Perfecting group schemes

We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a bijection with the set of classifying spaces of compact connected Lie groups topologically localised away from the characteristic. We also study the representations of perfectly reductive groups. We establish a highest weight classification of simple modules, the decomposition into blocks, and relate extension groups to those of the underlying abstract group.


Introduction
For a (group) scheme over a field k of characteristic p > 0, its 'perfection' is defined as the inverse limit over the Frobenius homomorphism.In this paper we study the perfection of group schemes and their representation theory.We place particular emphasis on reductive groups.We obtain an intrinsic characterisation ('perfectly reductive groups') and give a classification in terms of root data 'with p inverted'.We also give a highest weight classification of simple modules for perfectly reductive groups, establish the block decomposition, and make a first step towards the study of multiplicity questions.Finally, we prove that perfectly reductive groups and the classifying spaces of compact Lie groups localised away from p are classified by the same data.This result is the 'perfect analogue' of the fact that reductive groups over algebraically closed fields and compact Lie groups are both classified by root data.
The motivations for the current work were three-fold, and came from several directions.Before describing the structure of this paper in more detail, we outline these motivations and possible future directions.
Perfect representability.Sometimes functors on rings in characteristic p are only wellbehaved on perfect algebras (that is, algebras for which the Frobenius homomorphism is an isomorphism).A prominent example is the functor of Witt vectors and its relatives.An important observation (see for instance [BS, BD, Zh]) is that if a functor on the category of perfect commutative k-algebras can be represented by a scheme, it determines the scheme 'up to perfection'.An example of such a setting is the Witt vector affine Grassmannian, which plays a prominent role in recent advances in the Langlands program.
It is important that passage to the perfection gives an isomorphism of étale topoi.In particular, constructions built via étale sheaves (like étale cohomology, or categories of perverse sheaves in the étale topology) are insensitive to passage to the perfection.This fact plays an important role in [Zh], where a mixed characteristic analogue of the geometric Satake equivalence is obtained.Similarly, it plays an important role in [BD], where the 'Serre dual' of a unipotent group is shown to be the perfection of a unipotent group, and its character sheaves are studied.
By the above, also the étale homotopy type of a (simplicial) scheme only depends on its perfection.In [Fr], Friedlander used this homotopy type to construct interesting maps between topological localisations of classifying spaces of compact Lie groups, based on (exceptional) isogenies in positive characteristic.This is one of the main results we rely on to establish our bijection between perfectly reductive groups and localised classifying spaces.
Fractal representation theory.For a (reduced) group scheme defined over F p , the Frobenius twist realises its category of representations as a full subcategory of itself.This selfsimilarity induces a fractal-like structure.For example, Figure 1, shows a classic picture of the non-zero weight spaces of simple modules for SL 2 in characteristic 3. (For the reader unfamiliar with this picture, it may be helpful to note that it simply depicts the (non)-vanishing behaviour of Pascal's triangle modulo p, see for instance [Wi,§1].)This picture is fractal like, but not genuinely fractal: one can 'zoom out' but one cannot 'zoom in' indefinitely since the Frobenius homomorphism is not an isomorphism.By passing to the perfection, one gets a genuine fractal.One dream (not realised in the current paper) is to use this fractal structure to say something about important open questions in representation theory like dimensions and characters of simple modules.
Much of the difficulty in the representation theory of reductive groups in characteristic p remains after perfecting.We do observe two interesting simplifications.Firstly, the complexities of the block decomposition disappear after passage to the perfection (see Theorem 5.1.4).Secondly, perfect representation theory appears to provide the correct setting for results of [CPSV]: for a perfectly reductive group, extensions computed inside algebraic representations agree with extensions computed as abstract groups (see Theorem 5.2.2).Tensor categories in characteristic p.Over fields of characteristic zero, a famous theorem of Deligne classifies those tensor categories which admit a fibre functor to super vector spaces as precisely those of moderate growth [D].It is a fascinating open problem to find an analogue of this problem in characteristic p, with many potential applications to modular representation theory.Recently, this problem was solved in [CEO] for tensor categories with exact Frobenius functor.An important technical tool was a limit procedure in [CEO,§6] which, by restriction to representation (tensor) categories of affine group schemes, generalises perfection of group schemes.Remarkably, the 'perfection' of a Frobenius exact tensor category of moderate growth essentially returns the representation category of a perfect group scheme.In other words, up to perfection, all Frobenius exact tensor categories of moderate growth arise from (perfect) group schemes.We consider this as further evidence for their importance.
Structure of the paper.Motivated by the above considerations, we initiate a systematic study of perfecting group schemes.The paper is organised as follows: In Section 1 we investigate the purely combinatorial problem of describing root data over the ring Z[1/p], for later use in our classification results.In Section 2, we derive some general results on perfect schemes.In Section 3 we start studying the perfections of group schemes.We study perfect subgroups of perfect groups and their quotients.We also obtain criteria for when two group schemes perfect to isomorphic groups and derive some results on the perfections of the additive and multiplicative groups.In Section 4 we classify perfectly reductive groups by their Z[1/p]-root data.In Section 5 we study the representation theory of perfectly reductive groups.We classify simple modules and realise them as socles of induced modules from Borel subgroups.Then we show that the block decomposition simplifies considerably compared to the non-perfect case, in fact blocks are governed by the root lattice.We also show that extension groups for the perfected groups are given in terms of generic cohomology in the sense of [CPSV].This actually implies that extensions in the category of (rational) representations over the perfected reductive group can be computed in the category of representations of the abstract group of F p -points.In Section 6 we prove that Z[1/p]-root data also classify the localisations away from p of the classifying spaces of compact connected Lie groups.Finally, in Section 7, we present some explicit computations for extension groups, decomposition multiplicities and line bundle cohomology for perfected SL 2 .We also make explicit the fractal behaviour of perfected representation theory for SL 2 .

Root data over rings
For the entire section, we assume that D is a principal ideal domain of characteristic 0. By a D-lattice we understand a finitely generated free D-module.Because D is a PID we could replace 'free' by 'projective', so our definition agrees with standard terminology (e.g. in [CR].)For a lattice V , we have the dual lattice V * := Hom D (V, D).
1.1.Reflection groups and root data.We follow the definition of root data of for instance [Gr].
is a subgroup generated by reflections.We say (W, and {P σ } is a collection of rank one submodules of V , indexed by the set {σ} of reflections in W , satisfying (a) im(1 − σ) ⊂ P σ and (b) w(P σ ) = P wσw −1 for all w ∈ W .
An isomorphism of D-reflection groups (W, V ) Lemma 1.1.3.If there exists an embedding D → Q, then every D-root datum is the extension of scalars of a Z-root datum.
Proof.Let (W, V ) be a D-reflection group.Starting from a Z-lattice in V and acting on it with W shows there exists a finitely generated ZW -submodule V 0 ⊂ V with D ⊗ Z V 0 → V an isomorphism, see [CR,Corollary 23.14].For a D-root datum (W, V, {P σ }), we can then take the Z-root datum (W, V 0 , {P 0 σ }), with P 0 σ := P σ ∩ V 0 .Remark 1.1.4.
(1) Lemma 1.1.3does not imply that root data over D ⊂ Q are 'the same' as root data over Z, see Example 1.4.2.
(2) The condition D ⊂ Q is necessary in Lemma 1.1.3,as one observes by considering dihedral groups as real reflection groups on R 2 , or the complex reflection groups generated by a root of unity acting by multiplication on C.
We will use several times the following direct computation.
(2) If p > 2 and the further extension along Proof.It is well known and easy to show that for Z-root data, we either have (1.1) Hence the additional condition in the definition of an isomorphism of Z[1/2]-root data is trivially satisfied.Now assume p > 2. In this case, (1.1) shows that in Z[1/p] ⊗ V we have either ϕ(P σ ) = P ϕσϕ −1 , ϕ(P σ ) = 2P ϕσϕ −1 , or 2ϕ(P σ ) = P ϕσϕ −1 .By assumption, after extension to scalars to Z 2 , only the first option is possible.As 2 is not invertible in Z 2 , this means that also over Z[1/p] only the first option was possible.
1.2.Real-type root data.The following definition is closer to the classical definition of (reduced) root data.
(3) There are only finitely many D × orbits in R ∨ .
(4) For each α ∈ R, the reflection To a real-type root datum (X, R, Y, R ∨ ) over Z we can define a real-type D-root datum (D ⊗ X, D × R, D ⊗ Y, D × R ∨ ), with obvious bilinear pairing and bijection Remark 1.2.2.For D = Z, Definition 1.2.1 is equivalent to the definition of a 'donnée radicielle reduite' in [De,§3.6].We will show below in Lemma 1.2.5 that a real-type D-root datum (X, R, Y, R ∨ ) also satisfies: (2') If α ∈ R and a ∈ D, then aα ∈ R if and only if a ∈ D × .
(3') There are only finitely many D × orbits in R. In particular, the definition of real-type root data is closed under duality.
Example 1.2.3.Consider a D-root datum (W, V, {P σ }).Take a reflection σ ∈ W and a generator b ∈ P σ .By condition 1.1.1(2)(a),there exists (a unique) β ∈ V * such that (1.2) We then define R ∨ ⊂ V as the set of generators of the submodules {P σ } and R ⊂ V * as the set of elements β constructed by the above procedure.In particular, we have the defining map R ∨ R.
Theorem 1.2.4.Assume that for the D-root datum (W, V, {P σ }), every reflection in W has order 2, then Proof.The assumption σ 2 = 1 in (1.2) shows that β(b) = 2. Assume now that for two generators b 1 ∈ P σ 1 and b 2 ∈ P σ 2 , we obtain the same β ∈ V * .By finiteness of W , Lemma 1.1.5and the fact that D is of characteristic 0 we find b By the above paragraph, 1.2.1(1) is satisfied.To establish 1.2.1(2) it suffices to show that we cannot have non-trivial inclusions P σ 1 ⊂ P σ 2 .We can extend scalars along D → K, with K the field of fractions.Now σ 1 σ 2 acts as the identity on KP σ 1 = KP σ 2 , but also as the identity on KV /KP σ 1 .Since it has finite order and charK = 0, we find σ 1 σ 2 = 1.Property 1.2.1(3) follows immediately from the fact that W is finite.Property 1.2.1(4) is an immediate consequence of 1.1.1(2)(b).
By letting w ∈ W act on V * by w(f ) = f •w −1 , we also have a reflection group (W, V * ).We can define Q σ := Dβ, for each reflection σ ∈ W with β as in (1.2), since Q σ does not depend on our choice of generator b ∈ P σ .It follows immediately that (W, V * , {Q σ }) is a D-root datum.That 1.2.1(4') is satisfied follows by applying the proof for (4) to (W, V * , {Q σ }).
1.3.1.Hypotheses.Consider a finite-dimensional real vector space V (without fixed inner product/Euclidean structure), a reflection group W < Aut R (V ), in the sense of Definition 1.1.1(1),and a fixed finite generating set T of reflections in W such that • The map from T to the set of hyperplanes in V , s → H s := ker(1 − s), is injective.
• We have wT w −1 ⊂ T for all w ∈ W .
Theorem Proof.First we show that every W -orbit in V intersects A 0 .By continuity it suffices to show that every W -orbit in V \H intersects A 0 .For v ∈ V \H, there exists a sequence A 0 , A 1 , • • • , A l of distinct chambers where v ∈ A l , and for each 0 ≤ i < l there is t i ∈ T such that the intersections of H t i with A i and A i+1 cannot be contained in a codimension 2 hyperplane.If l = 0 there is nothing to prove, so assume l > 0. Then t 0 (A 1 ) = A 0 , and by assumption t 0 ∈ S. Now A 0 , A 1 = t 0 (A 2 ), A 2 = t 0 (A 3 ), • • • A l−1 = t 0 (A l ) forms a chain of distinct chambers as before and t 0 (v) ∈ A l−1 .We can thus perform induction on l to deduce the claim.
The proof can now be concluded exactly as in [Bo,V. §3.1 Lemma 2(iii)].Indeed, we have proved above the conclusion of Lemma 2(i) loc.cit.after which one can follow the proof using our assumption (trivial in the Euclidean context of [Bo]) that H t determines t.
Precisely as in [Bo,V. §3.2 Theorem 1], we then find the following consequence.
Proof of Theorem 1.3.2.The reflections in the Coxeter group (W, S) are by definition the elements of the set ∪ w∈W wSw −1 , which by assumption is included in T and hence finite.By [BB,Corollary 1.4.5]any Coxeter group (W, S) with finitely many reflections is finite.
Theorem 1.4.1.If there exists an embedding D → R, then the map in Theorem 1.2.4 is a bijection between the sets of isomorphism classes of D-root data and real-type D-root data.
Proof.Since we have D ⊂ R, the only roots of unity in D are ±1 and it follows that every reflection must have order two.Hence the map in Theorem 1.2.4 is defined on every D-root datum.
To each D-root datum (X, R, Y, R ∨ ) we will now associate a real-type D-root datum, in a way which is easily seen to be the inverse of the above map.Define the D-reflection group W < Aut D (Y) generated by {s α | α ∈ R}.To a reflection s γ , γ ∈ R, we associate the corresponding rank one submodule Dγ ∨ ⊂ Y.
We show that W is finite by considering the corresponding real reflection group acting on Y ⊗ D R. By Lemma 1.2.5, W is generated by a finite (see 1.2.1(2)) set of reflections {s α | α ∈ R}, such that the reflecting hyperplane ker(1 − s α ) = ker α, − determines s α .Moreover, we claim that for each α ∈ R and w ∈ W , we have ws α w −1 = s γ for some γ ∈ R. Clearly it suffices to consider the case w = s β , which is Lemma 1.2.6.We can now apply Theorem 1.3.2.Now it follows that the triple (W, Y, {Dγ ∨ }) is a D-root datum.Indeed, by Remark 1.3.3 and [Hu,Proposition 1.14], every reflection in W is equal to s γ for some γ ∈ R, and property (a) in 1.1.1(2)is automatic, while (b) follows from Lemma 1.2.6.
Clearly, the bijection in Theorem 1.4.1 exchanges the two notions of extensions of scalars of root data.We conclude this section with some examples of root data which become isomorphic after extension of scalars.
(1) The root datum of SO 2n+1 becomes isomorphic to its dual (the root datum of SP 2n ) after extension of scalars to D if and only if 2 is invertible in D.
(2) The root datum of SL n becomes isomorphic to its dual (the root datum of P GL n ) after extension of scalars to D if and only if n is invertible in D.
Remark 1.4.3.We expect that our methods can be used to prove that for D ⊂ C (a relevant case would be D = Z p ), there is a similar 1-to-1 correspondence between D-root data and the generalisation of real-type D-root data where 1.2.1( 1) is replaced by the condition that 1.5.Isogenies.We fix a prime p and define isogenies of Z[1/p]-root data.We will work with real-type root data, but refer to them simply as root data (this is justified by Theorem 1.4.1).
Example 1.5.2.An isogeny of Z-root data, with respect to some prime p, is defined in [St,§1].It follows immediately that the induction to Z[1/p] of such an isogeny yields an isogeny of Z[1/p]-root data.Note that Definition 1.5.1 is simpler than the definition in [St,§1], as the powers of p present in [St,§1] are subsumed by (2), since multiplication by p is invertible on Z[1/p]-root data.
Conversely, if for Z-root data RD 1 and RD 2 , there exists an isogeny ϕ : then for some l ∈ N, the map p j ϕ restricts to an isogeny RD 1 → RD 2 in the sense of [St] for all j ≥ l.

Perfection of schemes
2.1.Notation.We recall some basic set-up of algebraic geometry, see for instance [DG].
Fix a field k.Denote by Alg k the category of commutative k-algebras.We consider the categories (where the first two 'inclusions' are fully-faithful embeddings) Here Sch k is the category of k-schemes and Fun k is the category of functors Alg k → Set.The category Fais k stands for the full subcategory of such functors which are sheaves for the fpqc topology.In other words, a functor F is in Fais k if and only if is an equaliser for every faithfully flat A-algebra B, and F commutes with finite products.When k is clear, we will usually leave out the subscript in the above categories.
The inclusion I : Fais → Fun has a left adjoint S : Fun → Fais which commutes with finite limits (as well as all colimits).By a subgroup of an affine group scheme G we understand a closed subscheme which inherits a group structure, or in other words an affine group scheme represented by a quotient of the Hopf algebra representing G.

Perfection functors.
2.2.1.Frobenius morphisms.For a commutative F p -algebra A, we have the p-th power algebra morphism A → A, a → a p .For an F p -scheme X, we have the morphism Fr : X → X which is the identity on the underlying topological space and given by the p-th power map on the sheaf of algebras.For F ∈ Fun Fp , we can also define F → F by evaluation at the p-th power morphism.These Frobenius morphisms are compatible with the inclusions (2.1).
For an object F of Fun Fp or Alg Fp , the notation lim − → F or lim ← − F will always be used for the direct or inverse limit along the Frobenius morphism.
An F p -scheme (or an algebra or functor) is called perfect if the Frobenius map is an isomorphism, see [BS,Definition 3 Proof.It is a standard property that limits exist in a Grothendieck topos and can be computed in the presheaf category, which shows that perfection restricts to Fais.The remaining properties follow from the explicit realisation in Example 2.2.3 below.
Example 2.2.3.For an F p -algebra A, set (X, O) = SpecA.Using the basis of distinguished open subsets it follows easily that Spec(A perf ) = (X, O ), with O the sheafification of the presheaf U → O(U ) perf .It then follows that for an arbitrary F p -scheme X = (X, O), the scheme X perf can be realised as (X, O ) with O the sheafification of the presheaf U → O(U ) perf .
Remark 2.2.4.In 2.2.3 is essential to take O as the direct limit of O in the category of sheaves (as opposed to presheaves).For example: (1) For an infinite family of F p -algebras A i , consider the (non-affine) scheme (X, O) = i SpecA i .Then, for general A i , we have, by the sheaf axioms and Lemma 2.2.2, (2) Also for non-noetherian affine schemes this phenomenon occurs.
Remark 2.2.5.For F p -algebras A, B we have In particular, if B is perfect, we have Alg(A perf , B) ∼ = Alg(A, B).
(1) We have dim X = dim X perf and X perf is quasi-compact (resp.connected) if and only if X is quasi-compact (resp.connected).
(2) Any radical ideal I in a perfect F p -algebra A satisfies I 2 = I.
(3) If X is perfect, for x ∈ X we have T X,x = 0. (4) Perfect schemes are reduced.Moreover, − perf sends X red → X to an isomorphism.
Proof.Part (1) follows immediately from the fact that the underlying topological spaces of X and X perf are the same, see Example 2.2.3.Part (2) is obvious.By (2), it is clear that the Zariski cotangent space is zero, which proves (3).Alternatively, for (3), let A be a perfect F palgebra and κ a field.Every algebra morphism 2.3.1.For a fixed F p -scheme T, perfection naturally yields a functor from the category of T-schemes to the category of T perf -schemes.Using the canonical morphism T perf → T, we can also interpret perfection as an endofunctor of the category of T-schemes.We will take the latter point of view for T = Speck (in which case T perf → T is an isomorphism) and we will henceforth interpret the perfection functor as an endofunctor of Sch k .
2.3.2.Sometimes it will be beneficial to consider an alternative realisation of the perfection of k-schemes, in which the morphisms in the chain of which we take the limit are morphisms of k-schemes.For a k-scheme X, let X (1) denote the extension of scalars of X along the Frobenius automorphism k → k.The morphism Fr : X → X over F p from 2.2.1 then lifts to a morphism Fr : X → X (1) of k-schemes.For instance, for a k-algebra A this corresponds to the morphism A (1) → A, λ ⊗ a → λ a p , (2.2) with A (1) = k ⊗ A the extension of scalars along the Frobenius automorphism of k.
By taking iterates of the Frobenius automorphism and its inverse (k is perfect), we define X (i) for i ∈ Z. Then we have (over k) The advantage of the approach in this subsection is that it extends to Fun k , by setting By construction, perfection commutes with limits, for instance products, in Fun k .
(1) If f is an epimorphism in Fais k , then so is f perf .
(2) If f is an monomorphism in Fais k , then so is f perf .
Proof.For part (1), we can use the criterion from [DG,Corollaire 2.8] to describe that f is an epimorphism, which carries over to f perf by [BS,Lemma 3.4(xii)].Part (2) is a generality for limits of monomorphisms.
It is obvious that X → X perf loses a lot of information.For instance A more subtle example is given below.
Example 2.3.4.Assume p > 2. Consider the algebra given by x → z p , y → z 2 .Then X := SpecA is reduced, but perfection sends A 1 k → X to an isomorphism.
2.4.Perfect finite type.Recall that X ∈ Sch k is of finite type (over k) if the underlying topological space is quasi-compact and for every x ∈ X there exists an affine open neighbourhood isomorphic to the spectrum of a finitely generated k-algebra.
Lemma 2.4.1.For a perfect commutative k-algebra A, the following conditions are equivalent: (a) There is a finite subset S ⊂ A such that the set generates A as a k-algebra; (b) There exists a finitely generated k-algebra A 0 with A ∼ = (A 0 ) perf .If the conditions are satisfied we say that A is perfectly finitely generated. Proof.Exercise.
Proposition 2.4.2.For a perfect k-scheme X, the following are equivalent: (a) X is quasi-compact and for every x ∈ X there exists an affine open neighbourhood corresponding to a perfectly finitely generated k-algebra; (b) There exists scheme Y of finite type over k with X ∼ = Y perf .If the conditions are satisfied we say that X is of perfect finite type (over k).
2.5.Perfection of line bundle cohomology and quotients.All schemes and functors are assumed to be over k.
Lemma 2.5.1.Let X be a quasi-compact separated scheme over k and L a line bundle on X.For the pull-back p * L along p : X perf → X and i ∈ N, we have where the transition maps are induced from Proof.Since X is quasi-compact we can take a finite cover U by affine opens and, since X is separated, intersections of these opens are again affine.Moreover, the cohomology groups H i (X, −) are canonically isomorphic to the Čech cohomology groups Ȟi (U, −).It follows that H i (X, −) commutes with direct limits.Now, for a line bundle, we have (as sheaves on the underlying topological space of X perf or X) isomorphisms The conclusion follows from the combination of the two paragraphs.2.5.2.For X ∈ Fais and G a group object in Fais acting on X (on the right), we consider the corresponding co-equalisers in Fun and Fais of X × G ⇒ X, and denote them by X/ 0 G and X/ 1 G.In particular X/ 1 G = S(X/ 0 G), for the sheafification functor S : Fun → Fais.
It follows from the definitions that G perf is again a group object, and acts on X perf .We create the following commutative diagram in Fun (2.3) The vertical arrows are induced from the adjunction S I (either directly or via the action of the perfection functor).The left horizontal arrow is the defining one for the co-equaliser.
The remaining two arrows are uniquely defined from the coequaliser properties applied to the perfection of the morphisms X/ 1 G ← X → X/ 0 G.
Theorem 2.5.3.Assume that the action of G on X is free, then the morphism from (2.3) in Fais k is an isomorphism.
Proof.By Proposition 2.3.3(1), the composite morphism (from top left to bottom right) in (2.3) is an epimorphism in Fais.In particular the lower horizontal arrow is an epimorphism.Since isomorphisms in Grothendiek topoi are precisely morphisms which are both monomorphisms and epimorphisms, it now suffices to show this arrow is also a monomorphism.
Since sheafification sends monomorphisms to monomorphisms, it actually suffices to show that X perf / 0 G perf → (X/ 1 G) perf is a monomorphism in Fun.Using the assumption that the action is free, we can prove that Indeed, the first case follows from the fact that for an inverse system of sets X i with free is injective.By Proposition 2.3.3(2), for the second morphism, it suffices to demonstrate that X/ 0 G → X/ 1 G is a monomorphism.This is equivalent to the claim that the presheaf X/ 0 G is separated for the fpqc topology, meaning that is an injection for every faithfully flat A-algebra B (and the same for finite products of algebras).This is easily verified for free actions, see for instance [Ja,§I.5.5].

Perfection of group schemes
Let k be a perfect field of characteristic p > 0.
3.1.Perfection.For an affine group scheme G over k, clearly G perf is again an affine group scheme over k.Note also that, since k is perfect, G red is a subgroup of G.
3.1.1.The group p Z .We denote by p Z < Z[1/p] × the group of powers of p, an infinite cyclic group on one generator.Let G be an affine group scheme over k, which can be defined over F p .Then we can choose an isomorphism φ : G (1) ∼ − → G, which yields an automorphism and a corresponding group homomorphism p Z → Aut(G perf ), p → Φ. Examples are given in 3.4.5.
Theorem 3.1.3.Let G be a perfect affine group scheme over k.The following are equivalent: (a) The scheme G is of perfect finite type, i.e. k[G] is perfectly finitely generated; (b) The group scheme G is a subgroup of GL(V ) perf for a finite-dimensional vector space V ; (c) There exists an affine group scheme G of finite type with G ∼ = G perf ; (d) There exists a reduced affine group scheme G of finite type with G ∼ = G perf .
Proof.Clearly (d) implies (c).Any affine group scheme G of finite type is a subgroup of some GL(V ), see e.g.[DM,Corollary 2.5].It follows immediately that G perf < GL(V ) perf , so (c) implies (b).That (b) implies (a) follows by the observation that k[G] is a quotient of the perfectly finitely generated algebra lim − → k[GL(V )].Finally, we prove that (a) implies (d).Let S be a finite set which perfectly generates k[G] as in Lemma 2.4.1(a).It is possible to replace S by a finite set S 1 ⊃ S which generates a Hopf subalgebra, see [Ab,Lemma 3.4.5].Let G be the corresponding affine group scheme.By construction Lemma 3.1.4.Let G be a perfect affine group scheme and H an affine group scheme.Then with inverse given by perfection.Moreover, if H is of finite type and G an affine group scheme, then Proof.This is an immediate application of Remark 2.2.5, or the fact that the perfection functor on F p -schemes is right adjoint to the inclusion functor for perfect schemes.
3.1.5.For a subgroup H of an affine group scheme G, we denote by G/H, when it exists, the equaliser of Recall from [DG,III,§3 Théorème 5.4] that for G of finite type, the quotient G/ 1 H in Fais k is a scheme, of finite type over k, so in particular is equal to G/H.Theorem 3.1.6.
(1) For every perfect subgroup H of an affine group scheme G of perfect finite type, the quotient G/H exists, is of perfect finite type and isomorphic to G/ 1 H.
(2) For an affine group scheme G, every perfect subgroup of G perf is the perfection of a subgroup of G.More precisely, every perfect (normal) subgroup of G perf is the perfection of a reduced (normal) subgroup of G red < G.
(3) For an affine group scheme G of finite type with subgroup H, the quotient G perf /H perf exists and is isomorphic to (G/H) perf and G perf / 1 H perf .
Proof.We will freely use the results from [DG] recalled above.Part (1) is then an immediate consequence of parts (2) and (3).Now we prove part (2).Take a perfect subgroup H < G perf .We have a commutative square, where → denotes the inclusion of a subgroup and denotes a faithfully flat homomorphism where L is just defined to be the image of the composite diagonal homomorphism.By Lemma 3.1.4,perfecting the lower path in the square yields homomorphisms H L perf → G perf which compose to the original inclusion H → G perf .Clearly, H L perf must be an isomorphism.If H is a normal subgroup, it follows easily that so is L < G red .
Part (3) is an application of Theorem 2.5.3.
Remark 3.1.7.A perfect group scheme can have non-perfect subgroups, for instance For our purposes it is most convenient to define short exact sequences of affine group schemes as those sequences N → G → Q in which G → Q is faithfully flat and N is the kernel of the latter morphism.
Lemma 3.1.8.The perfection functor acting on a short exact sequence of affine group schemes Proof.Faithful flatness is preserved by perfection, see [BS,Lemma 3.4].Taking inverse limits of affine group schemes always respects kernels, alternatively we can apply Remark 2.2.5.
Corollary 3.1.9.A reduced affine group scheme G is solvable if and only if G perf is solvable.
Proof.Lemma 3.1.8shows that for any solvable affine group scheme, its perfection is again solvable.On the other hand, assume that G perf is solvable and G reduced.Applying iteratively Theorem 3.1.6(2)allows us to construct a finite chain of reduced normal subgroups such that the perfection of the quotients are abelian.A reduced affine group scheme with abelian perfection is clearly abelian itself.
3.2.Isomorphic perfections.We gather some examples and results about affine group schemes with isomorphic perfections.
(1) Let G be a finite group scheme (i.e.k[G] is finite dimensional).We have G perf ∼ = G red , so in particular, G is infinitesimal if and only if G perf is trivial.
(2) Reduced affine group schemes can also become isomorphic after perfection.For instance, if q is a power of p, then (SL q ) perf ∼ = (P GL q ) perf .This is an example of Lemma 3.2.2below, or follows from 4.2.3 below and Example 1.4.2.Moreover, the latter results also show that, conversely, (SL n ) perf ∼ = (P GL n ) perf implies that n must be a power of p.
(3) As follows from Remark 2.2.5, a necessary condition for affine group schemes G, H to have isomorphic perfections is that there exists an isomorphism G(k) ∼ = H(k) as abstract groups.
Recall that an isogeny is a faithfully flat homomorphism of affine group schemes with finite kernel N .An isogeny is infinitesimal if N is infinitesimal.An isogeny G → Q between reduced and connected affine group schemes is purely inseparable if the induced morphism k(Q) → k(G) between the fields of fractions is a purely inseparable field extension.Lemma 3.2.2.For an isogeny q : G → Q, the following conditions are equivalent: (a) q is infinitesimal; (b) The perfection of q is an isomorphism.Moreover, assuming that G, Q are reduced and connected, the above properties imply (c) q is purely inseparable.
Proof.The equivalence between (a) and (b) is an immediate application of Example 3.2.1(1)and Lemma 3.1.8.Condition (b) implies that for every a , from which (c) follows immediately.
Indeed, this follows from the equality k . The latter can be observed as follows.For S is finite dimensional.Clearly the action of G(k) on f factors through the canonical action of Q(k) and by our finite type assumption we have G An alternative argument considers the purely inseparable isogeny G → G/(ker q) 0 , which is also étale and therefore an isomorphism.Proposition 3.2.4.The following conditions are equivalent on two reduced affine group schemes G, H of finite type.
(a) G perf ∼ = H perf ; (b) There exists j ∈ N and an infinitesimal isogeny G → H (j) ; (c) There are i, j ∈ N and homomorphims φ : G → H (j) , ψ : H → G (i) for which the following triangles are commutative Proof.That (b) implies (a) is a special case of Lemma 3.2.2.Now we prove that (c) implies (b).The kernel of the Frobenius homomorphism is infinitesimal, hence the left diagram shows that the kernel of φ is infinitesimal.Since we assume that G and H are reduced, the Frobenius homomorphism is faithfully flat.The right diagram thus proves that φ is faithfully flat.
Applying Lemma 3.1.4to the isomorphism in (a) yields morphisms φ : G (−j) → H and ψ : H (−i) → G. Expressing that these induce mutually inverse homomorphisms on the perfected groups then states that there exists l ∈ N such that the composition is Fr i+j+l .Since Fr l is faithfully flat, we arrive precisely at the conditions in (c), so (a) implies (c).
3.3.Tannakian point of view.For an affine group scheme G we denote by RepG its category of (rational) representations which are finite dimensional over k.Its category of all representations will be denoted by Rep ∞ G ∼ = IndRepG.

Let us interpret
where the k-linear (exact) tensor functors in the chain are given by the pullback along G (−i−1) → G (−i) .These functors fit into commutative diagrams: (3.2) The non-horizontal arrows are only k-linear up to twist.By definition, 2).The downwards arrow is given by applying − (1) to both vector space and co-action.Note that we can equivalently realise the G-representation V (1) from the bottom right in (3.2) as the subquotient of ⊗ p V given by the image of Γ p V → S p V .This gives a more palatable definition of the Frobenius twist from the Tannakian point of view.

Diagram (3.2) allows us to realise RepG perf alternatively as
in the sprit of interpretation 2.3.1.Compared to (3.1) we no longer have to work with twists of G, but we do have the drawback that the the defining functors are not k-linear.
3.3.2.Notation.For M ∈ RepG, we denote by M [i] the object of lim − → RepG where M is placed in the i-th copy of RepG in the chain, and use the same notation for the corresponding object in Rep(G perf ), via (3.3).Note that every object in Rep(G perf ) is of this form and furthermore Lemma 3.3.3.[CEO, Remark 6.5] Let G be an affine group scheme over k.
Proof.If G is reduced, the p-th power map is injective on k[G] and the fullness in (1) follows.On the other hand, if − (1) is not full then (by applying adjunction) there is a G-representation V with a vector v ∈ V which is not G-invariant, but for which 1 Via diagram (3.2), the functor is an equivalence if and only if G (−1) → G is an isomorphism, which is equivalent to G being perfect.
Remark 3.3.4.As for any direct limit of abelian categories, for objects ). 3.4.Additive, multiplicative and unipotent groups.For convenience we let k be algebraically closed in this section.(1) Let G be a connected affine group scheme of perfect finite type and of dimension 1,

To lighten expressions, we introduce the following notation
(2) Let G be a reduced affine group scheme of finite type with Proof.By [Sp,Theorem 3.4.9],for any connected reduced affine group scheme G of finite type and of dimension 1, we must have G ∼ = G a or G ∼ = G m .This implies part (2), by Lemma 2.2.6(1).Part (1) follows similarly, using characterisation 3. 1.3(d).
By a torus we mean an affine group scheme isomorphic to G ×n m for some n ∈ N. Similarly, perfect tori are the affine group schemes isomorphic to G ×n m for some n ∈ N. Lemma 3.4.3.For a connected reduced affine group scheme G of finite type, the following are equivalent: (a) G is a torus; (b) G perf is a perfect torus.
Proof.Clearly (a) implies (b).On the other hand, (b) implies that G is connected and, since RepG → RepG perf is exact and fully faithful, see Lemma 3.3.3, it follows that RepG is semisimple and pointed (every simple representation has dimension one).That G is a torus then follows from [Sp,3.2.3 and 3.2.7(ii)].
Recall that an affine group scheme G is unipotent if and only if every representation has an invariant vector (equivalently every simple object in RepG is trivial).Lemma 3.4.4.For an affine group scheme G over k, G perf is unipotent if and only if G red is unipotent.
Proof.By equivalence (3.1), if G (or G red ) is unipotent, then so is G perf .On the other hand, if G perf is unipotent, then the fully faithful exact functor from RepG red to RepG perf shows that also G red is unipotent.
3.4.5.We have a ring isomorphism where λ stands for an element of G m (A) = lim ← − A × for a commutative k-algebra A. The restriction to p Z → Aut(G m ) is the homomorphism from 3.1.1.
For G a , the latter homomorphism extends to an isomorphism (3.5) 3.4.6.For a perfect affine group scheme G, Lemma 3.1.4shows that characters of G correspond to homomorphisms G → G m .As the latter formulation carries more structure, we define This is a Z[1/p]-module via (3.4).Consequently, we will define cocharacters of G to be the We have the obvious bilinear pairing It is non-degenerate if G is a perfect torus. 3.5.Induction.
3.5.1.For a homomorphism f : H → G of affine group schemes, we will sometimes abbreviate f * := Ind G H and f * := Res G H .This gives an adjoint pair f * f * of functors between Rep ∞ G and Rep ∞ H.

For a commutative square of group homomorphisms
the adjunction morphisms yield a natural transformation In particular, for M ∈ Rep ∞ A, the morphism ξ M is zero if and only if the composite

Now we consider two affine group schemes of the form A = lim
← − A i and B = lim ← − B i for inverse systems of affine group schemes (A i | i ∈ N) and (B i | i ∈ N).We label the homomorphisms p i : A → A i , q i : B → B i and a [i,j] : A i → A j for i > j.Assume also given homomorphisms A i → B i , leading to A → B. Proposition 3.5.4.
(1) For M i ∈ Rep ∞ A i and morphisms a * [i+1,i] M i → M i+1 , the evaluations at M i of the natural transformations in 3.5.2lead to an isomorphism Proof.The right-hand side in part ( 1) is given by the A-invariants in the vector space Since direct limits commute with co-equalisers, this is isomorphic to the direct limit of A iinvariants in the above spaces.This proves part (1).Note that the case n = 0 in part ( 2) is the special case of part (1) where we set M i := a * [i,j] M .We can prove the analogue of part (1) for derived functors, which similarly specialises to part (2), as follows.For a chain {M i } as in part (1), we consider injective hulls in Rep ∞ A i , yielding short exact sequences The defining property of injective modules gives a chain map from the action of a * [i+1,i] on the above sequence and the corresponding sequence for i + 1.In particular, this makes {I i } and {Q i } into chains of representations as in part ( 1), and we have a short exact sequence with notation as in 3.3.2.
Proof.We start by applying Proposition 3.5.4(2),using the interpretation G perf = lim ← − G (−i) , applied to M (−j) ∈ RepH (−j) .By assumption, all the representations appearing in the direct limit in 3.5.4(2)are finite dimensional.After passing from (3.1) to (3.3) this allows us to use the notation from 3.3.2 to rewrite the isomorphism in the desired form.
(1) For group schemes of finite type, we can prove Corollary 3.5.5 alternatively using Theorem 3.1.6(3)and (a generalisation from line bundles to general quasi-coherent sheaves with identical proof of) Lemma 2.5.1.
(2) Assume that M is one-dimensional, and Ind G H (M (j) ) = 0 for all j.It follows from 3.5.2that the morphisms in the directed system for the left-hand side in Corollary 3.5.5 for n = 0 are all non-zero.

Perfectly reductive groups
In this section we assume that k is algebraically closed of characteristic p > 0.
4.1.Definition.Recall that a reductive group over k is a connected reduced (smooth) affine group scheme of finite type which has no non-trivial normal reduced unipotent subgroup.
(1) For an affine group scheme G, the following are equivalent: (a) G is a connected affine group scheme of perfect finite type and has no non-trivial perfect normal unipotent subgroups; (b) G ∼ = G perf for a reductive group G; (c) G is an affine group scheme of perfect finite type and, if G ∼ = H perf for a reduced affine group scheme of finite type H, then H is reductive.If these conditions are satisfied, we call G perfectly reductive.
(2) For every connected affine group scheme of perfect finite type H, there exists a short exact sequence of perfect affine group schemes where U is unipotent and Q is perfectly reductive.
Proof.First we show that 1(b) implies 1(a).Set G = G perf for a reductive group G. Let U ¡ G be a perfect normal unipotent subgroup.Then by Theorem 3.1.6(2),there exists a reduced normal subgroup U ¡ G with U perf = U.By Lemma 3.4.4,U is unipotent, so U is trivial.Consequently U is trivial.That 1(c) implies 1(b) follows from Theorem 3.1.3.For H as in (2), we know that H = H perf for a connected reduced affine group scheme of finite type H by Theorem 3.1.3.By taking the perfection of the short exact sequence corresponding to the unipotent radical R u H ¡ H, see [Mi,§6.4.6], we get a short exact sequence as desired in (2) (provided we define, for now, perfectly reductive groups as the perfections of reductive groups), with U := (R u H) perf , by Lemma 3.1.8.
Since R u H is reduced, U is trivial if and only if R u H is trivial, which shows that 1(a) implies 1(c).
Remark 4.1.2.In addition to Theorem 4.1.1(2),we can also observe that every affine group scheme of perfect finite type G admits a short exact sequence H → G → Q where Q is a finite abstract group and H is a connected affine group scheme of perfect finite type.This follows from perfecting the classical theory, see [Mi,§2.g].
4.1.3.A perfect Borel subgroup B of a perfectly reductive group is a maximal solvable perfect subgroup.For a reductive group G, every perfect Borel subgroup of G perf is the perfection of a Borel subgroup of G by Theorem 3.1.6(2)and Corollary 3.1.9.In particular, every two perfect Borel subgroups are conjugate.
Similarly, by Theorem 3.1.6(2)and Lemma 3.4.3,every maximal perfect torus T in G perf is the perfection of a maximal torus T < G, and so maximal perfect tori are unique up to conjugation.
We henceforth use freely that every such choice T < B < G perf , for a reductive group G, can be obtained as the perfection of a corresponding choice T < B < G of Borel subgroup and maximal torus in G. Theorem 4.2.1.There is a canonical bijection between the set of isomorphism classes of perfectly reductive groups over k and the set of isomorphism classes of Z[1/p]-root data.
Remark 4.2.2.As in the classical case, this theorem can be extended to cover isogenies.We do this in 4.3 below.4.2.3.Idea of the proof.We will prove that, when characterising a reductive group in terms of its root datum, two reductive groups become isomorphic after perfection if and only if their root data become isomorphic after extension of scalars to Z[1/p].More explicitly: Denote by D-RD the set of isomorphism classes of D-root data.Furthermore, we let ReGr, resp.PeReGr, denote the set of isomorphism classes of reductive groups, resp.perfectly reductive groups, over k.We can exploit the classical bijection between Z-RD and ReGr, see for instance [De], and include it in the following (commutative) diagram: The upper surjection is given by the definition in 4.1.1(1)(b) of perfectly reductive groups.The lower surjection comes from Lemma 1.1.3.To prove Theorem 4.2.1, it suffices to show the dashed arrows in (4.1) exist.This is established in the following two propositions.Proposition 4.2.4.If the root data of two reductive groups G 1 , G 2 extend to isomorphic root data over Z[1/p], then (G 1 ) perf and (G 2 ) perf are isomorphic.In particular, the upwards dashed arrow in (4.1) exists.
, with i ∈ {1, 2}, denote the root datum of G i .Consider an isomorphism of root data given by ψ : Replacing ψ by p l ψ if necessary (as is allowed by Lemma 1.2.5), we can assume that ψ restricts to an embedding X 1 → X 2 .Furthermore, we get a bijection R 1 → R 2 , by associating to α ∈ R 1 the unique element α ∈ R 2 for which ψ(α) = p j α for some j ∈ Z. Again, after replacing ψ by p l ψ if necessary, we can assume j ∈ N. Now it follows quickly that this X 1 → X 2 satisfies the requirements to apply [St,Theorem 1.5], which yields an isogeny G 1 → G 2 .
We can apply the same procedure to ψ −1 to obtain an isogeny G 2 → G 1 .As we might need to replace ψ −1 again by a composition with multiplication by a power of p, our two isogenies will not necessarily be induced by mutually inverse maps X 1 ↔ X 2 , but by maps which compose to p l times the identity for some l ∈ N. Uniqueness of isogenies in [St,Theorem 1.5] then states that composition of the isogenies between G 1 and G 2 yield morphisms φ l • Fr l , for isomorphisms φ : G (1) i → G i as in 3.1.1(up to possible composition with inner automorphisms).That (G 1 ) perf and (G 2 ) perf are isomorphic now follows from Proposition 3.2.4.
Establishing the existence of the downwards dashed arrow will take more work.The following lemma can be proved by looking at the Hopf algebra morphisms.
Lemma 4.2.5.Consider a reduced affine group scheme H with a homomorphism φ : can be completed with dashed arrow to a commutative square.Then φ factors through Fr i : H (−i) → H. Definition 4.2.6.Let G be a perfectly reductive group with maximal perfect torus T.An rt-pair is a pair (x, α) of a subgroup inclusion x : G a → G and α ∈ X (i.e.α : T → G m ), for which the following square is commutative If we apply this definition to ordinary reductive groups (we replace every perfect group in (4.2) by its finite type analogue), we get precisely the pairs of inclusions of root subgroups and their corresponding root.Since root subgroups are unique, the inclusion of the root subgroup is unique up to scalar in k × ∼ = Aut(G a ).
Lemma 4.2.7.Let G be a perfectly reductive group with maximal perfect torus T.
(1) The group k × p Z acts on the set of rt-pairs as (See 3.4.5 for the definition of θ n κ and 3.4.6 for the action of (3) Consider a reductive group G with G perf ∼ = G, with maximal torus T which perfects to T. Consider an rt-pair (x, α) for which x is the perfection of some x 0 : G a → G, then α is the perfection of a root homomorphism T → G m and x 0 is an inclusion of the corresponding root subgroup.(4) For every rt-pair (x, α), there exists n ∈ p Z such that x • θ n 1 is the perfection of the inclusion of a root subgroup G a → G and n −1 α is the perfection of the corresponding root homomorphism T → G m .
Proof.Part (1) follows from a direct calculation.
For part (3), the homomorphism α : T → G m is induced from T (−i) → G m for some i ∈ N, as in Lemma 3.1.4.We find a diagram which 'perfects' to diagram (4.2).More precisely, after perfecting the above diagram and removing the automorphism which is the perfection of Fr i ×id, we recover (4.2).In particular, we find that the above diagram is commutative.It now follows from Lemma 4.2.5 that α comes from α 0 : T → G m and it follows immediately that α 0 is a root.Now we prove part (4).By Theorem 3.1.6(2)and Proposition 3.4.2(2),there exists a group monomorphism G a → G which perfects to x • φ n κ for some n ∈ p Z ,κ ∈ k × .By identifying k × with Aut(G a ), we might as well take κ = 1.The claim about α now follows from parts (1) and (3).
Finally, we prove part (2).Since roots of reductive groups cannot be multiples of one another, part (4) implies that there is n ∈ p Z for which both x • θ n 1 and z • θ n 1 are the perfections of inclusions of the same root subgroup.Those inclusions must be the same, up to a scalar in k × ∼ = Aut(G a ), from which the claim follows.
That sending a pair (θ, i) to the extension of scalars along Z → Z[1/p] of p i θ yields a bijection between (c) and (a) can be easily derived from Remark 1.5.2.
An isogeny G 1 → G yields a faithfully flat morphism G → G for high enough i, via Lemma 3.1.4.Its kernel is an affine group scheme of finite type which perfects to a finite group scheme.It must therefore be a finite group scheme and G → G is an isogeny.This principle allows us to establish a bijection between the set of isogenies G 1 → G and the set of equivalence classes of isogenies G → G, with equivalence generated by the condition that φ : G The above connection between isogenies G 1 → G and equivalence classes of isogenies G 1 → G allows us to use the classical Isogeny Theorem [St,1.5] to establish the bijection between (b) and (c).

Perfected representation theory
Let k be an algebraically closed field of characteristic p.
5.1.Simple and induced modules and block structure.Let G be a perfectly reductive group, B a perfect Borel subgroup and T < B a maximal perfect torus.We consider the set R + ⊂ R of positive roots, which are the ones for which the corresponding G a → G does not land in B (i.e.we let B be the negative Borel).
We set X = X(T) and X + ⊂ X the subset of λ ∈ X which satisfy λ, α ∨ ≥ 0 for all α ∈ R + .We have a canonical bijection {λ → k λ } between X and the set of isomorphism classes of simple B-representations (which are all one-dimensional).
(1) The representation , it has simple socle, which we denote by L(λ).
(2) The above association λ → L(λ) is a bijection between X + and the set of isomorphism classes of simple representations in RepG.
Proof.We choose a reductive group G with G perf ∼ = G and maximal torus and Borel subgroup T < B < G which perfect to T and B. We will use the notation in 3.3.2.By equivalence (3.3), every simple object in RepG is of the form L(µ)[i] for some µ ∈ X + and i ∈ N. Since L(µ) (1) ∼ = L(pµ), see [Ja,II.3],we can define unambiguously the simple object L(p −i µ) := L(µ)[i] for i ∈ N and µ ∈ X + .This clearly gives a bijection between X + and the set of isomorphism classes of simple objects in RepG.By Corollary 3.5.5,we have (5.1) where the chain of which we take the limit starts at i where p i λ ∈ X ⊂ X.It follows now from [Ja,II.2] that ∇(λ) = 0 whenever λ ∈ X + .Now consider λ ∈ X + .The morphisms in the direct system for (5.1) are not zero by Remark 3.5.6(2).Hence, the defining morphisms ∇(p i λ) (1) → ∇(p i+1 λ) are unique (up to scalar) and injective since ∇(p i λ) (1) has socle L(p i+1 λ) by Remark 5.1.3below and ∇(p i+1 λ) is the injective hull of this socle in a Serre subcategory containing ∇(p i λ) (1) .In particular, for λ, µ ∈ X + , we have We can therefore identify L(µ) with the socle of ∇(µ).
(1) We can alternatively construct the modules ∇(λ) as the global sections of line bundles on G/B ∼ = (G/B) perf .5.2.1.Let G be an affine group scheme over k, which is the extension of scalars of an affine group scheme over F p , which we also denote by G. Set F = F p .We can consider the forgetful functor from rational G-representations to representations over k of the abstract group G(F) (for instance using the homomorphism G(F) → G(k)), which induces comparison morphisms Ext i G (M, N ) → Ext i kG(F) (M, N ).Theorem 5.2.2.Let G be a perfectly reductive group, then the morphism ) is an isomorphism for all M, N ∈ RepG and i ∈ N.
We start the proof by pointing out a technical generality.
) is an isomorphism.Indeed, using group cohomology, Ext i kH (M, N ) is the cohomology of the inverse limit of chain complexes with cohomology Ext i kHn (M, N ).Since all vector spaces involved are finite-dimensional, the Mittag-Leffler property leads to the conclusion.
Proof of Theorem 5.2.2.Let G be a reductive group with perfection G and recall that G(F) = G(F) and RepG ∼ = lim − → RepG.Without loss of generality we assume that M, N factor over the natural map G → G.By Remark 3.3.4,we have a) , N (a) ).
It is proved in [CPSV] that the directed system in the above limit stabilises and moreover, for large enough a and q, the morphism Ext i G (M (a) , N (a) ) → Ext i G(Fq) (M, N ) is an isomorphism.Hence also the inverse system in lim ← − Ext i kG(F p n ) (M, N ) ∼ = Ext i kG(F) (M, N ) stabilises and we find the isomorphism in the theorem.
Strictly speaking, [CPSV] only deals with semisimple groups.However, for a general reductive group G, we have a short exact sequence N → G → G/N with G/N semisimple and N a torus.Since both N and N (F q ) ∼ = C ×r q−1 have semisimple representation theory over k, the result extends easily, for instance via a collapsing Hochschild-Serre spectral sequence.
Remark 5.2.4.The comparison morphism is not always an isomorphism for perfect groups, for instance Ext Question 5.2.5.The formulation of Theorem 5.2.2 suggests the question of whether the monomorphism (by Theorem 5.2.2) ) is also an isomorphism.This is equivalent to the question of whether the epimorphism is an isomorphism for all M, N ∈ RepG and i ∈ N. Note that (5.2) does not involve any perfection.
Example 5.2.6.The question in 5.2.5 has an affirmative answer for G = G m .Consider the short exact sequence By the Lyndon-Hochschild-Serre spectral sequence, showing (5.2) is an isomorphism can be quickly reduced to showing the group cohomology H i (Q, k) is zero for i > 0. Now the group structure on Q extends (uniquely) to a Q-vector space (since G m (F) < G m (k) is the group of roots of unity and k is algebraically closed), so we only need to show H i (Q, k) = 0 for i > 0. The case i = 1 is obvious.One can compute directly that H i (Q, −) = 0 for i > 1 (or via BQ, see [Su,(10) on p42]), hence H i (Q, −) = 0 for i > 2. Finally, ) must be an abelian group admitting both the structure of a Q-vector space as well as a k-vector space, hence it is zero.We conclude with an example showing that (5.2) being an isomorphism is also something which should not be expected to hold outside of reductive groups.
Example 5.2.7.If instead of a reductive group, we consider G = G a , as well as i = 1, M = N = k in (5.2), we obtain the morphism between spaces of group homomorphisms End(k + ) → Hom(F + , k + ), induced by restriction along F → k.This is not a bijection as soon as F = k.

Localisation of classifying spaces
Fix a prime p.
6.1.Main result.Following [Su], by a simple space we mean a connected topological space having the homotopy type of a CW complex and abelian fundamental group which acts trivially on the homotopy and homology of the universal covering space.Let F be a connected topological group of homotopy type of a CW complex (below we will consider more specifically complex Lie groups), then its classifying space BF is a simple space (note that π 1 (BF ) = π 0 (F ) is trivial).Following [Su,Chapter 2], to each simple space X we can associate a (simple) space X 1 p , the localisation of X away from p.Note that loc.cit.X 1 p is denoted by X where is the set of all primes different from p. Assume that the perfections of G k and H k are isomorphic.By Proposition 3.2.4,after replacing H k with an (isomorphic) Frobenius twisted version, there is an infinitesimal isogeny G k → H k .By Lemma 3.2.2, this isogeny is purely inseparable.It then follows from [Fr,Theorem 1.6] that BG(C) 1 p and BH(C) 1 p are homotopy equivalent.
The rest of this chapter is devoted to the proof of (b) ⇒ (c).6.2.Some useful facts.(a) For a complex reductive group F and a maximal compact subgroup K < F (the corresponding compact connected Lie group), the homomorphism K → F is a homotopy equivalence and hence BK BF .We will therefore henceforth replace BF by BK.(b) For a simple space X, the defining map X → X 1 p , see [Su,Chapter 2], induces an isomorphism The Weyl group W thus acts R-linearly on H 2 (BT, R).Moreover the image of takes values in the algebra of W -invariants, see [Bor,§27].(g) For a prime q and a connected CW complex Y , we denote by Y q the profinite completion at q of Y , see [Su,Chapter 3].If q = p, then the universality of X → X 1 p in the definition in [Su,Chapter 2] shows that the latter map induces a homotopy equivalence X q (X 1 p ) q.
(h) For G a compact connected Lie group, BG satisfies the requirement in (c), i.e. the homology groups H i (BG; Z) are finitely generated.One can observe this for instance via induction on i using the Serre fibration G → EG → BG.Note that EG is contractible, BG is simply connected and G is a finite cell complex.The Leray-Serre spectral sequence thus implies that the trivial group can be obtained, starting from H i (BG; Z) by a finite iteration of taking kernels of morphisms to finitely generated groups (subquotients of H a (BG; H b (G)), with a < i).Consequently, H i (BG; Z) must also be finitely generated.
6.3.Some results of Adams and Mahmud.We reformulate some results of Adams and Mahmud in the form we will need.
Theorem 6.3.1 (Adams -Mahmud).Let G and G be two compact connected Lie groups, with maximal tori T, T and Weyl groups W, W .
(1) For a map f : (BG)  2) is a reformulation of [AM,Theorem 1.7].The proof loc.cit.works for any field of characteristic zero.The case of integral domains follows from extension of scalars to the field of fractions, using (d).
The second statement of part (1) now follows from part (2) and fact (f), by using θ 1 = θ and θ 2 = wθ.Proof of (b) ⇒ (c).Assume first that p = 2. Then the result follows from Corollary 6.3.2 and Lemma 1.1.6(1).Similarly, for p > 2, by Lemma 1.1.6(2)it is sufficient to prove that the extension of scalars along Z[1/p] → Z 2 of t yields an isomorphism of Z 2 -root data.This allows us to resort to the established theory of 2-compact groups, see [AG].
Starting from a homotopy equivalence f as in Theorem 6.3.1, we have our θ from 6.3.1(1) which induces the isomorphism of reflection groups in Corollary 6.3.2, and using (g) we have a homotopy equivalence f 2 .We consider the diagram (BG) 2 O O Now (BT ) 2 → (BG) 2 is a maximal torus of the 2-compact group (BG) 2 , in the sense of [Gr,Theorem 2.2].By uniqueness of such maximal tori, see loc.cit., there exists a homotopy equivalence corresponding to the dashed arrow in the above diagram so that the diagram is commutative up to homotopy.
By [Su,Theorem 3.9] By uniqueness in Theorem 6.3.1(2)applied to D = Z 2 , we may assume that φ is actually induced from θ by extension of scalars Z[1/p] → Z 2 .Finally, the Z 2 -root data of the 2compact group (BG) 2 , as defined in [AG], is obtained from the map BT 2 → BG 2 and by construction yields the extension of scalars along Z 2 of the classical root datum of G.
The homotopy equivalence (BG) 2 (BG ) 2 with commutative diagram therefore indeed implies that our isomorphism of Z[1/p]-reflection groups extends to an isomorphism of Z 2root data.

Perfected SL 2
Let k be an algebraically closed field of characteristic p.For λ in N or N[1/p] := Z[1/p] ∩ R ≥0 , we consider its p-adic expansion λ = i λ i p i with 0 ≤ λ i < p. for some l ∈ N for which p l i, p l j ∈ N. By Lucas' theorem, this definition does not depend on the choice of l.
Here, the first map is given by the action of the Frobenius twist, so is injective by Remark 5.1.3.The second map comes from the inclusion ∇(p l µ) (1) → ∇(p l+1 µ).From the description of the cokernel of the inclusion in (7.1) it follows that the second map is also injective for p > 2. For p = 2 the second map need not be injective, but one shows easily that the composite is still injective.Assume p > 2, the case p = 2 can be proved similarly.It follows quickly from [Pa,Corollary 6.2], that for 0 ≤ i < p

Figure 1 .
Figure 1.Characters of simple modules for SL 2 in characteristic p = 3.

G
a := (G a ) perf and G m := (G m ) perf for the perfection of the additive and multiplicative group of k.Proposition 3.4.2.Assume that k is algebraically closed.
The claim now follows by induction on n, using long exact sequences in homology if we observe thatI := lim − → p * i I i is injective in Rep ∞ A.Exactness of Hom A (−, I) : RepA → Vec ∞ follows from the observation RepA ∼ = lim − → RepA i .The latter exactness is sufficient to conclude that I is injective.Indeed, we can consider an injective hull I ⊂ I and an intermediate module I ⊂ I ⊂ I for which I /I is finite dimensional.Now we must have I ∼ = I ⊕ I /I (apply Hom(−, I) to a finite submodule of I which still surjects onto I /I) which violates socI = socI unless I = I.Corollary 3.5.5.Let G be an affine group scheme with subgroup H

4. 2 .
Classification.We freely use the equivalence between D-root data and real-type Droot data for D = Z and D = Z[1/p] from Theorem 1.4.1.
Remark 5.2.3.Consider a chain of finite abstract groups {H n | n ∈ N} and set H = lim − → H n .For two finite dimensional H-representations M, N , the canonical morphism Ext Theorem 6.1.1.Let G, H be split (connected) reductive groups over Z.The following are equivalent: (a) The perfections of G k and H k are isomorphic for k = F p ; (b) The localisations BG(C) The root data of G and H become isomorphic after extension to Z[1/p].Proof of (c) ⇔ (a) ⇒ (b).The equivalence of (a) and (c) is already established in 4.2.3.

0 , 0 = 0
1.For i, j ∈ N, denote by i j the zero coefficient of the p-adic expansion of the binomial coefficient.By convention, i j if j > i.For i, j ∈ N[1/p], we set 1, • • • , p − 1},
and refer to the connected components of the complement of H in V as chambers.Fix one such chamber A 0 .Denote by S ⊂ T the set of reflections s for which the intersection of A 0 and H s cannot be contained in a codimension 2 hyperplane.Our assumptions in 1.3.1 imply that W acts on the (finite) set of chambers.
c) For a commutative ring D in which p is invertible, by (b) and the universal coefficient theorem, we have a natural isomorphism H * (X 1 d) For a flat morphism A → B of commutative algebras and a topological space X for which H i (X; Z) are all finitely generated, H * (X; A) ⊗ A B → H * (X; B) is an isomorphism.(e)For a torus T ∼ = (S 1 ) ×r , H * (BT ; Z) is a polynomial ring with r generators in degree 2.Consequently, for any commutative ring R, H (f) For a compact connected Lie group G, with maximal torus T , and a commutative ring R, consider canonical isomorphisms p ; D) ∼ = H * (X; D).In particular, a map * (Y ; D) → H * (X; D). (* (BT ; R) ∼ = H * (BT ; Z)⊗R is a polynomial ring and we have a bijection between R-linear morphisms θ : H 2 (BT ; R) → H 2 (BT ; R) and graded R-algebra morphisms θ : H * (BT ; R) → H * (BT ; R).