On the fundamental groups of commutative algebraic groups

Consider the abelian category ${\mathcal C}$ of commutative group schemes of finite type over a field $k$, its full subcategory ${\mathcal F}$ of finite group schemes, and the associated pro category ${\rm Pro}({\mathcal C})$ (resp. ${\rm Pro}({\mathcal F})$) of pro-algebraic (resp. profinite) group schemes. When $k$ is perfect, we show that the profinite fundamental group $\varpi_1 : {\rm Pro}({\mathcal C}) \to {\rm Pro}({\mathcal F})$ is left exact and commutes with base change under algebraic field extensions; as a consequence, the higher profinite homotopy functors $\varpi_i$ vanish for $i \geq 2$. Along the way, we describe the indecomposable projective objects of ${\rm Pro}({\mathcal C})$ over an arbitrary field $k$.


Introduction
Every real Lie group G gives rise to two exact sequences where G 0 denotes the identity component, G its universal cover, and π 0 (G), π 1 (G) are discrete groups; moreover, the second homotopy group π 2 (G) vanishes. This classical result has a remarkable analogue for commutative algebraic groups over an algebraically closed field k, as shown by Serre and Oort via a categorical approach (see [Ser60,Oor66]). More specifically, consider the category C of commutative k-group schemes of finite type, and the full subcategory F of finite group schemes; then C is an artinian abelian category, and F is a Serre subcategory. Let Pro(C) (resp. Pro(F)) denote the associated pro-category, consisting of pro-algebraic (resp. profinite) group schemes; recall that these categories have enough projectives, and C (resp. F) is equivalent to the full subcategory of Pro(C) (resp. Pro(F)) consisting of artinian objects. Assigning to each object of Pro(C) its largest profinite quotient yields a right exact functor The construction of the "profinite homotopy functors" i makes sense over an arbitrary field k; it is easy to extend the above exact sequence to this setting. The main result of this paper generalizes those of Serre and Oort as follows: Theorem 1.1. -When k is perfect, the functor 1 : Pro(C) → Pro(F) is left exact and commutes with base change under algebraic field extensions. As a consequence, the higher profinite homotopy functors i vanish for i 2.
Our approach is independent of the general theory of étale homotopy groups of schemes (see e.g. [AM69,Fri82]). We rather develop an ad hoc theory of homotopy groups in the setting of pairs (A, B), where A is an artinian abelian category, and B a Serre subcategory of A. For this, we build on constructions and results of Gabriel (see [Gab62,Chap. III]) and on further developments in [Bri19], recalled in Subsection 2.1. These may be conveniently formulated in terms of orthogonal or perpendicular categories (see [BR07,II.2] and [GL91] for these two notions). Homotopy groups are introduced in Subsection 2.2, which generalizes results of Demazure and Gabriel on the profinite homotopy groups of affine group schemes (see [DG70,V.3.3]). Subsection 2.4 investigates compatibility properties of homotopy groups in the presence of a Serre subcategory C of B.
In Section 3, we first apply this formalism to the category C of (commutative) algebraic groups, and its full subcategory L of linear algebraic groups, over an arbitrary field k; then Pro(L) is equivalent to the category of affine k-group schemes, in view of [DG70,V.2.2.2]. The resulting homotopy functor π C,L 1 turns out to be left exact (Proposition 3.3). We then consider the pair (C, F), and obtain the left exactness of 1 = π C,F 1 when k is perfect; in addition, we show that the profinite universal cover G has homological dimension at most 1 for any G ∈ Pro(C) (Theorem 3.5).
When G is an abelian variety over an arbitrary field k, we construct a minimal projective resolution of G (Theorem 3.10). We also describe the projective objects of Pro(C) (Proposition 3.11); for this, we use results of Demazure and Gabriel on the projectives of Pro(L) over a perfect field (see [DG70,V.3.7]), combined with properties of the isogeny category C/F (see [Bri17]) . We then show that the profinite homotopy functors commute with base change under separable algebraic field extensions (Proposition 3.15), thereby completing the proof of the main result.
As an application of the above developments, we obtain a spectral sequence à la Milne (see [Mil70]), which relates the extension groups in C and in the corresponding category over a Galois extension of k. Further applications, to the structure of homogeneous vector bundles over abelian varieties, are presented in [Bri18].
When the ground field k has characteristic p > 0, the prime-to-p part (p ) 1 of the profinite fundamental group commutes with arbitrary field extensions, and hence is left exact (Proposition 3.17). But over an imperfect field k, the functors 0 , 1 do not commute with purely inseparable field extensions, nor does the pro-étale p-primary part of 1 (see Remarks 3.19, 3.20 and 3.21). In this setting, it seems very likely that 2 is nontrivial, but we have no explicit example for this; also, the profinite fundamental group scheme 1 deserves further investigation, already for smooth connected unipotent groups.
Finally, it would be interesting to relate the above (affine, profinite or pro-étale) fundamental groups with further notions of fundamental group schemes considered in the literature. In this direction, note that the profinite fundamental group of any abelian variety A coincides with Nori's fundamental group scheme (defined in [Nor76,Nor82]), as shown by Nori himself in [Nor83]. Also, when k is algebraically closed, the affine fundamental group of A coincides with its S-fundamental group scheme introduced by Langer in [Lan11], as follows from [Lan12, Thm. 6.1].
the opposite category is a Grothendieck category. Moreover, A is equivalent to the Serre subcategory of Pro(A) consisting of artinian objects (see [DG70,V.2. 3.1]). Let B be a Serre subcategory of A; then we may view Pro(B) as a Serre subcategory of Pro(A), stable under inverse limits (see [Bri19,Lem. 2.11] ). We denote by ⊥ Pro(B) the full subcategory of Pro(A) with objects those X such that Hom Pro(A) (X, Y ) = 0 for all Y ∈ Pro(B) (this is the left orthogonal subcategory to Pro(B) in Pro(A) in the sense of [BR07, II.1]).
(2) X has a smallest subobject (2). -Let (X i ) i∈I be a family of subobjects of X such that X/X i ∈ Pro(B) for all i. Then X/(∩ i∈I X i ) is a subobject of i∈I X/X i , and hence an object of Pro(B). This shows the existence of X B .
If there exists a nonzero morphism f : X B → Y for some Y ∈ Pro(B), then X := Ker(f ) is a subobject of X B such that X B /X is a nonzero object of Pro(B). It follows that X/X ∈ Pro(B), contradicting the minimality of X B . So X B ∈ ⊥ Pro(B). ( . If in addition f is essential and Y ∈ ⊥ Pro(B), then Y = f (X B ) and hence X B = X.
In view of Lemma 2.1, every X ∈ Pro(A) lies in a unique exact sequence where X B ∈ ⊥ Pro(B) and X B ∈ Pro(B). Moreover, every f ∈ Hom Pro(A) (X, Y ) induces compatible morphisms This defines a functor Since Hom Pro(A) (X B , Y ) = 0 for any Y ∈ Pro(B), the natural map is an isomorphism. In other words, π 0 is left adjoint to the inclusion of Pro(B) in Pro(A). As a consequence, π 0 is right exact and sends any projective object of Pro(A) to a projective object of Pro(B).
Proof. -Consider a filtered inverse system (X i ) of objects of Pro(A). This yields a filtered inverse system (X B i ) of objects of ⊥ Pro(B); moreover, we have an isomorphism lim is an object of Pro(B); this yields the assertion.
We denote by which also commutes with inverse limits and sends projectives to projectives Hom Pro(A) (X, Y ) = 0 = Ext 1 Pro(A) (X, Y ) for all Y ∈ Pro(B) (these are the objects of the left perpendicular subcategory to Pro(B) in Pro(A), as defined in [GL91]). Moreover, for any X ∈ Pro(A), the adjunction map CQ(X) → X has its kernel and cokernel in Pro(B) (see [GL91,III.2.Prop. 3]). This yields an exact sequence in Pro(A) where we set X = X A,B := CQ(X) (in particular, X ∈ ⊥ Pro(B)), and we have Y 0 , Y 1 ∈ Pro(B). Note that the long exact sequence (2.3) depends functorially on X. Also, note the natural isomorphism for any Y ∈ Pro(A). In particular, if X, Y ∈ A then Lemma 2.3. -With the above notation, we have ρ( X) = X B and the induced epimorphism η : X → X B is essential. Also, there are functorial isomorphisms

TOME 3 (2020)
Proof. -In view of (2.2) and the exact sequence we obtain the vanishing of Hom Pro(A) (ρ( X), Y ) and an isomorphism It remains to show that η : X → X B is essential. Let Z be a subobject of X such that the composition Z → X → X B is an epimorphism. Then X = Y 1 + Z and hence Lemma 2.4. -With the notation of the exact sequence (2.3), the following conditions are equivalent for X ∈ Pro(A): ( Let X ∈ Pro(A) satisfy (3), and Y ∈ Pro(B). Then Hom Pro(A) (X, Y ) = 0 by Lemma 2.1 (1). Consider an essential epimorphism f : P → X, where P ∈ Pro(A) is projective (such a projective cover of X exists in view of [Gab62, II.6.Thm. 2]). Then P ∈ ⊥ Pro(B) by Lemma 2.1 (3). So the exact sequence In particular, Hom Pro(A) (X , Y ) = 0 for all Y ∈ B. It follows that X ∈ ⊥ Pro(B) by using Lemma 2.1 (1). Thus, Ext 1 Pro(A) (X, Y ) = 0 for all Y ∈ Pro(B).
for all X ∈ Pro(A) and i 1.
Proof. -The exact sequence (2.1) yields an isomorphism Q(X B ) → Q(X) in Pro(A)/ Pro(B), and hence an isomorphism CQ(X B ) → CQ(X) in Pro(A). In turn, this yields an isomorphism Y 1 (X B ) → Y 1 (X) in Pro(B), where Y 1 (X B ) denotes the kernel of the adjunction map CQ(X B ) → X B , and Y 1 (X) is defined similarly. Thus, we may assume that X ∈ ⊥ Pro(B). We then have an exact sequence which yields an exact sequence Moreover, π 0 ( X) = 0 by Lemma 2.4. So it suffices to show that π 1 ( X) = 0.
In view of Lemmas 2.3 and 2.6, the exact sequence (2.3) can be rewritten in a more suggestive way. Namely, with the assumption of Lemma 2.5, we have an exact sequence for any X ∈ Pro(A): In particular, when X ∈ ⊥ Pro(B), we obtain an extension Using Lemmas 2.3 and 2.6 again, this yields in turn: Corollary 2.7. -With the assumption of Lemma 2.5, let X ∈ ⊥ Pro(B) and Y ∈ Pro(B). Then Hom Pro(A) (π 1 (X), Y ) Pro(A) (X, Y ) via pushout by the extension (2.8). TOME 3 (2020) where the vertical arrows are isomorphisms. As a consequence, we may replace X with X B , and assume that π 0 (X) = 0 = Y 0 . Also, the induced morphism Q(X ) → Q(X) is an isomorphism, and hence so is CQ(X ) → CQ(X) = X. Since the adjunction CQ(X ) → X is an isomorphism, this yields an isomorphism X ∼ = X . Thus, we may further assume that (2.9) is of the form Then the associated map Hom is an isomorphism for all Y ∈ Pro(B), by Lemma 2. 4. In view of the uniqueness of the universal extension of X by an object of Pro(B), this completes the proof.
Next, we obtain two reformulations of the left exactness of the functor π 1 : Lemma 2.9. -With the assumption of Lemma 2.5, the following conditions are equivalent: (1) The cosection functor C : Pro(A)/ Pro(B) → Pro(A) is exact.

Proof. -
(1) ⇒ (2). -Consider an exact sequence in Pro(A). Then we have a commutative diagram of exact sequences In view of the exact sequence (2.7) and its analogues for X 1 , X 2 , the snake lemma yields an exact sequence In particular, π 1 is left exact. (2) ⇒ (3). -This is obtained by a standard argument that we recall for completeness. Let X ∈ Pro(A) and choose a projective cover As π i (P ) = 0 for all i 1, we obtain isomorphisms π i (X) ∼ = → π i−1 (X ) for all i 2. Since X is a subobject of P , we have π 1 (X ) = 0 by left exactness, hence π 2 (X) = 0. Iterating this argument completes the proof.
Finally, we record an easy and useful divisibility property of homotopy groups. For any X ∈ Pro(A) and any integer n, we denote by n X ∈ End A (X) the multiplication by n, and by X[n] its kernel. We say that X is divisible (resp. uniquely divisible) if n X is an epimorphism (resp. an isomorphism) for any n 1.
Lemma 2.10. -With the assumption of Lemma 2.5, let X be an object of Pro(A). Assume that X is divisible and X[n] ∈ Pro(B) for any n 1 (in particular, π i (X[n]) = 0 for any such n and any i 1). Then X and the π i (X) (i 2) are uniquely divisible. Moreover, there is an exact sequence for any n 1.
Proof. -By assumption, we have an exact sequence for any n 1. Thus, n X induces an automorphism of Q(X), and hence of CQ(X) = X. In other words, X is uniquely divisible. The remaining assertions follow from the homotopy exact sequence associated with (2.10).

Structure of projective objects
In this subsection, we consider an artinian abelian category A and a Serre subcategory B such that every projective object of Pro(B) is projective in Pro(A). Our aim is to describe the projectives of Pro(A) in terms of those of Pro(B) and Pro(A)/ Pro(B) ∼ = Pro(A/B). We first obtain a generalization of [DG70, V.3. 3.9]: Lemma 2.11. -For any projective object P ∈ Pro(A), there is an isomorphism P ∼ = P B ⊕ π 0 (P ) which is compatible with γ P : P → π 0 (P ). Moreover, P ∼ = P B .
Proof. -Recall that π 0 is left adjoint to the inclusion of Pro(B) in Pro(A). It follows that π 0 (P ) is projective in Pro(B), and hence in Pro(A) as well. This yields a compatible isomorphism P ∼ = P B ⊕ π 0 (P ). In particular, P B is projective, and hence in the essential image of C by (2.2). So the adjunction map CQ(P B ) → P B is an isomorphism. As CQ(P B ) ∼ = → CQ(P ) = P , this completes the proof.
Proof. -We may assume that X is projective. By Lemma 2.11, we may then choose an isomorphism X ∼ = X ⊕ π 0 (X) compatibly with γ X : X → π 0 (X). Since The above corollary asserts that the pair ( Next, recall from [DG70, V.2.4] that every projective object of Pro(A) is a product of indecomposable projectives, unique up to reordering; moreover, the indecomposable projectives are projective covers of objects of A. Also, given X ∈ Pro(A) such that Q(X) is projective in Pro(A/B), the adjunction map ρ : X = CQ(X) → X is the projective cover of X (indeed, C sends projectives to projectives, and ρ is essential by Lemma 2.3). Together with Lemma 2.11, this yields the following result (see also [Gab62,III.3

.Cor. 2]):
The latter indecomposable projectives can be constructed as follows: (1) Consider an exact sequence in Pro(A), Then f is essential if and only if Z ∈ Pro(B) and Y ∈ ⊥ Pro(B).
(2) Assume that Q(X) is projective in Pro(A)/ Pro(B). Then the essential epi- where Ker(f ) ∈ B, form a filtered inverse system with limit the projective cover of X in Pro(A).
This yields an exact sequence (where the morphism on the right is induced by f ), and by q : X → X/f (W ) the quotient morphism in Pro(A). Then p represents the identity endomorphism of X in Pro(A)/ Pro(B); thus, p − q represents zero there. Using again the assumption that X ∈ ⊥ Pro(B), it follows that p − q is zero in Pro(A). In particular, the composition h(X) → Y /W → X is an epimorphism. Since f is essential, h must be an epimorphism as well. So g is an isomorphism in Pro(A)/ Pro(B), hence Z/W ∈ Pro(B). We conclude that Z ∈ Pro(B).
Conversely, assume that Z ∈ Pro(B) and , and hence is zero. We conclude that f is essential.
(2). -Consider two exact sequences where f 1 , f 2 are essential and Z 1 , Z 2 ∈ B. Then the induced morphism is an epimorphism with kernel Z 1 × Z 2 . In view of (1), it follows that the composition Y B → Y → X is an essential epimorphism. Thus, these essential epimorphisms form a filtered inverse system. Given such an essential epimorphism f : Y → X, the map ρ : X → X lifts to a morphism ϕ Y : X → Y . Moreover, ϕ Y is unique (since Ker(f ) ∈ Pro(B) and X ∈ ⊥ Pro(B)), and is an epimorphism as well. So we obtain an epimorphism with an obvious notation. To show that ϕ is a monomorphism, consider the family (K i ) of subobjects of Ker(ρ) such that Ker(ρ)/K i ∈ B. Then X/K i ∈ A and ρ factors through an essential epimorphism X/K i → X; moreover, the corresponding morphism ϕ i : X → X/K i is just the quotient morphism. Since ∩K i is zero, this completes the proof.

Compatibility properties
Throughout this subsection, we consider an artinian abelian category A, a Serre subcategory B such that the pair (A, B) satisfies the lifting property, and in addition a Serre subcategory C of B. We first relate the homotopy functors associated to the three pairs (A, B), (B, C) and (A, C): (1) There is a natural isomorphism π A,C 0 (X) -This follows readily from the definitions.
(2). -Recall that π A,B 0 : Pro(A) → Pro(B) sends projectives to projectives; also, every projective in Pro(B) is obviously acyclic for π B,C 0 . In view of (1), this yields a Grothendieck spectral sequence as stated.
Remark 2. 16. -When X ∈ B, the above spectral sequence yields isomorphisms for all i 0, in view of Lemma 2.5. Alternatively, these isomorphisms follow from the obvious equality π B,C 0 (X) = π A,C 0 (X), since every projective object of Pro(B) is projective in Pro(A).
On the other hand, when X ∈ ⊥ Pro(B), the first terms of the spectral sequence yield a natural isomorphism . This can also be seen directly: consider the universal extension of X by an object of 1 (X). Then one may readily check that the induced exact sequence Next, let 0 → X 1 → X → X 2 → 0 be an exact sequence in A/C. Then there exists a commutative diagram in that category We now check that (A/C, B/C) satisfies the lifting property. Let ϕ : X → Y be an epimorphism in A/C. In view of [Bri19, Lem. 2.7] again, replacing Y with an isomorphic object in A/C, we may assume that ϕ is represented by a morphism f : X → Y in A; then Coker(f ) is an object of C by [Gab62, III.1. Lem. 2] again. Next, we may replace X, Y with X C , Y C , and hence assume that f is an epimorphism in A. Then there exists a subobject Y of X such that Y ∈ B and the composition Y → X → Y is an epimorphism in A, hence in A/C.  Let X ∈ Pro(A) and consider the exact sequence . This sequence is still exact in Pro(A/C); thus, it suffices to show that X B ∈ ⊥ Pro(B/C). In view of Lemma 2.1, it suffices in turn to show that every morphism ϕ : In Pro(A), we have X B = lim ← X i , where X i ∈ A and the projections X B → X i are epimorphisms. Hence this also holds in Pro(A/C). Since

The affine fundamental group
Let k be a field. As in the introduction, we consider the artinian abelian category C of commutative k-group schemes of finite type, and the associated pro-category Pro(C) of pro-algebraic groups. We denote by L the full subcategory of C with objects the affine (or equivalently, linear) algebraic groups. Then L is a Serre subcategory of C, as follows from fpqc descent (see e.g. [Sta18,34.20.18]). Also, recall that Pro(L) is equivalent to the category of commutative affine k-group schemes.
By the results of Subsection 2.1, every object of Pro(C) has a largest affine quotient; this yields a right exact functor which commutes with filtered inverse limits and extends the affinization functor C → L considered for example in [DG70, III.3.8]. The results of Subsection 2.2 also apply to this setting, in view of the following observation: Proof. -Let G ∈ C. By a variant of Chevalley's structure theorem for algebraic groups (see [Bri17,Thm. 2.3]) that we will use repeatedly, there is an exact sequence   -Recall that C commutes with inverse limits, and hence with products. Since the category Pro(C)/ Pro(L) is semi-simple (Lemma 3.2), this yields the assertion.
(2). Let A be an abelian variety. Since every affine quotient of A is trivial, the adjunction map ρ : P (A) → A is an epimorphism. Also, ρ is essential by Lemma 2.3; thus, P (A) is a projective cover of A in Pro(C). The unique divisibility assertion follows from Lemma 2.10, since A is divisible and its n-torsion subgroup schemes are finite for all n 1.

The profinite fundamental group
We now consider the Serre subcategory F of L with objects the finite group schemes. As in the introduction, we denote by the profinite homotopy functors. For any G ∈ Pro(C), the exact sequence (2.7) may be rewritten as where G denotes the profinite universal cover of G F := Ker(G → 0 (G)).
The pair (C, F) satisfies the lifting property in view of [Bri15, Thm. 1.1]; thus, we may again use the constructions and results of Section 2.
(3) G is the limit of the filtered inverse system (G, n G ) n 1 , where the positive integers are ordered by divisibility. Also, G is uniquely divisible. (2). -Consider an epimorphism G → H, where H ∈ F. Then H is divisible (as a quotient of G) and torsion (as a finite group scheme), hence zero. This yields the assertion.
(3). -Let G := lim ← G (limit over the above system). For any H ∈ C and i 0, we have The projection π : G → G associated with n = 1, lies in an exact sequence So we may identify G with G. Then (3.2) is identified with the universal profinite extension of G, in view of Lemma 2.8.
(4). -The first assertion has just been proved; the second one follows from Lemma 2.10 in view of the vanishing of 0 (G).
(5). -By Lemma 2.10 again, the profinite group scheme i (G) is uniquely divisible for any i 2. As a consequence, every finite quotient of i (G) is divisible, hence zero. This yields the assertion.
We may now prove a large part of our main result: Theorem 3.5. -Assume that k is perfect.
(3) The profinite universal cover G has projective dimension at most 1, for any G ∈ Pro(C).

Proof. -
(1). -In view of the homotopy exact sequence and the fact that i commutes with filtered inverse limits, it suffices to show that i (G) = 0 for any G ∈ C and any i 2. This follows from Lemma 3.4 when G is an abelian variety. On the other hand, when G ∈ L, we have i (G) = π L,F i (G) in view of Remark 2.16 and Lemma 3.1. So the assertion follows from [DG70, V.3. 6.8] in that case. In the general case, just recall that every G ∈ C is an extension of an abelian variety by a linear algebraic group.
) for all G ∈ Pro(C) and all i 0. As a consequence, the pro-étale fundamental group π 1 is left exact when k is perfect.

Projective covers of abelian varieties
Consider an abelian variety A, and its projective cover P (A) in Pro(C). By Proposition 3.3, we have an exact sequence in Pro(C) where L(A) is affine. Also, recall that (3.3) is the universal affine extension of A, that is, the pushout by this extension yields an isomorphism  Arguing as in (1) completes the proof. We may summarize the main results of this subsection in the following: Theorem 3.10. -Let A be an abelian variety over a field k with characteristic p 0 and separable closure k s .
(1) The universal profinite cover A is the limit of the filtered inverse system of multiplication maps (A, n A ) n 1 .

Structure of indecomposable projectives
We still consider an arbitrary ground field k, of characteristic p 0.
Proposition 3.11. -The indecomposable projectives of Pro(C) are exactly: (1) the P (A), where A is a simple abelian variety, (2) the universal profinite covers of the simple tori, (3) the additive group G a if p = 0, resp. the universal profinite cover of the Witt group scheme W := lim ← W n if p > 0, (4) the indecomposable projectives of Pro(F).
Proof. -Applying Corollary 2.13 to the pair (C, F), we see that the indecomposable projectives of Pro(C) are exactly those of Pro(F) and the universal profinite covers P , where P is an indecomposable projective of Pro(C/F). Also, every object of C/F has finite length (see [Bri17,Prop. 3.2]). In view of [DG70,V.2.4.6], it follows that every indecomposable projective of Pro(C/F) is the projective cover of a simple object of C/F, unique up to isomorphism.
Next, the simple objects of C/F are exactly G a , the simple tori and the simple abelian varieties (see [Bri17,Prop. 3.2] again). Moreover, every torus is projective in C/F, and hence in Pro(C/F); also, G a is projective if and only if p = 0 (see [Bri17,Thm. 5.14]. The universal profinite cover of a torus T is the group of multiplicative type with character group X(T ) ⊗ Z Q, in view of [DG70, V.3.5.2]. Also, G a = G a if p = 0, as follows e.g. from Lemma 3.4. If p > 0 and k is perfect, then the projective cover of G a in L (or equivalently, in C) is the universal profinite cover W (see [DG70,V.3.7.5]); equivalently, W is the projective cover of G a in Pro(C/F). But the category C/F is invariant under base change by purely inseparable field extensions (see [Bri17,Thm. 3.11]); moreover, W is obtained by base change of a group scheme of finite type over Z, and hence makes sense over an arbitrary field k. Thus, W is the projective cover of G a in that setting, too.
Remark 3.12. -We now describe the indecomposable projectives of the profinite category Pro(F) in terms of those of the pro-étale category Pro(E). For this, we may assume that p > 0, since F = E if p = 0.
We will adapt the arguments in the proof of Proposition 3.11 twice. First, consider the pair (F, I), where I denotes the full subcategory of F consisting of the infinitesimal algebraic groups; then I is a Serre subcategory of F, and the pair (F, I) satisfies the lifting property in view of [Bri17,Lem. 2.2]. Also, the quotient category F/I is equivalent to the category E of étale algebraic groups, by assigning to any finite algebraic group its largest étale quotient. It follows that the functor yields an equivalence of categories Pro(F/I) ∼ = Pro(E). Thus, the indecomposable projectives of Pro(F) are exactly those of Pro(I) and the universal pro-infinitesimal covers P , where P is an indecomposable projective of E.
Next, consider the pair (I, I m ), where I m denotes the full subcategory of I consisting of (infinitesimal algebraic) groups of multiplicative type. Then again, I m is a Serre subcategory; moreover, I/I m ∼ = I u , the full subcategory of I consisting of unipotent groups (see [DG70,IV.3 .1.1]). Also, I u has a unique simple object α p , the kernel of the Frobenius endomorphism of G a (see [DG70,IV.2

.2.5]).
We now show that the pair (I, I m ) satisfies the lifting property. Consider an epimorphism f : G → H in I, where H is multiplicative. Denote by M the largest multiplicative subgroup of G; then G/M is unipotent, hence so is H/f (M ). It follows that H/f (M ) = 0, i.e., the composition M → G → H is an epimorphism as well.
As a consequence, we see that the indecomposable projectives of Pro(I) are exactly those of Pro(I m ) and the universal multiplicative cover P , where P is the projective cover of α p in I u .
The above results take a much simpler form when k is perfect: then we have an equivalence of categories IV.3.5.9]. Thus, the indecomposable projectives of Pro(F) are exactly those of Pro(I m ), Pro(I u ) and Pro(E).

Field extensions
For any field extension k /k, we denote by the associated base change functor. Then ⊗ k k is exact and faithful; hence it extends uniquely to an exact functor Pro(C k ) → Pro(C k ) which commutes with filtered inverse limits (see e.g. [KS06, Prop. 6.1.9, Cor. 8. 6.8]). We still denote this extension by ⊗ k k .
Lemma 3.13. -The functor ⊗ k k : Pro(C k ) → Pro(C k ) is faithful. If k /k is separable algebraic, then ⊗ k k sends projectives to projectives.
Proof. -Let X, Y ∈ Pro(C k ) and f ∈ Hom Pro(C k ) (X, Y ) such that f k = 0. Then Im(f k ) = 0. Since ⊗ k k is exact, this means that Im(f ) k = 0. Let Z := Im(f ), then Z = lim ← Z i (filtered inverse limit), where Z i ∈ C k and Z → Z i is an epimorphism for all i. Thus, Z k is the filtered inverse limit of the (Z i ) k , and Z k → (Z i ) k is an epimorphism for all i as well. As Z k = 0, it follows that (Z i ) k = 0 for all i. So Z i = 0 and Z = 0, that is, f = 0. This proves that ⊗ k k is faithful.
Next, assume that k /k is separable algebraic and let P ∈ Pro(C k ) be projective. To show that P k is projective in Pro(C k ), it suffices to check that given an epimorphism f : G → H and a morphism g : 3.5]). As above, we have P = lim ← P i (filtered inverse limit), where P i ∈ C k and P → P i is an epimorphism for all i. So g lies in Thus, g is represented by a morphism g i : (P i ) k → H for some i. Since the schemes G, H, (P i ) k are of finite type over k , the morphisms f : G → H and g i : (P i ) k → H are "defined over some finite subextension K/k", i.e., there exist such a subextension and morphisms f K : where R K/k denotes the Weil restriction (see e.g. [DG70, I.1. 6.6] or [CGP15, App. B]). As K/k is finite and separable and f K : is an epimorphism as well (see [DG70,III.5.7.9]). Since P is projective, it follows that (g i ) K lifts to a morphism for some j. This yields a lift f j ∈ Hom C k ((P j ) k , G k ) of g i , and in turn the desired lift f ∈ Hom C k (P k , G k ) of g.
Remark 3.14. -In the setting of affine group schemes, the fact that the base change functor ⊗ k k preserves projectives for any separable algebraic extension k of k is due to Demazure and Gabriel (see [DG70, V.3.2.1]). For arbitrary group schemes, this fact is stated and used in [Mil70,p. 437], but the argument sketched there is flawed.
We may now complete the proof of the main theorem: Thus, so does π C,L 0 , since it commutes with filtered inverse limits. By Lemma 3.13, it follows that π C,L i commutes with base change under separable algebraic field extensions for any i 1. In view of Lemma 2.8, the universal affine cover satisfies the same property.
We now show that 0 (the largest profinite quotient) commutes with ⊗ k k , where k /k is any separable algebraic field extension; this will imply the assertions on the profinite homotopy groups and profinite universal cover by arguing as above. For any X ∈ ⊥ Pro(F k ), we have to check that X k ∈ ⊥ Pro(F k ), i.e., Hom C k (X k , Y ) = 0 for any Y ∈ F k . But this follows by a Weil restriction argument as in the proof of Lemma 3. 13.
More specifically, let X = lim ← X i (filtered inverse limit), where X i ∈ C k and the natural map X → X i is an epimorphism for all i. Then X k = lim ← (X i ) k (filtered inverse limit), where (X i ) k ∈ C k and the natural map X k → (X i ) k is an epimorphism for all i as well. Thus, for any morphism f : In turn, there exist a finite subextension K/k and a morphism ( Moreover, R K/k (Y K ) ∈ F k , since Y is a finite k -group scheme and hence Y K is a finite K-group scheme. It follows that Hom C k (X i , R K/k (Y K )) = 0, as X ∈ ⊥ Pro(F k ).

M. BRION
Remark 3. 16. -One may check similarly that the functors π F ,I i and the universal pro-infinitesimal cover (considered in Remark 3.12) also commute with base change under separable algebraic field extensions. Indeed, being infinitesimal is preserved under Weil restriction associated with finite separable field extensions.
Likewise, the functors π I,Im i and the universal multiplicative cover commute with such base change, since being of multiplicative type is preserved under Weil restriction as above.
By Proposition 3.15, the profinite fundamental group 1 commutes with base change under algebraic field extensions in characteristic 0. Yet this does not extend to an imperfect ground field, see Remark 3.21 below. To remedy this, we now recall the definition of the prime-to-p part of 1 , and show that it satisfies the assertions of the main theorem.
Every finite group scheme G decomposes into a product G p × G p , where G p is a p-group, and G p has order prime to p; moreover, G p is étale. This decomposition is clearly functorial, and yields an equivalence of categories F ∼ = F p × F p with an obvious notation. In turn, we obtain an equivalence of categories where every object of Pro(F p ) is pro-étale. Composing the resulting exact functor Pro(F) → Pro(F p ) (the prime-to-p part) with the profinite homotopy functors i , we obtain functors Proposition 3.17. -With the above notation and assumptions, the functor   • Q C,I is identified with the pro-étale homotopy functor π i discussed in Remark 3.6. Thus, π C/I,E 1 is left exact and its prime-to-p part is (p ) 1 . The latter assertion extends to an imperfect field k, since π C/I,E 1 may be identified with the pro-étale fundamental group over its perfect closure.
Remark 3. 19. -The functor 0 does not commute with base change under purely inseparable field extensions. Consider indeed an imperfect field k, and choose t ∈ k \ k p . Let G denote the kernel of the morphism Then G is connected and reduced; thus, 0 (G) is connected and reduced as well, hence zero. Let k := k(t 1/p ), then the map (x, y) → (x, x − t 1/p y) yields an isomorphism G k ∼ = G a,k × α p,k , where α p,k denotes the kernel of the Frobenius endomor- Remark 3.20. -The functor 1 does not commute with base change under purely inseparable field extensions either. Consider indeed a smooth connected algebraic group G and a finite group scheme H. Then 0 (G) = 0, hence we obtain canonical isomorphisms H). If 1 commutes with base change under a field extension k /k, then the map is injective in view of the above isomorphisms and the faithfulness of ⊗ k k (obtained in Lemma 3.13). Now assume that k is separably closed, but not algebraically closed; then there exist nontrivial k-forms of G a , and Ext 1 C (G, G m ) = 0 for any such form G (see [Tot13,Lem. 9.4]). As G is killed by p, so is Ext 1 C (G, G m ). It follows that the natural map The above examples show that the "pro-infinitesimal part" of i (the largest pro-infinitesimal subobject) does not commute with base change under purely inseparable field extensions for i = 0, 1. One may wonder whether the "pro-étale part" (the largest pro-étale quotient of i ) is better behaved. The answer is affirmative for 0 , which commutes with arbitrary field extensions (see [DG70,II.5.1]). Also, the answer is affirmative for the prime-to-p part of 1 by Proposition 3. 17. But the answer is negative for its pro-étale p-primary part, as we now show in the case of the additive group G a .
Since G a is killed by p, so are 1 (G a ) and its largest pro-étale quotient Q. Denoting by ν p the constant k-group scheme associated with Z/pZ, it follows that the natural map Hom Pro(C) (Q, ν p ) → Hom Pro(C) ( 1 (G a ), ν p ) is an isomorphism. So it suffices to show that the formation of Hom Pro(C) ( 1 (G a ), ν p ) does not commute with purely inseparable field extensions.
As in 3.20 above, we have an isomorphism Hom Pro(C) ( 1 (G a ), ν p ) ∼ = Ext 1 C (G a , ν p ) of modules over End C (G a ). Also, recall that End C (G a ) consists of the additive polynomials (also known as p-polynomials), x −→ a 0 x + a 1 x p + · · · + a n x p n , where a 0 , . . . , a n ∈ k (see e.g. [ End C (G a ) P −→ End C (G a ) −→ Ext 1 C (G a , ν p ) −→ 0 of End C (G a )-modules, where End C (G a ) acts on its two copies by right multiplication, and P(f )(x) := f (x) p − f (x) for any f ∈ End C (G a ) and x ∈ G a . We claim that the exact sequence (3.7) can also be obtained as follows: consider a nontrivial extension Then G is smooth and unipotent; also, the composition G 0 → G → G a is an epimorphism, where G 0 denotes the neutral component. It follows that G is connected, and hence is a k-form of G a . By [Rus70, Lem. 1.3], there is an exact sequence where I is infinitesimal; moreover, we have y = F n G for n 0. Then the morphism (x, y) : G → G a × G a has a trivial kernel; its cokernel is a quotient of G a × {0} for dimension reasons, and hence is isomorphic to G a in view of [DG70, IV.2.1.1]. This yields an exact sequence where f, g ∈ End C (G a ). So we may view G as the zero scheme V(f (x) + g(y)) in G a × G a ; this identifies ν p = Ker(x : G → G a ) with Ker(g). We may thus assume that g(y) = y p − y, so that G = V(y p − y + f (x)). This defines a map ), which is surjective as f = 0 gives the trivial extension. One may readily check that u is a morphism of End C (G a )-modules; also, u(f ) = 0 if and only if f (x) = h(x) p −h(x) for some h ∈ End C (G a ), that is, f = P(h). This completes the proof of the claim.
Clearly, we have Ker(P) = Hom C (G a , ν p ) = 0. To describe Coker(P), we first consider the case where k is perfect. Then a x p n = P(a 1/p x p n−1 ) + a 1/p x p n−1 for all a ∈ k and all integers n 1. It follows that Coker(P) ∼ = k via the map k → End C (G a ) given by scalar multiplication. For an arbitrary field k, we obtain by using a p-basis In particular, the natural map k → Coker(P) is not surjective if k is imperfect. This shows that Ext 1 C (G a , ν p ) does not commute with purely inseparable field extensions. The above construction may be interpreted in terms of the exact sequence which yields an exact sequence If k is perfect, then Ext 1 C (G a , G a ) is a free module over End C (G a ) acting on the left (see [DG70, V.1.5.2]). Thus, we obtain an isomorphism of End C (G a )-modules This isomorphism does not extend to an imperfect field k, as the image of ι * may be identified with ∞ n=1 k/k p .

The Milne spectral sequence
We first record a variant of a result obtained by Demazure and Gabriel in the setting of affine group schemes (see [DG70,V.3

.2.3]):
Lemma 3.22. -Let k /k be a separable field extension. Then there are canonical isomorphisms for any G ∈ Pro(C), H ∈ C and j 0: where K/k runs over the filtered direct system of finite subextensions of k /k.
Proof. -Let g ∈ Hom A (X, Z). If g = 0, then the composition Ker(g) → X → Y is not an epimorphism, since f is essential. As Y is simple, this composition is zero, i.e., Ker(g) ⊂ Ker(f ). This yields an exact sequence As Z is simple, we have X/ Ker(g) ∼ = Z. So Z ∼ = Y and Ker(f ) = Ker(g), i.e., g factors uniquely through f . Applying Lemma 3.25 to the abelian category Pro(C /F ) and to the essential epimorphism W → G a , we see that Hom Pro(C /F ) (W , H ) = 0 unless H ∼ = G a , and Hom Pro(C /F ) (W , G a ) ∼ = End C /F (G a ) as Γ-modules.
We now make a further reduction to the case where k is perfect: indeed, the Galois group Γ is invariant under purely inseparable field extensions of k, and the same holds for the isogeny category C/F by [Bri17,Thm. 3.11]. Recall that End C (G a ) is the noncommutative polynomial ring k [F ], and End C /F (G a ) is its fraction skewfield k (F ), as follows e.g. from [DG70, V.3. 6.7]. To show that k (F ) is acyclic, it suffices to check that it is the direct limit of its Γ-submodules g −1 k [F ] over all nonzero g ∈ k[F ], since every such submodule is isomorphic to k [F ] ∼ = k ⊗ k k[F ], hence is acyclic. For this, we adapt a standard argument of commutative algebra.
Since the left k[F ]-module k [F ] is finitely generated and the ring k[F ] is left Noetherian, the increasing sequence of submodules k[F ] + k[F ] g + · · · + k[F ] g n stops. So there exist an integer n 1 and a 1 , . . . , a n ∈ k[F ] such that g n + a 1 g n−1 + · · · + a n = 0. Since k [F ] is a domain and g = 0, we may further assume that a n = 0. Then g g = −a n ∈ k[F ] \ {0}, where g := g n−1 + a 1 g n−2 + · · · + a n−1 . Thus, g −1 f = (g g) −1 g f is as desired.
This completes the proof of the proposition for G = W , and leaves us with the case where G is profinite (and k is arbitrary). We now prove: We may therefore assume that G ∈ F; then Im(f ) is a finite k -subgroup of H . Let I ⊂ Im(f ) denote the largest infinitesimal subgroup, then I is contained in some Frobenius kernel Ker(F n H /k ). Hence I ⊂ Ker(F n H/k ) =: J , where J ⊂ H is infinitesimal. Thus, I = J ∩ Im(f ), and Im(f )/I is a finite étale k -subgroup of H /J = (H/J) . So we may assume that Im(f ) is étale; then we may view Im(f ) as a finite subgroup of H(k s ), stable under Gal(k s /k ). In that case, the (finitely many) conjugates of Im(f ) under Gal(k s /k) generate the desired finite k-subgroup F ⊂ H.
By Lemma 3.26, we have Hom Pro(C ) (G , H ) = lim → Hom Pro(F ) (G , F ), where the limit runs over all the finite subgroups F ⊂ H. Since taking Γ-cohomology commutes with direct limits, it suffices to show that the Γ-module Hom Pro(F ) (G , H ) is acyclic whenever G is the projective cover of a finite simple group, and H is finite.
We may further assume H simple.
Consider the Serre subcategory I of F, and recall that F/I ∼ = E. By Remark 3.12, the indecomposable projective objects of Pro(F) are exactly those of Pro(I) and the universal pro-infinitesimal covers P , where P ∈ Pro(E) is indecomposable and projective. Also, the universal pro-infinitesimal cover commutes with base change under separable algebraic field extensions by Remark 3. 16. Thus, we obtain Hom Pro(F ) ( P , H ) ∼ = Hom Pro(E ) (P , Q(H )), where Q := Q F ,I .
To show that the above Γ-module is acyclic, we may assume H ∈ E. We now adapt the argument in the proof of [Bri19, Lem. 3.10], by using results of Galois cohomology from [Ser97, Chap. II]. Consider the Galois groups Γ k := Gal(k s /k) and Γ k := Gal(k s /k ); these fit in an exact sequence By [DG70, II.5. 1.7], E is equivalent to the category Γ k − mod of finite commutative groups equipped with a discrete action of Γ k . The latter category has a duality given by M → Hom(M, Q/Z), where the right-hand side denotes the group homomorphisms on which Γ k acts via its given action on M and the trivial action on Q/Z. This yields an anti-equivalence between E and Γ k − mod, which extends uniquely to an anti-equivalence between Pro(E) and the category Γ k − Mod of all discrete Γ k -modules (the latter is the ind-category of Γ k − mod). Under this anti-equivalence, the base change functor ⊗ k k : Pro(E) → Pro(E ) corresponds to the restriction from Γ k to Γ k . So it suffices to check that Hom Γ k (M, N ) is Γ-acyclic for any object M of Γ k − mod and any injective object N of Γ k − Mod.
We have an injective morphism of discrete Γ k -modules where the right-hand side denotes the group of continuous maps Γ k → N , equipped with the action Γ k via right multiplication on itself. Since the Γ k -module N is injective, it is identified with a summand of Hom cont (Γ k , N ) via ι; thus, the Γ-module Hom Γ k (M, N ) is a summand of Hom Γ k (M, Hom cont (Γ k , N )) ∼ = Hom Hom(M, N )). So it suffices in turn to show that the latter Γ-module is acyclic. Let P := Hom(M, N ); this is a discrete Γ k -module, and hence we have an isomorphism The inverse isomorphism sends ϕ : Γ → P to the map Γ k −→ P, g −→ gϕ(ḡ), whereḡ denotes the image of g in Γ k /Γ k = Γ k . Moreover, Hom(Γ, P ) is an acyclic Γ-module as desired. This journal is a member of Centre Mersenne.