{u} G_{2} ⟶^{v} G_{3} ⟶ 0,

where G_{1}, G_{2}, G_{3} are smooth and connected. Then ξ splits in \underline{C} if and only if the push-out f_{∗}ξ splits in C for some epimorphism with finite kernel f : G_{1} → H.

Proof. The exact sequence ξ splits in \underline{C} if and only if there exists a \underline{C}-morphism \underline{g} : G_{2} → G_{1} such that \underline{g} ∘ u = id in \underline{C}. Equivalently, there exists a finite subgroup G′_{1} ⊂ G_{1} and a C-morphism g : G_{2} → G_{1}/G′_{1} such that g ∘ u is the quotient map f_{1} : G_{1} → G_{1}/G′_{1}. If such a pair (G′_{1}, g) exists, then g factors through a C-morphism G_{2}/u(G′_{1}) → G_{1}/G′_{1}, which splits the bottom exact sequence in the push-out diagram

0 ⟶ G_{1}  ⟶^{u}  G_{2}  ⟶^{v} G_{3} ⟶ 0
f_{1}↓  ^{f_{2}}↓  ^{id}↓
0 ⟶ G_{1}/G′_{1} ⟶^{u′} G_{2}/u(G′_{1}) ⟶^{v′} G_{3} ⟶ 0.

Replacing G′_{1} by a larger finite subgroup, we may assume that G_{1}/G′_{1} is smooth and connected (Lemma 2.2). Conversely, a splitting of the bottom exact sequence in the above diagram is given by a C-morphism g′ : G_{2}/u(G′_{1}) → G_{1}/G′_{1} such that g′ ∘ u′ = id in C. Let

g : G_{2} → G_{1}/G′_{1} denote the composition G_{2} ⟶^{f_{2}} G_{2}/u(G′_{1}) ⟶^{g′} G_{1}/G′_{1}. Then

g ∘ u = g′ ∘ f_{2} ∘ u = g′ ∘ u′ ∘ f_{1} = f_{1} as desired.

We may now construct non-split exact sequences in \underline{C}, thereby showing that hd(\underline{C}) ≥ 1:

Examples 3.4. (i) Consider an exact sequence

ξ : 0 ⟶ 𝔾_{a} ⟶ G ⟶ A ⟶ 0,

where A is an abelian variety. Then ξ, viewed as an extension of A by 𝔾_{a} in C, is classified by an element η ∈ H^{1}(A, O_{A}) (see [Ro58] or [MM74, §1.9]). If char(k) = 0, then every epimorphism with finite kernel f : 𝔾_{a} → H may be identified with the multiplication by some t ∈ k^{∗}, viewed as an endomorphism of 𝔾_{a}; then the push-out f_{∗}ξ is classified by tη. In view of Lemma 3.3, it follows that ξ is non-split in \underline{C} whenever η ≠ 0. In contrast, if char(k) = p > 0, then ξ splits in \underline{C}. Indeed, the multiplication map p_{A} yields an isomorphism in \underline{C}, and p_{𝔾_{a}} = 0 whereas (p_{𝔾_{a}})_{∗}ξ = p_{A}^{∗}ξ. (ii) Assume that char(k) = p > 0 and consider the algebraic group W_{2} of Witt vectors of length 2. This group comes with an exact sequence

ξ : 0 ⟶ 𝔾_{a} ⟶ W_{2} ⟶ 𝔾_{a} ⟶ 0,

see e.g. [DG70, §V.1.1.6]. Every epimorphism with finite kernel f : 𝔾_{a} → H may be identified with a non-zero endomorphism of 𝔾_{a}. In view of [DG70, Cor. V.1.5.2], it follows that the push-forward f_{∗}ξ is non-split. Thus, ξ does not split in \underline{C}.

Proposition 3.5. Consider an exact sequence

0 ⟶ G_{1} ⟶^{\underline{u}_{1}} G_{2} ⟶^{\underline{u}_{2}} ⋯ ⟶^{\underline{u}_{n−1}} G_{n} ⟶ 0

in \underline{C}. Then there exists an exact sequence

0 ⟶ H_{1} ⟶^{v_{1}} H_{2} ⟶^{v_{2}} ⋯ ⟶^{v_{n−1}} H_{n} ⟶ 0

in C, and epimorphisms with finite kernels f_{i} : G_{i} → H_{i} (i = 1, 2, …, n), such that the diagram

0 ⟶ G_{1} ⟶^{\underline{u}_{1}} G_{2} ⟶^{\underline{u}_{2}} ⋯ ⟶^{\underline{u}_{n−1}} G_{n} ⟶ 0
\underline{f}_{1}↓  \underline{f}_{2}↓  \underline{f}_{n}↓
0 ⟶ H_{1} ⟶^{\underline{v}_{1}} H_{2} ⟶^{\underline{v}_{2}} ⋯ ⟶^{\underline{v}_{n−1}} H_{n} ⟶ 0

commutes in \underline{C}.

Proof. We argue by induction on the length n. If n = 2, then we just have an isomorphism \underline{u} : G_{1} → G_{2} in \underline{C}. Then \underline{u} is represented by an isomorphism u : G_{1}/G′_{1} → G_{2}/G′_{2} in C, for some finite groups G′_{1} ⊂ G_{1} and G′_{2} ⊂ G_{2}. If n = 3, then the \underline{C}-morphism \underline{u}_{2} : G_{2} → G_{3} is represented by an epimorphism u_{2} : G_{2} → G_{3}/G′_{3} in C, where G′_{3} is a finite subgroup of G_{3}. We may thus replace G_{3} with G_{3}/G′_{3}, and assume that u_{2} is an epimorphism in C. Next, \underline{u}_{1} : G_{1} → G_{2} is represented by a morphism u_{1} : G_{1} → G_{2}/G′_{2} with finite kernel, where G′_{2} is a finite subgroup of G_{2}. We may thus replace G_{1} (resp. G_{2}, G_{3}) with G_{1}/Ker(u_{1}) (resp. G_{2}/G′_{2}, G_{3}/u_{2}(G′_{2})) and assume that u_{1} is a monomorphism in C. Then u_{2} ∘ u_{1} has finite image, and hence is zero since G_{1} is smooth and connected. We now have a complex in C

0 ⟶ G_{1} ⟶^{u_{1}} G_{2} ⟶^{u_{2}} G_{3} ⟶ 0,

where u_{1} is a monomorphism, u_{2} an epimorphism, and Ker(u_{2})/Im(u_{1}) is finite. By Lemma 2.2, we may choose a finite subgroup F ⊂ Ker(u_{2}) such that Ker(u_{2}) = Im(u_{1}) + F. This yields a commutative diagram in C

0 ⟶  G_{1}  ⟶^{u_{1}} G_{2} ⟶^{u_{2}} G_{3} ⟶ 0
q_{1}↓  ^{q_{2}}↓  ^{id}↓
0 ⟶ G_{1}/u_{1}^{−1}(F) ⟶^{v_{1}} G_{2}/F ⟶^{v_{2}} G_{3} ⟶ 0,

where q_{1}, q_{2} denote the quotient maps. Clearly, v_{1} is a monomorphism, and v_{2} an epimorphism. Also, v_{2} ∘ v_{1} = 0, since q_{1} is an epimorphism. Finally, q_{2} restricts to an epimorphism Ker(u_{2}) → Ker(v_{2}), and hence Ker(v_{2}) = Im(v_{1}). This completes the proof in the case where n = 3. For an arbitrary length n ≥ 4, we cut the given exact sequence into two exact sequences in \underline{C}

0 ⟶ G_{1} ⟶^{\underline{u}_{1}} G_{2} ⟶ K ⟶ 0,

0 ⟶ K ⟶ G_{3} ⟶^{\underline{u}_{3}} ⋯ ⟶^{\underline{u}_{n−1}} G_{n} ⟶ 0.

By the induction assumption, there exists a commutative diagram in \underline{C}

0 ⟶ K ⟶ G_{3} ⟶^{\underline{u}_{3}} ⋯ ⟶^{\underline{u}_{n−1}} G_{n} ⟶ 0
\underline{f}↓  \underline{f}_{3}↓  \underline{f}_{n}↓
0 ⟶ L ⟶^{\underline{v}} H_{3} ⟶^{\underline{v}_{3}} ⋯ ⟶^{\underline{v}_{n−1}} H_{n} ⟶ 0,

where f, f_{3}, …, f_{n} are epimorphisms with finite kernels, and the bottom sequence comes from an exact sequence in C. Since \underline{f} is an isomorphism in \underline{C}, we have an exact sequence

0 ⟶ G_{1} ⟶^{\underline{u}_{1}} G_{2} ⟶ L ⟶ 0

in \underline{C}, and hence another commutative diagram in \underline{C}

0 ⟶ G_{1} ⟶^{\underline{u}_{1}} G_{2} ⟶ L ⟶ 0
\underline{f}_{1}↓  \underline{f}_{2}↓  \underline{g}↓
0 ⟶ H_{1} ⟶^{\underline{v}_{1}} H_{2} ⟶ M ⟶ 0,

where again f_{1}, f_{2}, g are epimorphisms with finite kernels, and the bottom sequence comes from an exact sequence in C. Let H′_{3} := H_{3}/v(F), and denote by F the kernel of g : L → M; then we have a commutative diagram in \underline{C}

0 ⟶ L ⟶ G_{3} ⟶^{\underline{u}_{3}} G_{4} ⟶^{\underline{u}_{4}} ⋯ ⟶^{\underline{u}_{n−1}} G_{n} ⟶ 0
\underline{g}↓  \underline{h}↓  \underline{f}_{4}↓  \underline{f}_{n}↓
0 ⟶ M ⟶ H′_{3} ⟶^{\underline{w}} H_{4} ⟶^{\underline{v}_{4}} ⋯ ⟶^{\underline{v}_{n−1}} H_{n} ⟶ 0,

satisfying similar properties. By concatenating the two latter diagrams, we obtain the desired commutative diagram

0 ⟶ G_{1} ⟶^{\underline{u}_{1}} G_{2} ⟶^{\underline{u}_{2}} G_{3} ⟶^{\underline{u}_{3}} ⋯ ⟶^{\underline{u}_{n−1}} G_{n} ⟶ 0
↓  ↓  ↓  ↓
0 ⟶ H_{1} ⟶^{\underline{v}_{1}} H_{2} ⟶ H′_{3} ⟶^{\underline{w}} ⋯ ⟶^{\underline{v}_{n−1}} H_{n} ⟶ 0.

3.2 Divisible groups

Given a divisible algebraic group G and a non-zero integer n, the morphism n_{G} : G → G factors through an isomorphism G/G[n] ⟶^{≅} G. We denote the inverse isomorphism by

u_{n} : G ⟶^{≅} G/G[n].

By construction, we have a commutative triangle

\xymatrix{G \ar[d]_{n_{G}} \ar[dr]^{q} \\ G \ar[r]^{u_{n}} & G/G[n],}

where q denotes the quotient morphism. Since q yields the identity morphism in \underline{C}, we see that u_{n} yields the inverse of the \underline{C}-automorphism n_{G} of G. As a consequence, End\underline{C}(G) is a ℚ-algebra. More generally, we have the following:

Proposition 3.6. Let G, H be smooth connected algebraic groups, and assume that H is divisible.

(i) Every extension group Extn ; \underline{C}(G, H) is a ℚ-vector space.

(ii) The natural map Q : Hom_{C}(G, H) → Hom\underline{C}(G, H) is injective and induces an isomorphism

γ : ℚ ⊗_{ℤ} Hom_{C}(G, H) ⟶ Hom\underline{C}(G, H), \frac{1}{n} ⊗ f ⟼ Q(u_{n} ⊗ f).

(iii) If G is divisible as well, then the natural map

Q^{1} : Ext_{C}^{1}(G, H) → Ext1 ; \underline{C}(G, H)

induces an isomorphism

γ^{1} : ℚ ⊗_{ℤ} Ext_{C}^{1}(G, H) ⟶ Ext1 ; \underline{C}(G, H).

Proof. (i) Just note that Extn ; \underline{C}(G, H) is a module over the ℚ-algebra End\underline{C}(H). (ii) Let f ∈ Hom_{C}(G, H), and n a positive integer. If γ(\frac{1}{n} ⊗ f) = 0, then γ(f) = 0, i.e., \underline{f} = 0. Thus, f = 0 in view of Lemma 3.1; this shows the injectivity of γ. For the surjectivity, consider a \underline{C}-morphism \underline{f} : G → H represented by a Cmorphism f : G → H/H′, where H′ is a finite subgroup of H. Then H′ ⊂ H[n] for some positive integer n, which we may take to be the order of H′. Thus, we may assume that H′ = H[n]. Then the C-morphism φ := u_{n}^{−1} ∘ f : G → H satisfies \underline{φ} = n\underline{f}, i.e., \underline{f} = γ(\frac{1}{n} ⊗ φ). (iii) Consider η ∈ Ext_{C}^{1}(G, H) such that γ^{1}(\frac{1}{n} ⊗η) = 0 for some positive integer n. Then of course γ^{1}(η) = 0, i.e., η is represented by an exact sequence in C

0 ⟶ H ⟶^{u} E ⟶^{v} G ⟶ 0,

which splits in \underline{C}. By Lemma 3.3 and the divisibility of H, it follows that the push-out by m_{H} of the above extension splits in C for some m > 0. But (m_{H})_{∗}η = mη (see e.g. [Oo66, Lem. I.3.1]), and hence mη = 0. This shows the injectivity of γ^{1}. For the surjectivity, we adapt the proof of Proposition 3.5. Let η ∈ Ext1 ; \underline{C}(G, H) be represented by an exact sequence in \underline{C}

0 ⟶ H ⟶^{\underline{u}} E ⟶^{\underline{v}} G ⟶ 0.

Since G is divisible, \underline{v} is represented by a C-morphism v : E → G/G[m] for some positive integer m. Replacing η with its pull-back u_{m}^{∗}η = (m_{G}^{∗})^{−1}η = γ^{1}(\frac{1}{m}⊗η), we may thus assume that \underline{v} is represented by a C-epimorphism v : E → G. Likewise, as H is divisible, \underline{u} is represented by a C-morphism u : H → E/E[n]. Then η is represented by the exact sequence in \underline{C}

0 ⟶ H ⟶^{\underline{u}} E/E[n] ⟶^{\underline{v}_{n}} G ⟶ 0,

where v_{n} : E/E[n] → G/G[n] is the C-epimorphism induced by v. So we may further assume that u is represented by a C-morphism u : H → E. By Lemma 3.1, we then have v ∘ u = 0; moreover, Ker(u) and Ker(v)/Im(u) are finite. In view of Lemma 2.2, we have Ker(v) = Im(u) + E′ for some finite subgroup E′ ⊂ E. This yields a commutative diagram in C

0 ⟶  H  ⟶^{u} E ⟶^{v} G ⟶ 0
↓  ↓  ^{id}↓
0 ⟶ H/u^{−1}(E′) ⟶^{u_{1}} E/E′ ⟶^{v_{1}} G ⟶ 0,

where the bottom sequence is exact, and u^{−1}(E′) is finite. We may thus choose a positive integer r such that u^{−1}(E′) ⊂ H[r]. Taking the push-out by the quotient map H/u^{−1}(E′) → H/H[r] yields a commutative diagram in C

0 ⟶ H ⟶^{u}  E  ⟶^{v} G ⟶ 0
↓  ↓  ^{id}↓
0 ⟶ H/H[r] ⟶^{u_{2}} E/E′ + u(H[r]) ⟶^{v_{2}} G ⟶ 0,

where the bottom sequence is exact again. Thus, rη = (r_{H})_{∗}η is represented by an exact sequence in C.

Remarks 3.7. (i) Given two divisible groups G, H, the map

Q^{1} : Ext_{C}^{1}(G, H) ⟶ Ext1 ; \underline{C}(G, H)

is not necessarily injective. Indeed, the group Ext_{C}^{1}(A, 𝔾_{m}) has non-zero torsion for any non-zero abelian variety A over (say) a separably closed field, as follows from the Weil-Barsotti isomorphism.

(ii) We may also consider the natural maps

γ^{n} : ℚ ⊗_{ℤ} Ext_{C}^{n}(G, H) ⟶ Extn ; \underline{C}(G, H)

for n ≥ 2. But these maps turn out to be zero for any algebraic groups G, H, since Extn ; \underline{C}(G, H) = 0 (Lemma 4.11) and Ext_{C}^{n}(G, H) is torsion (Remark 4.12).

As a first application of Proposition 3.6, we obtain:

Proposition 3.8. Assume that char(k) = 0.

(i) The composition of the inclusion U → C with the quotient Q : C → \underline{C} identifies U with a full subcategory of \underline{C}.

(ii) The unipotent radical functor yields an exact functor

\underline{R}_{u} : \underline{C} ⟶ U,

which is right adjoint to the inclusion. Moreover, \underline{R}_{u} commutes with base change under field extensions.

(iii) Every unipotent group is a projective object in \underline{C}.

Proof. (i) Recall that a morphism of unipotent groups is just a linear map of the associated k-vector spaces. In view of Proposition 3.6, it follows that the natural map Hom_{C}(U, V) → Hom\underline{C}(U, V) is an isomorphism for any unipotent groups U, V. (ii) The functor R_{u} : C → U is exact by Theorem 2.9, and sends every finite group to 0. By the universal property of Q, there exists a unique exact functor S : C/F → U such that R_{u} = S ∘ Q. Since R_{u} commutes with base change under field extensions (Theorem 2.9 again), so does S by uniqueness. Thus, composing S with the inclusion \underline{C} → C/F yields the desired exact functor. For any unipotent group U and any algebraic group G, the natural map

Hom_{U}(U, R_{u}(G)) ⟶ Hom_{C}(U, G)

is an isomorphism. By Proposition 3.6 again, the natural map

Q : Hom_{C}(U, G) ⟶ Hom\underline{C}(U, G)

is an isomorphism as well. It follows that \underline{R}_{u} is right adjoint to the inclusion. (iii) Let U be a unipotent group. Then the functor on C defined by

G ⟼ Hom_{U}(U, R_{u}(G))

is exact, since the unipotent radical functor is exact and the category U is semisimple. Thus, G ↦ Hom\underline{C}(U, G) is exact as well; this yields the assertion.

3.3 Field extensions

Let k′ be a field extension of k. Then the assignment G ↦ G_{k′} := G ⊗_{k} k′ yields the base change functor

⊗_{k}k′ : C_{k} ⟶ C_{k′}.

Clearly, this functor is exact and faithful; also, note that an algebraic group G is connected (resp. smooth, finite, linear, unipotent, a torus, an abelian variety, a semi-abelian variety) if and only if so is G_{k′}.

Lemma 3.9. With the above notation, the functor ⊗_{k}k′ yields an exact functor

\underline{⊗}_{k}k′ : \underline{C}_{k} ⟶ \underline{C}_{k′}.

Proof. As the composite functor Q_{k′} ∘⊗_{k}k′ : C_{k} → \underline{C}_{k′} is exact and sends every finite k-group to 0, it factors through a unique exact functor C_{k}/F_{k} → \underline{C}_{k′}. This yields the existence and exactness of \underline{⊗}_{k}k′.

Lemma 3.10. Let k′ be a purely inseparable field extension of k, and G′ a k′-group.

(i) There exists a smooth k-group G and an epimorphism f : G′ → G_{k′} such that Ker(f) is infinitesimal.

(ii) If G′ ⊂ H_{k′} for some k-group H, then there exists a k-subgroup G ⊂ H such that G′ ⊂ G_{k′} and G_{k′}/G′ is infinitesimal.

Proof. (i) Let n be a positive integer and consider the nth relative Frobenius morphism

F_{G′/k′}^{n} : G′ ⟶ G^{′(p^{n})}.

Recall that the quotient G′/Ker(F_{G′/k′}^{n}) is smooth for n ≫ 0. Since Ker(F_{G′/k′}^{n}) is infinitesimal, we may assume that G′ is smooth. Then F_{G′/k′}^{n} is an epimorphism in view of [SGA3, VIIA, Cor. 8.3.1]. Next, note that G′ is defined over some finite subextension k′′ of k′, i.e., there exists a k′′-subgroup G′′ such that G′ = G′′ ⊗_{k′′} k′. By transitivity of base change, we may thus assume that k′ is finite over k. Let q := [k′ : k], then q = p^{n}, where p = char(k) and n is a positive integer; also, k′^{q} ⊂ k. Consider again the morphism F_{G′/k′}^{n}; then by construction, G^{′(p^{n})} ≅ G′ ⊗_{k′} k′, where k′ is sent to itself via the qth power map. Thus, G^{′(p^{n})} ≅ G_{k′}, where G denotes the k-group G′ ⊗_{k′} k; here k′ is sent to k via the qth power map again. So the induced map G′ → G_{k′} is the desired morphism. (ii) As above, we may reduce to the case where k′ is finite over k. Then the statement follows by similar arguments, see [Br17, Lem. 4.3.5] for details.

We now are ready to prove Theorem 1 (iv):

Theorem 3.11. Let k′ be a purely inseparable field extension of k. Then the base change functor \underline{⊗}_{k}k′ : \underline{C}_{k} → \underline{C}_{k′} is an equivalence of categories.

Proof. By Lemma 3.10, every k′-group G′ is isogenous to G_{k′} for some smooth k-group G. It follows that \underline{⊗}_{k}k′ is essentially surjective. Next, let G, H be smooth connected k-groups, and \underline{f} : G → H a \underline{C}_{k}-morphism, represented by a C_{k}-morphism f : G → H/H′ for some finite k-subgroup H′ of H. If \underline{f}_{k′} : G_{k′} → H_{k′} is zero in \underline{C}_{k′}, then the image of f_{k′} : G_{k′} → H_{k′}/H′_{k′} is finite. By Lemma 2.1, it follows that f_{k′} = 0. This shows that \underline{⊗}_{k}k′ is faithful. We now check that \underline{⊗}_{k}k′ is full. Let again G, H be smooth connected k-groups, and let \underline{f} ∈ Hom_{\underline{C}_{k′}} (G_{k′}, H_{k′}). We show that there exists a finite k-subgroup H′ ⊂ H and a k-morphism φ : G → H/H′ such that φ_{k′} represents \underline{f}. For this, we may replace H with its quotient by any finite k-subgroup. Choose a representative f : G_{k′} → H_{k′}/H′′ of \underline{f}, where H′′ ⊂ H_{k′} is a finite k′-subgroup. By Lemma 3.10, there exists a k-subgroup I ⊂ H such that H′′ ⊂ I_{k′} and I_{k′}/H′′ is finite; then I_{k′} is finite as well, and hence so is I. We may thus replace H by H/I, and f by its composition with the quotient morphism H_{k′}/H′′ → H_{k′}/I_{k′} = (H/I)_{k′}. Then \underline{f} is represented by a morphism f : G_{k′} → H_{k′}. Consider the graph Γ(f) ⊂ G_{k′} ×_{k′} H_{k′}. By Lemma 3.10 again, there exists a k-subgroup Δ ⊂ G × H such that Γ(f) ⊂ Δ_{k′} and Δ_{k′}/Γ(f) is finite. Then the intersection Δ_{k′} ∩ (0 × H_{k′}) is finite, since Γ(f) ∩ (0 × H_{k′}) is zero. Thus, Δ ∩ (0 × H) is finite as well; equivalently, the k-group H′ := H ∩ (0 × id)^{−1}(Δ) is finite. Denoting by Γ the image of Δ in G × H/H′, we have a cartesian square

Δ ⟶ G × H
↓  ↓
Γ ⟶ G × H/H′,

where the horizontal arrows are closed immersions, and the left (resp. right) vertical arrow is the quotient by Δ ∩ (0 × H) (resp. by H′ acting on H via addition). So Γ is a k-subgroup of G × H/H′, and Γ ∩ (0 × H/H′) is zero; in other words, the projection π : Γ → G is a closed immersion. Since G is smooth and connected, and dim(Γ) = dim(Δ) = dim Γ(f) = dim(G), it follows that π is an isomorphism. In other words, Γ is the graph of a k-morphism φ : G → H/H′. Since the above cartesian square lies in a push-out diagram,

0 ⟶ Δ ⟶ G × H ⟶^{φ−q} H/H′ ⟶ 0
↓  ↓  ^{id}↓
0 ⟶ Γ ⟶ G × H/H′ ⟶^{φ−id} H/H′ ⟶ 0,

where q : H → H/H′ denotes the quotient morphism, it follows that Δ = ker(φ − q). As Γ(f) ⊂ Δ_{k′}, we see that φ_{k′} = q_{k′} ∘ f. Thus, \underline{f} is represented by φ_{k′}; this completes the proof of the fullness assertion.

Remarks 3.12. (i) Likewise, the base change functor induces equivalences of categories \underline{U}_{k} → \underline{U}_{k′}, \underline{T}_{k} → \underline{T}_{k′}, \underline{L}_{k} → \underline{L}_{k′}, and \underline{A}_{k} → \underline{A}_{k′}. For tori, this follows much more directly from the anti-equivalence of \underline{T}_{k} with the category of rational representations of the absolute Galois group, see Proposition 4.1. (ii) In particular, the category \underline{U}_{k} is equivalent to \underline{U}_{k_{i}}, where k_{i} denotes the perfect closure of k in k̄. Recall from [DG70, Thm. V.1.4.3, Cor. V.1.4.4] that the category U_{k_{i}} is anti-equivalent to the category of finitely generated modules over the Dieudonné ring 𝔻 = 𝔻_{k_{i}} which are killed by some power of the Verschiebung map V. Moreover, the category \underline{U}_{k_{i}} is anti-equivalent to the category of finitely generated modules over the localization 𝔻_{(V)} which are killed by some power of V; see [DG70, §V.3.6.7]. By work of Schoeller, Kraft, and Takeuchi (see [Sc72, Kr75, Tak75]), the category U_{k} is anti-equivalent to a category of finitely generated modules over a certain k-algebra, which generalizes the Dieudonné ring but seems much less tractable.

4 Tori, abelian varieties, and homological dimension

4.1 Tori

Denote by T (resp. M, FM) the full subcategory of C with objects the tori (resp. the groups of multiplicative type, the finite groups of multiplicative type). Then T is stable under taking quotients and extensions, but not subobjets; in particular, T is an additive subcategory of C, but not an abelian subcategory. Also, F and FM are stable under taking subobjects, quotients and extensions. Thus, we may form the quotient abelian category M/FM, as in §3.1. One may readily check that M/FM is a full subcategory of C/F. Let \underline{T} be the full subcategory of M/FM with objects the tori. Since these are the smooth connected objects of M, one may check as in Lemma 3.1 that the inclusion of \underline{T} in M/FM is an equivalence of categories. The remaining statements of Lemma 3.1 also adapt to this setting; note that we may replace the direct limits over all finite subgroups with those over all n-torsion subgroups, since tori are divisible. Also, Proposition 3.6 yields natural isomorphisms

ℚ ⊗_{ℤ} Hom_{C}(T_{1}, T_{2}) ⟶^{≅} Hom\underline{T} (T_{1}, T_{2})

for any tori T_{1}, T_{2}. By assigning with each group of multiplicative type G its character group,

X(G) := Hom_{k_{s}}(G_{k_{s}}, 𝔾_{m,k_{s}}),

one obtains an anti-equivalence between M (resp. FM) and the category of finitely generated (resp. finite) abstract commutative groups equipped with the discrete topology and a continuous action of the Galois group Γ; see [DG70, Thm. IV.1.3.6]. Thus, the assignment

X_{ℚ} : G ⟼ ℚ ⊗_{ℤ} X(G) =: X(G)_{ℚ}

yields a contravariant exact functor from M to the category Rep_{ℚ}(Γ) of finitedimensional ℚ-vector spaces equipped with a continuous representation of Γ as above; moreover, every finite group of multiplicative type is sent to 0. This yields in turn a contravariant exact functor

\underline{X}_{ℚ} : \underline{T} ⟶ Rep_{ℚ}(Γ).

Proposition 4.1. The functor \underline{X}_{ℚ} is an anti-equivalence of categories. In particular, the category \underline{T} is semi-simple, and Hom\underline{T} (T_{1}, T_{2}) is a finitedimensional ℚ-vector space for any tori T_{1}, T_{2}.

Proof. Given a finite-dimensional ℚ-vector space V equipped with a continuous action of Γ, there exists a finitely generated Γ-stable subgroup M ⊂ V which spans V; thus, V ≅ X(T)_{ℚ}, where T denotes the torus with character group M. So \underline{X}_{ℚ} is essentially surjective. Given two tori T_{1}, T_{2}, the natural isomorphism

Hom_{M}(T_{1}, T_{2}) ≅ Hom^{Γ}(X(T_{2}), X(T_{1}))

yields an isomorphism

Hom\underline{T} (T_{1}, T_{2}) ≅ Hom^{Γ}(X(T_{2})_{ℚ}, X(T_{1})_{ℚ})

in view of Proposition 3.6. It follows that \underline{X}_{ℚ} is fully faithful.

Lemma 4.2. (i) Every algebraic group G has a unique maximal torus, T (G).

(ii) Every morphism of algebraic groups u : G → H sends T (G) to T (H).

(iii) The formation of T (G) commutes with base change under field extensions.

Proof. (i) This follows from the fact that T_{1} + T_{2} is a torus for any subtori T_{1}, T_{2} ⊂ G. (ii) Just note that the image of a torus under any morphism is still a torus. (iii) Consider an algebraic group G, its maximal torus T, and a field extension k′ of k. If char(k) = 0, then Theorem 2.9 implies that G/T is a an extension of an abelian variety by a product M × U, where M is finite and U unipotent. As a consequence, a similar assertion holds for G_{k′}/T_{k′}; it follows that G_{k′}/T_{k′} contains no non-zero torus, and hence T_{k′} is the maximal torus of G_{k′}. On the other hand, if char(k) > 0, then G/T is a 3-step extension of a unipotent group by an abelian variety by a finite group, in view of Theorem 2.11. It follows similarly that T_{k′} is the maximal torus of G_{k′}.

By Lemma 4.2, the assignment G ↦ T (G) yields a functor

T : C ⟶ T ,

the functor of maximal tori. This functor is not exact, as seen from the exact sequence

0 ⟶ G[n] ⟶ G ⟶^{n_{G}} G ⟶ 0,

where G is a non-zero torus and n a non-zero integer. But T is exact up to finite groups, as shown by the following;

Lemma 4.3. Every exact sequence in C

0 ⟶ G_{1} ⟶^{u} G_{2} ⟶^{v} G_{3} ⟶ 0

yields a complex in C

0 ⟶ T (G_{1}) ⟶^{T (u)} T (G_{2}) ⟶^{T (v)} T (G_{3}) ⟶ 0,

where T (u) is a monomorphism, T (v) an epimorphism, and Ker T (v)/Im T (u) is finite.

Proof. We argue as in the proof of Theorem 2.11 (iii). Clearly, T (u) is a monomorphism. Also, the group

Ker T (v)/Im T (u) = T (G_{2}) ∩ u(G_{1})/u(T (G_{1}))

is the quotient of a group of multiplicative type by its maximal torus, and hence is finite. To show that T (v) is an epimorphism, we may replace G_{2} with v^{−1}(T (G_{3})), and hence assume that G_{3} is a torus. Next, we may replace G_{2} with G_{2}/T (G_{2}), and hence assume that T (G_{2}) is zero. We then have to check that G_{3} is zero. If char(k) = 0, then there is an exact sequence

0 ⟶ M × U ⟶ G_{2} ⟶ A ⟶ 0

as in Theorem 2.9, where M is finite. Thus, every morphism G_{2} → G_{3} has finite image. Since v : G_{2} → G_{3} is an epimorphism, it follows that G_{3} = 0. On the other hand, if char(k) > 0, then there are exact sequences

0 ⟶ H ⟶ G_{2} ⟶ U ⟶ 0, 0 ⟶ M ⟶ H ⟶ A ⟶ 0

as in Theorem 2.11, where M is finite. This implies again that every morphism G_{2} → G_{3} has finite image, and hence that G_{3} = 0.

Proposition 4.4. (i) The functor of maximal tori yields an exact functor

\underline{T} : \underline{C} ⟶ \underline{T} ,

right adjoint to the inclusion \underline{T} → \underline{C}. Moreover, \underline{T} commutes with base change under field extensions.

(ii) Every torus is a projective object in \underline{C}.

Proof. (i) Composing T with the functor T → \underline{T} induced by the quotient functor Q, we obtain an exact functor C → \underline{T} (Lemma 4.3), which sends every finite group to 0. This yields an exact functor \underline{T} : \underline{C} → \underline{T} . The adjointness assertion follows from the natural isomorphism

Hom_{C}(T, G) ≅ Hom_{T} (T, T (G))

for any torus T and any algebraic group G, which yields a natural isomorphism

Hom\underline{C}(T, G) ≅ Hom\underline{T} (T, T (G))

by using Lemmas 3.1 and 4.3. Finally, the assertion on field extensions is a direct consequence of Lemma 4.2. (ii) This follows by arguing as in the proof of Proposition 3.8 (iii).

4.2 Abelian varieties

Denote by A (resp. P) the full subcategory of C with objects the abelian varieties (resp. the proper groups, i.e., those algebraic groups G such that the structure map G → Spec(k) is proper). Like the categorye of tori, A is stable under taking quotients and extensions, but not subobjects; so A is an additive subcategory of C, but not an abelian subcategory. Also, P is stable under taking subobjects, quotients and extensions; it also contains the category F of finite groups. We may thus form the quotient abelian category P/F, which is a full subcategory of C/F. Next, let \underline{A} be the full subcategory of P/F with objects the abelian varieties. As in §4.1, the inclusion of \underline{A} in P/F is an equivalence of categories, and the remaining statements of Lemma 3.1 adapt to this setting. Also, Proposition 3.6 yields natural isomorphisms

ℚ ⊗_{ℤ} Hom_{C}(A_{1}, A_{2}) ⟶^{≅} Hom\underline{A}(A_{1}, A_{2})

for any abelian varieties A_{1}, A_{2}. Since the abelian group Hom_{C}(A_{1}, A_{2}) is free of finite rank (see [Mi86, Thm. 12.5]), Hom\underline{A}(A_{1}, A_{2}) is a finite-dimensional ℚvector space. Moreover, the category \underline{A} is semi-simple, in view of the Poincaré complete reducibility theorem (which holds over an arbitrary field, see [Co06, Cor. 3.20] or [Br17, Cor. 4.2.6]).

Lemma 4.5. (i) Every smooth connected algebraic group G has a largest abelian variety quotient,

α = α_{G} : G ⟶ A(G).

Moreover, Ker(α) is linear and connected.

(ii) Every morphism u : G → H, where H is smooth and connected, induces a unique morphism A(u) : A(G) → A(H) such that the square

G ⟶^{u} H
α_{G}↓  ^{α_{H}} ↓
A(G) ⟶^{A(u)} A(H)

commutes.

(iii) For any field extension k′ of k, the natural morphism A(G_{k′}) → A(G)_{k′} is an isomorphism if char(k) = 0, and an isogeny if char(k) > 0.

Proof. (i) and (ii) The assertions are direct consequences of Theorem 2.3 (ii). (iii) By (i), we have an exact sequence

0 ⟶ L(G) ⟶ G ⟶^{α} A(G) ⟶ 0,

where L(G) is linear and connected. This yields an exact sequence

0 ⟶ L(G)_{k′} ⟶ G_{k′} ⟶ A(G)_{k′} ⟶ 0.

Thus, L(G_{k′}) ⊂ L(G)_{k′} and we obtain an exact sequence

0 ⟶ L(G)_{k′}/L(G_{k′}) ⟶ A(G_{k′}) ⟶ A(G)_{k′} ⟶ 0.

Sinc L(G)_{k′} is linear, the quotient L(G)_{k′}/L(G_{k′}) must be finite; this yields the assertion when char(k) > 0. When char(k) = 0, we may characterize L(G) as the largest connected linear subgroup of G. It follows that L(G)_{k′} ⊂ L(G_{k′}); hence equality holds, and

A(G_{k′}) →^{≅} A(G)_{k′}.

Remarks 4.6. (i) An arbitrary algebraic group G may admit no largest abelian variety quotient, as shown by the following variant of [Br17, Ex. 4.3.8]: let A be a non-zero abelian variety, and choose an integer n ≥ 2. Let

G := (A × A[n^{2}])/diag(A[n]),

where A[n] is viewed as a subgroup of A[n^{2}]. Consider the subgroups of G

H_{1} := (A[n] × A[n^{2}])/diag(A[n]), H_{2} := diag(A[n^{2}])/diag(A[n]).

Then G/H_{1}, G/H_{2} are both isomorphic to A. Also, H_{1} ∩ H_{2} = 0, and G is not an abelian variety. (ii) The assignment G ↦ A(G) does not preserve exactness of sequences of smooth connected algebraic groups. For example, consider an elliptic curve E equipped with a k-rational point of prime order ℓ ≥ 2. Assume that k contains a nontrivial ℓth root of unity; this identifies μ_{ℓ} with the constant group scheme ℤ/ℓℤ. Consider the quotient

G := (E × 𝔾_{m})/diag(μ_{ℓ}),

with an obvious notation. Then G is a smooth connected algebraic group, which lies in an exact sequence

0 ⟶ E ⟶ G ⟶ 𝔾_{m} ⟶ 0.

Moreover, the induced map A(E) → A(G) is just the quotient map E → E/μ_{ℓ}.

We now show that the assignment G ↦ A(G) is exact up to finite groups:

Lemma 4.7. Consider an exact sequence in C

0 ⟶ G_{1} ⟶^{u} G_{2} ⟶^{v} G_{3} ⟶ 0,

where G_{1}, G_{2}, G_{3} are smooth and connected. Then we have a commutative diagram in C

0 ⟶ G_{1} ⟶^{u} G_{2} ⟶^{v} G_{3} ⟶ 0
α_{1}↓  ^{α_{2}}↓  ^{α_{3}}↓
0 ⟶ A(G_{1}) ⟶^{A(u)} A(G_{2}) ⟶^{A(v)} A(G_{3}) ⟶ 0,

where A(v) is an epimorphism, and Ker A(u), Ker A(v)/Im A(u) are finite.

Proof. Clearly, A(v) is an epimorphism. Let L_{i} := Ker(α_{i}) for i = 1, 2, 3; then each L_{i} is connected and linear by Theorem 2.3. We have isomorphisms Ker A(u) ≅ u^{−1}(L_{2})/L_{1}, Im A(u) ≅ u(G_{1})/u(G_{1}) ∩ L_{2} and Ker A(v) ≅ v^{−1}(L_{3})/L_{2}. Since u^{−1}(L_{2}) is linear, Ker A(u) is linear as well; it is also proper, and hence finite. Also, Ker A(v)/Im A(u) ≅ v^{−1}(L_{3})/L_{2}+u(G_{1}) is a quotient of v^{−1}(L_{3})/u(G_{1}) ≅ L_{3}. It follows similarly that Ker A(v)/Im A(u) is finite.

Next, we obtain a dual version of Proposition 4.4:

Proposition 4.8. (i) The Albanese functor yields an exact functor

\underline{A} : \underline{C} ⟶ \underline{A},

which is left adjoint to the inclusion \underline{A} → \underline{C}. Moreover, \underline{A} commutes with base change under field extensions.

(ii) Every abelian variety is an injective object in \underline{C}.

Proof. (i) Consider a \underline{C}-morphism \underline{f} : G → H and choose a representative by a C-morphism f : G → H/H′, where H′ is a finite subgroup of H. This yields an A-morphism

A(f) : A(G) ⟶ A(H/H′) = A(H)/α_{H}(H′),

and hence an \underline{A}-morphism \underline{A}(f) : A(G) → A(H). One may readily check that \underline{A}(f) depends only on \underline{f}, in a covariant way. By Proposition 3.5 and Lemma 4.5, the resulting functor \underline{A} is exact and commutes with base change under field extensions. To show the adjointness assertion, consider a smooth connected algebraic group G and an abelian variety A. Then the map

Hom_{A}(A(G), A) ⟶ Hom_{C}(G, A), f ⟼ f ∘ α_{G}

is an isomorphism by Lemma 4.5 again. In view of Proposition 3.6, it follows that the analogous map

Hom\underline{A}(A(G), A) ⟶ Hom\underline{C}(G, A)

is an isomorphism as well. (ii) This follows formally from (i).

Remarks 4.9. (i) Denote by Â the dual of an abelian variety A. Then the assignment A ↦ Â yields a contravariant endofunctor of A, which is involutive and preserves isogenies and finite products. As an easy consequence, we obtain a contravariant endofunctor of \underline{A}, which is involutive and exact. Note that each abelian variety is (non-canonically) \underline{A}-isomorphic to its dual, via the choice of a polarization. (ii) Let k′ be a field extension of k. Then the assignment

A ⟼ ℚ ⊗_{ℤ} A(k′) =: A(k′)_{ℚ}

yields a functor from A to the category of ℚ-vector spaces (possibly of infinite dimension), which preserves finite products. Moreover, each isogeny

f : A ⟶ B

yields an isomorphism

A(k′)_{ℚ} ⟶^{≅} B(k′)_{ℚ},

since this holds for the multiplication maps n_{A}. Thus, the above assignment yields an exact functor from \underline{A} to the category of ℚ-vector spaces.

4.3 Vanishing of extension groups

In this subsection, we prove the assertion (v) of Theorem 1: hd(\underline{C}) = 1. We first collect general vanishing results for extension groups in \underline{C}:

Lemma 4.10. Let G be a smooth connected algebraic group, U a smooth connected unipotent group, A an abelian variety, and T a torus.

(i) Extn ; \underline{C}(T, G) = 0 = Extn ; \underline{C}(G, A) = 0 for all n ≥ 1.

(ii) If char(k) = 0, then Extn ; \underline{C}(U, G) = 0 for all n ≥ 1.

(iii) If char(k) = p > 0 and G is divisible, then Extn ; \underline{C}(U, G) = 0 = Extn ; \underline{C}(G, U) for all n ≥ 0.

(iv) If G is linear, then Extn ; \underline{C}(G, T) = 0 for all n ≥ 1. If moreover char(k) = 0, then Extn ; \underline{C}(G, U) = 0 for all n ≥ 1 as well.

Proof. (i) Just recall that T is projective in \underline{C} (Proposition 4.4), and A is injective in \underline{C} (Proposition 4.8). (ii) Likewise, U is projective in \underline{C} by Proposition 3.8. (iii) Since U is unipotent, there exists a positive integer m such that p_{G}^{m} = 0. It follows that both groups Extm ; \underline{C}(U, G) and Extm ; \underline{C}(G, U) are p^{m}-torsion (see e.g. [Oo66, Lem. I.3.1]). But these groups are also modules over End\underline{C}(G), and hence ℚ-vector spaces by Proposition 3.6. This yields the assertion. (iv) By Proposition 2.8 and the long exact sequence for Ext groups, we may assume that G is unipotent or a torus. In the latter case, both assertions follows from (i); in the former case, the first assertion follows from (ii) and (iii), and the second assertion, from the fact that unipotent groups are just vector spaces.

Next, recall that hd(\underline{C}) ≥ 1 in view of Examples 3.4. So, to conclude that hd(\underline{C}) = 1, it suffices to check the following:

Lemma 4.11. For any smooth connected algebraic groups G, H and any integer n ≥ 2, we have Extn ; \underline{C}(G, H) = 0.

Proof. Let η ∈ Extn ; \underline{C}(G, H), where n ≥ 3. Then η is represented by an exact sequence in \underline{C}

0 ⟶ H ⟶ G_{1} ⟶ ⋯ ⟶ G_{n} ⟶ G ⟶ 0,

which we may cut into two exact sequences in \underline{C}

0 ⟶ H ⟶ G_{1} ⟶ G_{2} ⟶ K ⟶ 0,
0 ⟶ K ⟶ G_{3} ⟶ ⋯ ⟶ G_{n} ⟶ G ⟶ 0.

Thus, η can be written as a Yoneda product η_{1} ∪ η_{2}, where η_{1} ∈ Ext2 ; \underline{C}(G, K) and η_{2} ∈ Extn−2 ; \underline{C}(K, H). So it suffices to show the assertion when n = 2. Using the long exact sequences for Ext groups, we may further reduce to the case where G, H are simple objects in \underline{C}, i.e., 𝔾_{a}, simple tori T, or simple abelian varieties A (Proposition 3.2). In view of Lemma 4.10, it suffices in turn to check that

(i) Ext2 ; \underline{C}(A, T) = 0,

(ii) Ext2 ; \underline{C}(A, 𝔾_{a}) = 0 when char(k) = 0,

(iii) Ext2 ; \underline{C}(𝔾_{a}, 𝔾_{a}) = 0 when char(k) > 0.

For (i), we adapt the argument of [Oo66, Prop. II.12.3]. Let η ∈ Ext2 ; \underline{C}(A, T) be represented by an exact sequence in \underline{C}

0 ⟶ T ⟶^{u} G_{1} ⟶^{v} G_{2} ⟶^{w} A ⟶ 0.

As above, η = η_{1} ∪ η_{2}, where η_{1} denotes the class of the extension

0 ⟶ T ⟶^{u} G_{1} ⟶ K ⟶ 0,

and η_{2} that of the extension

0 ⟶ K ⟶ G_{2} ⟶^{w} A → 0.

Also, note that \underline{A}(w) : \underline{A}(G_{2}) → A is an epimorphism in \underline{A}, and hence has a section, say s. Denoting by H_{2} the pull-back of s(A) under the Albanese morphism G_{2} → \underline{A}(G_{2}), we have a monomorphism ι : H_{2} → G_{2} in \underline{C}, such that \underline{A}(w ∘ ι) : \underline{A}(H_{2}) → A is an isomorphism. This yields a commutative diagram of exact sequences in \underline{C}

0 ⟶ T ⟶ H_{1} ⟶ H_{2} ⟶ A ⟶ 0
id↓  ↓  ^{ι}↓  ^{id}↓
0 ⟶ T ⟶^{u} G_{1} ⟶^{v} G_{2} ⟶^{w} A ⟶ 0.

Thus, η is represented by the top exact sequence, and hence we may assume that \underline{A}(w) is an isomorphism. Then K is linear in view of Theorem 2.3. Thus, κ_{1} = 0 by Lemma 4.10 (iv). So η = 0; this completes the proof of (i). For (ii), we replace T with 𝔾_{a} in the above argument, and use the vanishing of Ext1 ; \underline{C}(L, 𝔾_{a}) for L linear (Lemma 4.10 (iv) again). Finally, for (iii), it suffices to show that Ext2 ; \underline{U}(𝔾_{a}, 𝔾_{a}) = 0. Also, we may assume that k is perfect, in view of Theorem 3.11. Let η ∈ Ext2 ; \underline{U}(𝔾_{a}, 𝔾_{a}) be represented by an exact sequence in \underline{U},

0 ⟶ 𝔾_{a} ⟶ G_{1} ⟶ G_{2} ⟶ 𝔾_{a} ⟶ 0.

Then Proposition 3.5 yields an exact sequence in U,

0 ⟶ 𝔾_{a} ⟶ H_{1} ⟶ H_{2} ⟶ 𝔾_{a} ⟶ 0,

and a commutative diagram in \underline{U},

0 ⟶ 𝔾_{a} ⟶ G_{1} ⟶ G_{2} ⟶ 𝔾_{a} ⟶ 0
↓  ↓  ↓  ↓
0 ⟶ 𝔾_{a} ⟶ H_{1} ⟶ H_{2} ⟶ 𝔾_{a} ⟶ 0.

where the vertical arrows are isomorphisms in \underline{U} (here we use the fact that the quotient of 𝔾_{a} by a finite subgroup is isomorphic to 𝔾_{a}). As Ext_{U}^{2}(𝔾_{a}, 𝔾_{a}) = 0 (see [DG70, V.1.5.1, V.1.5.2]), the bottom exact sequence is equivalent to the trivial exact sequence in U, and hence in \underline{U}. Thus η = 0 as desired.

Remark 4.12. When k is perfect, the groups Ext_{C}^{n}(G, H) are torsion for all n ≥ 2 and all algebraic groups G, H, in view of [Mi70, Cor., p. 439]. In fact, this assertion extends to an arbitrary field k: indeed, it clearly holds when G or H is finite, or more generally m-torsion for some positive integer m. Using Proposition 2.8, one may thus reduce to the case when G, H are simple objects of \underline{C}. Then the assertion is obtained by combining Proposition 3.6, Lemma 4.10, and the proof of Lemma 4.11.

5 Structure of isogeny categories

5.1 Vector extensions of abelian varieties

In this subsection, we assume that char(k) = 0. Recall that a vector extension of an abelian variety A is an algebraic group G that lies in an extension

ξ : 0 ⟶ U ⟶ G ⟶ A ⟶ 0,

where U is unipotent. Then U = R_{u}(G) and A = A(G) are uniquely determined by G; also, the extension ξ has no non-trivial automorphisms, since Hom_{C}(A, U) = 0. Thus, the data of the algebraic group G and the extension ξ are equivalent. We denote by V the full subcategory of C with objects the vector extensions (of all abelian varieties). By Theorems 2.3 and 2.4, the objects of V are those smooth connected algebraic groups that admit no non-zero subtorus. In view of Lemmas 4.2 and 4.3, this readily implies:

Lemma 5.1. (i) Let 0 → G_{1} → G_{2} → G_{3} → 0 be an exact sequence in C, where G_{2} is connected. Then G_{2} is an object of V if and only if so are G_{3} and G_{1}^{0}.

(ii) Let f : G → H be an isogeny of connected algebraic groups. Then G is an object of V if and only if so is H.

(iii) Let k′ be a field extension of k, and G an algebraic k-group. Then G is an object of V_{k} if and only if G_{k′} is an object of V_{k′}.

In particular, V is stable under taking quotients and extensions, but not subobjects; like T and A, it is an additive subcategory of C, but not an abelian subcategory. Next, recall from [Ro58] or [MM74, §1.9] that every abelian variety A has a universal vector extension,

ξ(A) : 0 ⟶ U(A) ⟶ E(A) ⟶ A ⟶ 0,

where U(A) is the additive group of the vector space H^{1}(A, O_{A})^{∗}; moreover, dim U(A) = dim A. Also, E(A) is anti-affine, i.e., every morphism from E(A) to a linear algebraic group is zero (see e.g. [Br17, Prop. 5.5.8]).

Proposition 5.2. (i) The assignments A ↦ E(A), A ↦ U(A) yield addi-

tive functors

E : A ⟶ V, U : A ⟶ U,

which commute with base change under field extensions.

(ii) For any morphism f : A → B of abelian varieties, the induced morphism U(f) : U(A) → U(B) is the dual of the pull-back morphism f^{∗} : H^{1}(B, O_{B}) → H^{1}(A, O_{A}). Moreover, U(f) is zero (resp. an isomorphism) if and only if f is zero (resp. an isogeny).

(iii) E is left adjoint to the Albanese functor A : V → A.

Proof. We prove (i) and (ii) simultaneously. Let f : A → B be a morphism of abelian varieties. Consider the pull-back diagram of exact sequences

0 ⟶ U(B) ⟶ F ⟶ A ⟶ 0
id↓  ↓  ^{f}↓
0 ⟶ U(B) ⟶ E(B) ⟶ B ⟶ 0.

The universal property of ξ(A) yields a commutative diagram of exact sequences

0 ⟶ U(A) ⟶ E(A) ⟶ A ⟶ 0
↓  ↓  ^{id}↓
0 ⟶ U(B) ⟶ F ⟶ A ⟶ 0,

and hence another such diagram,

0 ⟶ U(A) ⟶ E(A) ⟶ A ⟶ 0
U(f)↓  ^{E(f)}↓  ^{f}↓
0 ⟶ U(B) ⟶ E(B) ⟶ B ⟶ 0,

which defines morphisms E(f) and U(f). Next, let η ∈ H^{1}(B, O_{B}), so that we have a push-out diagram of extensions

0 ⟶ U(B) ⟶ E(B) ⟶ B ⟶ 0
η↓  ↓  ^{id}↓
0 ⟶ 𝔾_{a} ⟶ E_{η} ⟶ B ⟶ 0.

By construction, the pull-back of E_{η} by f is the push-out of ξ(A) by the map η∘U(f) : U(A) → 𝔾_{a}. Hence U(f) is the dual of f^{∗} : H^{1}(B, O_{B}) → H^{1}(A, O_{A}). As a consequence, U is a covariant functor, and hence so is E. Since the formation of the universal vector extension commutes with base change under field extensions, the functors E and U commute with such base change as well. Clearly, they are additive; this completes the proof of (i). To complete the proof of (ii), recall the canonical isomorphism

H^{1}(A, O_{A}) ≅ Lie(Â),

where the right-hand side denotes the Lie algebra of the dual abelian variety (see [Mi86, Rem. 9.4]). This isomorphism identifies f^{∗} with Lie(f̂), where f̂: B̂ → Â denotes the dual morphism of f. As a consequence,

U(f) = 0 ⇔ f^{∗} = 0 ⇔ f̂ = 0 ⇔ f = 0,

where the second equivalence holds since char(k) = 0, and the third one follows from biduality of abelian varieties. Likewise, U(f) is an isomorphism if and only if f^{∗} is an isomorphism; equivalently, f̂ is an isogeny, i.e., f is an isogeny. (iii) Given a vector extension 0 → U → G → A(G) → 0, we check that the map

α : Hom_{V}(E(A), G) ⟶ Hom_{A}(A, A(G)), u ⟼ A(u)

is an isomorphism. Consider a morphism u : E(A) → G such that A(u) = 0. Then u factors through a morphism E(A) → R_{u}(G), and hence u = 0 as E(A) is anti-affine. Next, consider a morphism v : A → A(G). By (i), we have a commutative square

E(A) ⟶^{E(v)} E(A(G))
↓  ↓
A ⟶^{v} A(G).

Also, the universal property of ξ(A) yields a commutative square

E(A(G)) ⟶^{δ} G
↓  ^{α_{G}}↓
A(G) ⟶^{id} A(G).

Thus, w := E(v) ∘ δ ∈ Hom_{V}(E(A), G) satisfies α(v) = u.

Denote by \underline{V} the isogeny category of vector extensions, that is, the full subcategory of \underline{C} with the same objects as V. Then \underline{V} is an abelian category in view of Lemma 5.1. Also, Proposition 3.6 yields natural isomorphisms

ℚ ⊗_{ℤ} Hom_{C}(G_{1}, G_{2}) →^{≅} Hom\underline{V}(G_{1}, G_{2}), ℚ ⊗_{ℤ} Ext_{C}^{1}(G_{1}, G_{2}) →^{≅} Ext1 ; \underline{V}(G_{1}, G_{2})

for any objects G_{1}, G_{2} of \underline{V}.

Corollary 5.3. (i) The functors E : A → V, U : A → U yield exact

functors

\underline{E} : \underline{A} ⟶ \underline{V}, \underline{U} : \underline{A} ⟶ U,

which commute with base change under field extensions. Moreover, \underline{E} is left adjoint to the Albanese functor \underline{A} : \underline{V} → \underline{A}.

(ii) The universal vector extension of any abelian variety is a projective object of \underline{V}.

Proof. (i) This follows from Propositions 3.5 and 5.2. (ii) We have canonical isomorphisms for any object G of \underline{V}:

Hom\underline{V}(E(A), G) ≅ ℚ ⊗_{ℤ} Hom_{V}(E(A), G) ≅ ℚ ⊗_{ℤ} Hom_{A}(A, A(G)),

where the first isomorphism follows from Proposition 3.6, and the second one from Proposition 5.2. Using Proposition 3.6 again, this yields a canonical isomorphism

Hom\underline{V}(E(A), G) ≅ Hom\underline{A}(A, \underline{A}(G)).

Since the Albanese functor \underline{A} is exact (Proposition 4.8), it follows that the functor G ↦ Hom\underline{V}(E(A), G) is exact as well.

Next, let G be an object of V. Form and label the commutative diagram of exact sequences in C

0 ⟶ U(A) ⟶^{ι} E(A) ⟶ A ⟶ 0
γ↓  ^{δ}↓  ^{id}↓
0 ⟶ U ⟶ G ⟶ A ⟶ 0,

where U = U(G), A = A(G), and γ = γ_{G} classifies the bottom extension. This yields an exact sequence in C

ξ : 0 ⟶ U(A) ⟶^{γ−ι} U × E(A) ⟶ G ⟶ 0.

Proposition 5.4. Keep the above notation.

(i) ξ yields a projective resolution of G in \underline{V}.

(ii) For any object H of V, we have an exact sequence

0 ⟶ Hom\underline{V}(G, H) ⟶^{φ} Hom_{U}(U(G), U(H)) × Hom\underline{A}(A(G), A(H))

⟶^{ψ} Hom_{U}(U(A(G)), U(H)) ⟶ Ext1 ; \underline{V}(G, H) ⟶ 0,

where φ(\underline{f}) := (U(\underline{f}), \underline{A}(\underline{f})), and ψ(u, v) := u ∘ γ_{G} − γ_{H} ∘ U(v).

Proof. (i) This holds as U, U(A) are projective in C (Theorem 2.9), and E(A) is projective in \underline{C} (Proposition 5.2). (ii) In view of (i), this follows readily from the long exact sequence of extension groups

0 → Hom\underline{V}(G, H) → Hom\underline{V}(U × E(A), H) → Hom\underline{V}(U, H) → Ext1 ; \underline{V}(G, H) → 0

associated with the short exact sequence ξ.

As a direct consequence of Proposition 5.4, we obtain:

Corollary 5.5. The following conditions are equivalent for an object G of V:

(i) G ≅ E(A) in \underline{V} for some abelian variety A.

(ii) Hom\underline{V}(G, 𝔾_{a}) = Ext1 ; \underline{V}(G, 𝔾_{a}) = 0.

As a further consequence, we describe the projective or injective objects of \underline{V}:

Corollary 5.6. (i) The projective objects of \underline{V} are the products V × E(A), where V is unipotent, and A is an abelian variety.

(ii) The injective objects of \underline{V} are the abelian varieties.

Proof. Let G be an extension of an abelian variety A by a unipotent group U.

(i) If G is projective in \underline{V}, then Ext1 ; \underline{V}(G, 𝔾_{a}) = 0. In view of Proposition 5.4, it follows that the map Hom_{U}(U, 𝔾_{a}) → Hom_{U}(U(A), 𝔾_{a}), u ↦ u∘γ is surjective. Equivalently, γ is injective; hence so is δ : E(A) → G. Identifying E(A) with a subgroup of G, it follows that G = U + E(A), and U(A) ⊂ U. We may choose a complement V ⊂ U to the subspace U(A) ⊂ U; then G ≅ V × E(A). Conversely, every such product is projective by Proposition 5.2. This yields the assertion. (ii) If G is injective in \underline{V}, then Ext1 ; \underline{V}(B, G) = 0 for any abelian variety B. Thus, we have an exact sequence

0 ⟶ Hom\underline{V}(B, U) ⟶ Hom\underline{V}(B, G) ⟶ Hom\underline{V}(B, A) ⟶^{∂} Ext1 ; \underline{V}(B, U) ⟶ 0.

Moreover, Ext1 ; \underline{V}(B, U) ≅ Hom_{U}(U(B), U), as follows e.g. from Proposition 5.4. Since Hom\underline{V}(B, A) = Hom\underline{A}(B, A) is a finite-dimensional ℚ-vector space, so is Hom_{U}(U(B), U). When k is not a number field, i.e., k is an infinite-dimensional ℚ-vector space, this forces U = 0, since U(B) ≠ 0 for any non-zero abelian variety B. Thus, G is an abelian variety. On the other hand, when k is a number field, there are only finitely many isomorphism classes of abelian varieties that are isogenous to any prescribed abelian variety (see [MW93] for a quantitative version of this finiteness result). As a consequence, we may choose a simple abelian variety B, not isogenous to any simple factor of A. Then Hom\underline{A}(B, A) = 0; as above, this yields U = 0, i.e., G is an abelian variety. Conversely, every abelian variety is injective in \underline{V} by Proposition 4.8.

We now describe the structure of V and \underline{V} in terms of linear algebra. Let D be the category with objects the triples (A, U, γ), where A is an abelian variety, U a unipotent group, and γ : U(A) → U a morphism; the D-morphisms from (A_{1}, U_{1}, γ_{1}) to (A_{2}, U_{2}, γ_{2}) are those pairs of C-morphisms u : U_{1} → U_{2}, v : A_{1} → A_{2} such that the square

U(A_{1}) ⟶^{U(v)} U(A_{2})
γ_{1}↓  ^{γ_{2}}↓
U_{1} ⟶^{u} U_{2}

commutes. We also introduce the ‘isogeny category’ \underline{D}, by allowing v to be a \underline{C}-morphism in the above definition (this makes sense in view of Corollary 5.3). Next, define a functor

D : V ⟶ D

by assigning to each object G the triple (A(G), R_{u}(G), γ), where γ : U(A(G)) → R_{u}(G) denotes the classifying map, and to each morphism f : G_{1} → G_{2}, the pair (A(f), U(f)). By Corollary 5.3 again, we may define similarly a functor

\underline{D} : \underline{V} ⟶ \underline{D}.

Proposition 5.7. With the above notation, the functors D and \underline{D} yield equivalences of categories.

We omit the easy proof.

5.2 Semi-abelian varieties

Recall that a semi-abelian variety is an algebraic group G that lies in an extension

ξ : 0 ⟶ T ⟶ G ⟶ A ⟶ 0,

where T is a torus, and A an abelian variety. We now adapt part of the results of §5.1 to this setting, leaving the (easy) verifications to the motivated reader. The algebraic groups T = T (G) and A = A(G) are uniquely determined by G, and the extension ξ has no non-trivial automorphisms. Thus, the data of G and of the extension ξ are equivalent. Moreover, recall the natural isomorphism

c : Ext_{C}^{1}(A, T) ⟶^{≅} Hom^{Γ}(X(T), Â(k_{s})),

which arises from the Weil-Barsotti isomorphism

Ext_{C_{ks}}^{1} (A_{k_{s}}, 𝔾_{m,k_{s}}) ⟶^{≅} Â(k_{s})

together with the pairing

Ext_{C}^{1}(A, T) × X(T) ⟶ Ext_{C_{ks}}^{1} (A_{k_{s}}, 𝔾_{m,k_{s}})

given by push-out of extensions via characters of T. Denote by S the full subcategory of C with objects the semi-abelian varieties. Then the analogue of Lemma 5.1 holds in view e.g. of [Br17, §5.4] (but there is no analogue of the universal vector extension in this setting). Thus, the isogeny category of semi-abelian varieties, \underline{S}, is an abelian category. As for vector extensions of abelian varieties, we have natural isomorphisms

ℚ ⊗_{ℤ} Hom_{C}(G_{1}, G_{2}) →^{≅} Hom\underline{S}(G_{1}, G_{2}), ℚ ⊗_{ℤ} Ext_{C}^{1}(G_{1}, G_{2}) →^{≅} Ext1 ; \underline{S}(G_{1}, G_{2})

for any objects G_{1}, G_{2} of \underline{S}. This yields a natural isomorphism

Ext1 ; \underline{C}(A, T) ⟶^{≅} Hom^{Γ}(X(T)_{ℚ}, Â(k_{s})_{ℚ}).

Note that the assignment

A ⟼ Â(k_{s})_{ℚ}

yields an exact functor from \underline{A} to the category of ℚ-vector spaces equipped with the discrete topology and a continuous representation of Γ, as follows e.g. from Remarks 4.9. Next, we obtain a description of \underline{S} in terms of linear algebra. Let \underline{E} be the category with objects the triples (A, M, c), where A is an abelian variety, M a finite-dimensional ℚ-vector space equipped with a continuous action of Γ, and c : M → Â(k_{s})_{ℚ} a Γ-equivariant linear map; the \underline{E}-morphisms from (A_{1}, M_{1}, c_{1}) to (A_{2}, M_{2}, c_{2}) are those pairs (\underline{u}, v), where \underline{u} : A_{1} → A_{2} is a \underline{A}-morphism and v : M_{2} → M_{1} a Γ-equivariant linear map, such that the square

M_{2} ⟶^{c_{2}} Â_{2}(k_{s})_{ℚ}
v↓  ˆ  \underline{u}↓
M_{1} ⟶^{c_{1}} Â_{1}(k_{s})_{ℚ}

commutes. Then one may check that the assignment

G ⟼ (A(G), X(T (G))_{ℚ}, c(G)_{ℚ})

yields an equivalence of categories \underline{S} → \underline{E}. Moreover, the sequence

0 ⟶ Hom\underline{S}(G_{1}, G_{2}) ⟶^{φ} Hom^{Γ}(M_{2}, M_{1}) × Hom\underline{A}(A_{1}, A_{2})

⟶^{ψ} Hom^{Γ}(M_{2}, Â_{1}(k_{s})_{ℚ}) ⟶ Ext1 ; \underline{S}(G_{1}, G_{2}) ⟶ 0

is exact for any semi-abelian varieties G_{1}, G_{2}, where \underline{E}(G_{i}) := (A_{i}, M_{i}, c_{i}) for i = 1, 2, φ(\underline{f}) := (\underline{X}_{ℚ} ∘ \underline{T})(\underline{f}), \underline{A}(\underline{f})), and ψ(u, v) := c_{1} ∘ u − v̂ ∘ c_{2}. Yet there are important differences between the isogeny categories of vector extensions and semi-abelian varieties. For example, the latter does not have enough projectives in general:

Proposition 5.8. (i) If k is not locally finite, then the projective objects of \underline{S} are the tori.

(ii) If k is locally finite, then the product functor \underline{T} × \underline{A} → \underline{S} yields an equivalence of categories.

Proof. (i) Let G be a semi-abelian variety, extension of an abelian variety A by a torus T. Denote by c : X(T) → Â(k_{s}) the classifying map, and by

c_{ℚ} : X(T)_{ℚ} ⟶ Â(k_{s})_{ℚ}

the corresponding ℚ-linear map; recall that c and c_{ℚ} are Γ-equivariant. If G is projective in \underline{C}, then Ext1 ; \underline{C}(G, T′) = 0 for any torus T′. Thus, we have an exact sequence

0 ⟶ Hom\underline{C}(A, T′) ⟶ Hom\underline{C}(G, T′) ⟶ Hom\underline{C}(T, T′) ⟶^{∂} Ext1 ; \underline{C}(A, T′) ⟶ 0.

Moreover, the boundary map ∂ may be identified with the map

Hom^{Γ}(X(T′)_{ℚ}, X(T)_{ℚ}) ⟶ Hom^{Γ}(X(T′)_{ℚ}, Â(k_{s})_{ℚ}). f ⟼ c_{ℚ} ∘ f.

Since ∂ is surjective, and X(T′)_{ℚ} may be chosen arbitrarily among finitedimensional ℚ-vector spaces equipped with a continuous representation of Γ, the map c_{ℚ} is surjective as well. In particular, the abelian group Â(k_{s}) has finite rank. In view of [FJ74, Thm. 9.1], this forces A to be zero, i.e., G is a torus. (ii) This follows readily from Proposition 2.5 and Corollary 2.13.

5.3 Product decompositions

In this subsection, we first prove the remaining assertions (ii) and (iii) of Theorem 1. Then we describe the isogeny category \underline{C} in characteristic 0, and its projective or injective objects in arbitrary characteristics.

Proposition 5.9. (i) If k is perfect, then the product functor M × U → L yields an equivalence of categories.

(ii) For any field k, the product functor \underline{T} × \underline{U} → \underline{L} yields an equivalence of categories.

Proof. (i) This follows readily from Theorem 2.4 and Proposition 2.5. (ii) This is a consequence of (i) in view of Theorem 3.11.

Proposition 5.10. If char(k) > 0, then the product functor \underline{S} × \underline{U} → \underline{C} yields an equivalence of categories.

Proof. Let G be a smooth connected algebraic group. By Corollary 2.13, there exists a finite subgroup F ⊂ G such that G/F ≅ S × U, where S is a semiabelian variety, and U is unipotent. Thus, the product functor Π is essentially surjective. Next, let S_{1}, S_{2} be semi-abelian varieties, and U_{1}, U_{2} smooth connected unipotent groups. We check that Π induces an isomorphism

Hom\underline{C}(S_{1}, S_{2})×Hom\underline{C}(U_{1}, U_{2}) ⟶ Hom\underline{C}(S_{1} ×U_{1}, S_{2} ×U_{2}), (\underline{φ}, \underline{ψ}) ⟼ \underline{φ}×\underline{ψ}.

Assume that \underline{φ} × \underline{ψ} = 0. Choose representatives φ : S_{1} → S_{2}/S′_{2} and ψ : U_{1} → U_{2}/U′_{2}, where S′_{2}, U′_{2} are finite. Then φ × ψ : S_{1} × U_{1} → (S_{2} × U_{2})/(S′_{2} × U′_{2}) has finite image, and hence is zero by Lemma 2.1. So \underline{φ} = \underline{ψ} = 0. Let \underline{γ} ∈ Hom\underline{C}(S_{1} × U_{1}, S_{2} × U_{2}) be represented by γ : S_{1} × U_{1} → (S_{2} × U_{2})/F, where F is finite. Then F ⊂ S′_{2} × U′_{2} for some finite subgroups S′_{2} ⊂ S_{2}, U′_{2} ⊂ U_{2}. Thus, we may assume that F = S′_{2} × U′_{2}. Then the composite morphisms

S_{1} ⟶ S_{1} × U_{1} ⟶^{γ} S_{2}/S′_{2} × U_{2}/U′_{2} ⟶ U_{2}/U′_{2},

U_{1} ⟶ S_{1} × U_{1} ⟶^{γ} S_{2}/S′_{2} × U_{2}/U′_{2} ⟶ S_{2}/S′_{2}

are zero by Lemma 2.1 and Proposition 2.5. Thus, γ = φ × ψ for some morphisms φ : S_{1} → S_{2}/S′_{2}, ψ : U_{1} → U_{2}/U′_{2}.

Combining Propositions 5.8 (i) and 5.10, we obtain readily:

Corollary 5.11. If k is locally finite, then the product functor

\underline{T} × \underline{A} × \underline{U} ⟶ \underline{C}

yields an equivalence of categories.

Remarks 5.12. (i) With the notation of the above corollary, each of the categories \underline{T} , \underline{A}, \underline{U} admits a description of its own. By Proposition 4.1, \underline{T} is equivalent to the category of ℚ-vector spaces equipped with an automorphism of finite order. Also, the isomorphism classes of abelian varieties over a finite field are classified by the Honda-Tate theorem (see [Ho68, Tat66]); their endomorphism rings are investigated in [Wa69]. Finally, the structure of \underline{U} (obtained in [DG70, §V.3.6.7]) has been described in Remark 3.12. (ii) Combining Lemma 2.2, Theorem 2.4 and Lemma 3.10, one may show that the product functor M/IM × U/IU → L/I yields an equivalence of categories. Here I denotes the category of infinitesimal algebraic groups, and IM (resp. IU) the full subcategory of infinitesimal groups of multiplicative type (resp. unipotent).

Next, assume that char(k) = 0. Then every algebraic group is isogenous to a fibered product E ×_{A} S, where E is a vector extension of the abelian variety A, and S is semi-abelian with Albanese variety isomorphic to A (see e.g. Remark 2.10). This motivates the consideration of the fibered product \underline{V} ×\underline{A} \underline{S}: this is the category with objects the triples (E, S, \underline{f}), where E is a vector extension of an abelian variety, S a semi-abelian variety, and \underline{f} : A(E) → A(S) an \underline{A}- isomorphism. The morphisms from (E_{1}, S_{1}, \underline{f}_{1}) to (E_{2}, S_{2}, \underline{f}_{2}) are those pairs of \underline{C}-morphisms \underline{u} : E_{1} → E_{2}, \underline{v} : S_{1} → S_{2} such that the square

A(E_{1}) ⟶^{\underline{A}(\underline{u})} A(E_{2})
\underline{f}_{1}↓  \underline{f}_{2}↓
A(S_{1}) ⟶^{\underline{A}(\underline{v})} A(S_{2})

commutes in \underline{A}.

Proposition 5.13. If char(k) = 0, then \underline{C} is equivalent to \underline{V} ×\underline{A} \underline{S}.

The proof is similar to that of Proposition 5.10, and will be omitted. Note that the descriptions of \underline{V} and \underline{S} in terms of linear algebra, obtained in §5.1 and §5.2, can also be reformulated in terms of fibered products of categories. Returning to an arbitrary field k, we obtain:

Theorem 5.14. The projective objects of \underline{C} are:

• the linear algebraic groups, if char(k) = 0.

• the semi-abelian varieties, if k is locally finite.

• the tori, if char(k) > 0 and k is not locally finite.

Proof. Let G be a smooth connected algebraic group. As a consequence of Theorem 2.9 and Proposition 5.10, we have an exact sequence in \underline{C}

0 ⟶ U ⟶ G ⟶ S ⟶ 0,

where U is smooth, connected, and unipotent, and S is a semi-abelian variety. If G is projective in \underline{C}, then Ext1 ; \underline{C}(G, T′) = 0 for any torus T′. Since Hom\underline{C}(U, T′) = 0 (as a consequence of Proposition 2.5) and Ext1 ; \underline{C}(U, T′) = 0 (by Lemma 4.10), the long exact sequence for Ext groups yields Ext1 ; \underline{C}(S, T′) = 0 as well. By arguing as in the proof of Proposition 5.8, this forces either A(S) to be zero, or k to be locally finite. If A(S) = 0, then G is linear, and hence G ≅ T × U in \underline{C} by Proposition 5.9. Moreover, tori are projective in \underline{C} by Proposition 4.4; thus, we may assume that G is unipotent. If char(k) = 0, then every unipotent group is projective, as follows e.g. from Lemma 4.10. If char(k) > 0 and G ≠ 0, then there exists an exact sequence

0 ⟶ H ⟶ G ⟶ 𝔾_{a} ⟶ 0

in C. Since Ext1 ; \underline{C}(G, 𝔾_{a}) = 0 = Ext2 ; \underline{C}(H, 𝔾_{a}), it follows that Ext1 ; \underline{C}(𝔾_{a}, 𝔾_{a}) = 0. But this contradicts Example 3.4 (ii), hence G = 0. On the other hand, if k is locally finite, then G ≅ T × A × U in \underline{C} (by Corollary 5.11) and it follows as above that U is zero. Conversely, every semi-abelian variety is projective in \underline{C}, by Corollary 5.11 again.

Corollary 5.15. If char(k) > 0, then C has no non-zero projective objects.

Proof. Let G be a projective object of C. By the claim in the proof of Theorem 2.9 (iii), the (abstract) group Hom_{C}(G, H) is divisible for any divisible group H. By arguing as in that proof, it follows that G is linear. Also, for any torus T, the group Hom_{C}(G, T) is finitely generated and divisible, hence zero.

Next, we show that G is connected. Indeed, the quotient G/G^{0} is finite and étale, hence contained in a torus T. As Hom_{C}(G, T) = 0, this yields the assertion. In view of Theorem 2.4, we obtain an exact sequence

0 ⟶ M ⟶ G ⟶ U ⟶ 0,

where M is of multiplicative type, and U is unipotent and connected. We now show that U is projective in the unipotent category U. Indeed, given an exact sequence U_{1} → U_{2} → 0 in U and a morphism φ : U → U_{2}, we may lift the composition ψ : G → U → U_{2} to a morphism γ : G → U_{1}. Then γ(M) = 0 (Proposition 2.5) and hence γ factors through a morphism δ : U → U_{1}, which lifts φ. Let F be an infinitesimal subgroup of U such that U/F is smooth; then U/F is an object of U, and one easily checks by using Proposition 3.5 that U/F is projective in \underline{U}. In view of Proposition 5.10 and Theorem 5.14, it follows that U/F is zero, hence U is infinitesimal. If U ≠ 0 then there exists an epimorphism U → α_{p} and hence a non-zero morphism U → A for some (supersingular) abelian variety A. Since Hom_{C}(G, A) = 0, this yields a contradiction. Thus, U = 0, i.e., G is of multiplicative type, hence contained in a torus, hence zero.

Theorem 5.16. The injective objects of \underline{C} are the semi-abelian varieties if k is locally finite, and the abelian varieties otherwise.

Proof. Let G be an injective object of \underline{C}, and

0 ⟶ G_{1} ⟶ G ⟶ G_{2} ⟶ 0

an exact sequence in \underline{C}. Since hd(\underline{C}) = 1, the map Ext1 ; \underline{C}(H, G) → Ext1 ; \underline{C}(H, G_{2}) is an epimorphism for any object H of \underline{C}. Thus, every quotient of G is injective. In particular, so is the largest semi-abelian quotient S (in view of Remark 2.10 and Theorem 2.11). We now adapt the argument of Corollary 5.6. Recall that S lies in a unique extension

0 ⟶ T ⟶ S ⟶ A ⟶ 0,

classified by a Γ-equivariant morphism c : X(T) → Â(k_{s}). Since S is injective in \underline{C}, we have Ext1 ; \underline{C}(B, S) = 0 for any abelian variety B, and hence the connecting homomorphism ∂ : Hom\underline{C}(B, A) ⟶ Ext1 ; \underline{C}(B, T) is surjective. Under the isomorphisms

Hom\underline{C}(B, A) ≅ ℚ ⊗_{ℤ} Hom_{C}(B, A), Ext1 ; \underline{C}(B, T) ≅ ℚ ⊗_{ℤ} Hom^{Γ}(X(T), B̂(k_{s}))

(Proposition 3.6 and §5.2), the map ∂ is identified with the composition of the natural map

ℚ ⊗_{ℤ} Hom_{C}(B, A) ⟶ ℚ ⊗_{ℤ} Hom^{Γ}(Â(k_{s}), B̂(k_{s})) with the map

γ : ℚ ⊗_{ℤ} Hom^{Γ}(Â(k_{s}), B̂(k_{s})) ⟶ ℚ ⊗_{ℤ} Hom^{Γ}(X(T), B̂(k_{s})), u ⟼ u ∘ c.

Next, consider a free abelian group M of finite rank, equipped with a continuous action of Γ. Then for any abelian variety B, the tensor product B_{k_{s}} ⊗_{ℤ} M is a k_{s}-abelian variety equipped with the diagonal Γ-action, and descends to a unique k-abelian variety B(M). Replacing B with B(M), it follows that the corresponding map

∂(M) : ℚ ⊗_{ℤ} Hom_{C_{ks}}^{Γ} (B_{k_{s}} ⊗_{ℤ} M, A_{k_{s}}) ⟶ ℚ ⊗_{ℤ} Hom^{Γ}(X(T) ⊗_{ℤ} M, B̂(k_{s}))

is surjective as well. As ℚ⊗_{ℤ}Hom_{C_{ks}} (B_{k_{s}}, A_{k_{s}}) and ℚ⊗_{ℤ}Hom(X(T), B̂(k_{s})) are direct sums of continuous, finite-dimensional Γ-modules, and M is arbitrary, it follows that the natural map

Hom_{C_{ks}} (B_{k_{s}}, A_{k_{s}}) ⟶ Hom(X(T), B̂(k_{s}))

is surjective over the rationals. In the case where k is not locally finite, recall from [FJ74, Thm. 9.1] that the group B̂(k_{s}) has infinite rank for any non-zero abelian variety B. Since Hom_{C_{ks}} (B_{k_{s}}, A_{k_{s}}) has finite rank, it follows that X(T) = 0, i.e., S is an abelian variety. We now distinguish between two subcases. If char(k) = 0, then G is a vector extension of A, and hence is injective in the category \underline{V}. Thus, G = A in view of Corollary 5.6. If char(k) > 0, then G ≅ A × U in \underline{C} for some split unipotent group U, which must be injective in \underline{C}. If in addition U ≠ 0, then it follows that 𝔾_{a} (a quotient of U) is injective in \underline{C} as well. But Ext1 ; \underline{C}(𝔾_{a}, 𝔾_{a}) ≠ 0 in view of Example 3.4, a contradiction. Thus, G = A again. Finally, in the case where k is locally finite, we have G ≅ T × A × U in \underline{C} with an obvious notation. As above, we obtain that U = 0; on the other hand, T and A are injective in \underline{C} by Corollary 5.11. This completes the proof.

Remark 5.17. The category C has no non-zero injective objects. This result should be well-known, but we could not find it in the literature; also, it does not seem to follow from Theorem 5.16, as the relation between injective objects in C and \underline{C} is unclear. So we sketch a direct proof: let G be an injective object of C. For any positive integer n which is prime to char(k), the n-torsion subgroup G[n] is finite and étale, hence isomorphic to a subgroup of some torus T_{n}. The inclusion G[n] ⊂ G extends to a morphism T_{n} → G, and hence G[n] is contained in the maximal torus T (G). Likewise, G[n] is isomorphic to a subgroup of an abelian variety A_{n}, and hence is contained in the largest abelian subvariety of G. Since this abelian variety intersects T (G) along a finite subgroup, it follows that G[n] = 0 for n ≫ 0. If char(k) = 0, then there is an exact sequence

0 ⟶ R_{u}(G) ⟶ G ⟶ H ⟶ 0,

where H^{0} is a semi-abelian variety. Since n_{R_{u}(G)} is an isomorphism, we have G[n] ≅ H[n] for all n; it follows that H is finite. Thus, G = R_{u}(G) + F for some finite group F, which embeds into some torus T. As above, it follows that F = 0, i.e., G is unipotent. If G ≠ 0, then Ext_{C}^{1}(A, G) ≠ 0 for any abelian variety A, a contradiction. If char(k) = p > 0, then there is an exact sequence

0 ⟶ H ⟶ G ⟶ U ⟶ 0,

where H^{0} is again a semi-abelian variety, H/H^{0} is of multiplicative type, and U is unipotent. Thus, H[n] ≅ G[n] for all n prime to p, and hence H is finite. It follows as above that H = 0, i.e., G is unipotent. If G ≠ 0, then G contains a copy of α_{p}, which embeds into some abelian variety; this yields a contradiction.

Acknowledgements. I warmly thank Claire Amiot, Brian Conrad, Hélène Esnault, Stéphane Guillermou, Bruno Kahn, Giancarlo Lucchini Arteche, Gaël Rémond, Jean-Pierre Serre, and Catharina Stroppel for very helpful discussions or e-mail exchanges. Special thanks are due to an anonymous referee for a careful reading and valuable comments.

References

[BLR90] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse Math. Grenzg. (3) 21, Springer, 1990.

[Br15] M. Brion, On extensions of algebraic groups with finite quotient, Pacific J. Math. 279 (2015), 135–153.

[Br17] M. Brion, Some structure theorems for algebraic groups, Proc. Symp. Pure Math. 94 (2017), 53–125.

[Ch55] W.-L. Chow, Abelian varieties over function fields, Trans. Amer. Math. Soc. 78 (1955), 253–275.

[Co06] B. Conrad, Chow’s K/k-image and K/k-trace, and the Lang-Néron theorem, Enseign. Math. (2) 52 (2006), no. 1-2, 37–108.

[CGP15] B. Conrad, O. Gabber, G. Prasad, Pseudo-reductive groups. Second edition, New Math. Monogr. 26, Cambridge Univ. Press, Cambridge, 2015.

[CP15] B. Conrad, G. Prasad, Classification of pseudo-reductive groups, Ann. of Math. Stud. 191, Princeton Univ. Press, 2015.

[DG70] M. Demazure, P. Gabriel, Groupes algébriques, Masson, Paris, 1970.

[FJ74] G. Frey, M. Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc. (3) 28 (1974), 112–128.

[Ga62] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

[GZ67] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Ergeb. Math. Grenzgeb. 35, Springer-Verlag, New York, 1967.

[Ha01] D. Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), no. 2, 381–398.

[Ho68] T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95.

[Kr75] H. Kraft, Kommutative algebraische Gruppen und Ringe, Lecture Notes in Math. 455, Springer-Verlag, Berlin-New York, 1975.

[LA15] G. Lucchini Arteche, Extensions of algebraic groups with finite quotient, preprint, arXiv:1503:06582.

[MM74] B. Mazur, W. Messing, Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Math. 370, Springer-Verlag, Berlin-New York, 1974.

[MW93] D. Masser, G. Wüstholz: Isogeny estimates for abelian varieties, and finiteness theorems, Ann. Math. (2) 137 (1993), no. 3, 459–472.

[Ma10] N. Mazzari, Cohomological dimension of Laumon 1-motives up to isogenies, J. Théor. Nombres Bordeaux 22 (2010), no. 3, 719–726.

[Mi70] J. S. Milne, The homological dimension of commutative group schemes over a perfect field, J. Algebra 16 (1970), 436–441.

[Mi86] J. S. Milne, Abelian varieties, in: Arithmetic geometry, 103–150, Springer-Verlag, New York, 1986.

[Oo64] F. Oort, Yoneda extensions in abelian categories, Math. Annalen 153 (1964), no. 3, 227–235.

[Oo66] F. Oort, Commutative group schemes, Lecture Notes in Math. 15, Springer-Verlag, Berlin-New York, 1966.

[Or04] F. Orgogozo, Isomotifs de dimension inférieure ou égale à un, Manuscripta Math. 115 (2004), no. 3, 339–360.

[Ra70] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Math. 119, Springer-Verlag, Berlin-New York, 1970.

[Ro56] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443.

[Ro58] M. Rosenlicht, Extensions of vector groups by abelian varieties, Amer. J. Math. 80 (1958), 685–714.

[Ru70] P. Russell, Forms of the affine line and its additive group, Pacific J. Math. 32 (1970), 527–539.

[SGA3] M. Demazure, A. Grothendieck, Schémas en groupes I, II, III (SGA 3), Springer Lecture Notes in Math. 151, 152, 153 (1970); revised version edited by P. Gille and P. Polo, vols. I and III, Soc. Math. de France, Paris, 2011.

[Sc72] C. Schoeller, Groupes affines, commutatifs, unipotents sur un corps non parfait, Bull. Soc. Math. France 100 (1972), 241–300.

[Se59] J.-P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959.

[Se60] J.-P. Serre, Groupes proalgébriques, Publ. Math. IHÉS 7 (1960).

[SP16] Stacks Project, http://stacks.math.columbia.edu, 2016.

[Tat66] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144.

[Tak75] M. Takeuchi, On the structure of commutative affine group schemes over a non-perfect field, Manuscripta Math. 16 (1975), no. 2, 101–136.

[To13] B. Totaro, Pseudo-abelian varieties, Ann. Sci. Éc. Norm. Sup. (4) 46 (2013), no. 5, 693–721.

[Wa69] W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521–560.

Michel Brion

Université Grenoble Alpes Institut Fourier 100 rue des maths 38610 Gières France Michel.Brion@ujf-grenoble.fr 726

Documenta Mathematica 22 (2017)