On unipotent algebraic G-groups and 1-cohomology

In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction map H^1(G,Q)\to H^1(B,Q) is an isomorphism. We also show that this situation admits a notion of rational stability and generic cohomology. We use these results to obtain corollaries about complete reducibility and subgroup structure.


Introduction
The low degree cohomology of a group G with coefficients in another group Q on which G acts gives rise to important invariants in group theory. For instance in degree 0, H 0 (G, Q) is the group of fixed points of the action of G on Q, H 1 (G, Q) is the set of conjugacy classes of complements to Q in the semidirect product GQ, and H 2 (G, Q) (where defined) measures the number of non-equivalent group extensions 1 → Q → E → G → 1.
In this paper we consider the situation where Q is in general, a non-abelian G-group. In such a situation H 1 (G, Q) is just a pointed set (in contrast to the situation that Q = V is a representation for G, when H 1 (G, V) is a vector space). One needs to work harder to employ the usual tools of cohomology since exact sequences do not, unfortunately, say very much about pointed sets; for instance, if 1 → A → B, is exact, the map A → B need not be an injection, merely that anything mapping to the distinguished element in B, should have been the distinguished element in A. However one can usually rescue the situation for non-abelian 1-cohomology by employing 'twisting', a technique familiar in Galois cohomology.
Before we describe our results, let us say at the outset, that we will be consider in detail the situation where G is a (possibly reductive) algebraic group over an algebraically closed field k, and Q is a unipotent algebraic G-group. However up to the end of §3.1 all our results apply equally in the category of abstract groups. For clarity we give our results up to this point in the abstract setting and then in §3.2 indicate the appropriate modifications to use these results in the rational (algebraic) situation; the proofs are the same, mutatis mutandis.
Letting first G be an abstract group and Q an abstract G-group, we give in §2.1, the well-known definitions of the 0th, 1st and 2nd cohomology groups of G with coefficients in Q in terms of cocycles (where these definitions exist) as well as the notion of twisting an action by a cocycle. When Q fits into a short exact sequence 1 → R → Q → S → 1 and where the image of R in Q is contained in the centre Z(Q) of Q, there is an exact sequence of non-abelian cohomology (Proposition 2.2.1(i)). This exact sequence fits into various commutative diagrams (Proposition 2.2.1(ii)-(x)). In particular we consider the restriction of this exact sequence to a subgroup B of We apply this in the special situation where G is an algebraic group acting (morphically) on a unipotent algebraic group Q. We first show that Q has a central filtration by G-modules (Theorem 3.3.5). This filtration allows us to go further than the results obtained in [Ric82,§6] where this general set-up was first considered; in particular, one can bring second cohomology into play. 1 Perhaps surprisingly, it turns out that unipotent algebraic groups are just as good as vector spaces as far as low-degree cohomology is concerned. By way of demonstration, we can use our work to generalise several of the results in the paper [CPSvdK77]. Consider the case where G is connected and reductive; and let P be a parabolic subgroup of G. In [CPSvdK77], it is proved that H n (G, V) H n (P, V) for any finite dimensional rational G-module V. We use our filtration to extend this result by applying our Theorem 3.1.1 inductively. We show that H 1 (G, Q) H 1 (P, Q) for any unipotent algebraic group Q on which G acts morphically.
The main result of [CPSvdK77] is the proof of the existence of rational stability and generic cohomology. Let G be additionally defined and split over the prime subfield F p . If V [r] denotes the rth Frobenius twist of V then it shown that for each m ∈ N and V a finite dimensional module there is some r 0 with H m (G, V [r] ) H m (G, V [r 0 ] ) for any r ≥ r 0 . In addition, there is some s 0 such that 1 We remark that after this paper was submitted it came to our attention that George McNinch has a preprint establishing the existence of such a central filtration, while working in the more general context of algebraic groups over arbitrary fields k. In such a situation one can have unipotent radicals of algebraic groups which are not defined over k. McNinch defines a condition (RS): the unipotent radical of G is defined and split over k. He then proves that when (RS) holds, such a filtration of Q exists. Now, since (RS) holds automatically for k =k our result can be recovered from his. Our proofs, however, appear to be substantively different. The one given here is significantly shorter, no doubt owing to the technical difficulties involved in McNinch's more general result.
is an isomorphism. We extend each of these theorems in the case m = 1, replacing V with a unipotent algebraic group (Theorem 3.5.6).
The above has a number of corollaries on subgroup structure related to Serre's notion of G-complete reducibility. If H is a closed, connected, reductive subgroup of a parabolic subgroup P of G, one can find its imageH under the map π : P → L for L a Levi subgroup of P. One need not have H a complement to Q inHQ, even though it is when considered as an abstract group. Where it is a complement, it corresponds, up to Q-conjugacy, to an element of H 1 (H, Q); we show in Lemma 3.6.1 in almost all cases, H is a complement, and even when it is not, it still corresponds to an element of H 1 (H, Q [1] ). This gives a cohomological characterisation of conjugacy classes of subgroups of parabolics. We can then prove results on G-complete reducibility using this characterisation, via a result of Bate

Definitions
We start with some standard material. Let G be a group and let Q be a G-set, that is, a set equipped with a right action of G.
Then we define H 0 (G, Q) as the set Q G of fixed points of G acting on Q, i.e. H 0 (G, Q) = {q ∈ Q : q g = q, ∀g ∈ G}.
If further Q is a G-group, then we may define the 1-cohomology H 1 (G, Q) as follows.
Firstly, a 1-cocycle is a map γ : G → Q satisfying the cocycle condition, i.e. γ(gh) = γ(g) h γ(h). The set of all 1-cocycles is denoted by Z 1 (G, Q). We say that two 1-cocycles γ, δ are cohomologous and write γ ∼ δ if δ(g) = q −g γ(g)q for some element q ∈ Q. Then we define H 1 (G, Q) as the set of equivalence classes There is a distinguished point in Z 1 (G, Q) given by the trivial cocycle 1 : G → Q; 1(g) = 1. The equivalence class of the trivial cocycle, [1] ⊆ Z 1 (G, Q) is the set of coboundaries and we denote this distinguished point in H 1 (G, Q) by B 1 (G, Q).
H 2 (G, R) := Z 2 (G, R)/ ∼. Again there is a distinguished point in Z 2 (G, R) given by the trivial 2-cocycle 1 and its class in H 2 (G, R) is denoted by B 2 (G, R).

Twisting, sequences and commutative diagrams
Recall from [Ser94, I. §5.3] the notion of twisting the action of G on a G-group Q by a cocycle in Z 1 (G, Q). To wit, fix an arbitrary 1-cocycle γ ∈ Z 1 (G, Q) and define q * g = (q g ) γ(g) = γ(g) −1 q g γ(g) for each q ∈ Q. 2 One checks this is a new action of G on Q. We denote the new G-group by Q γ , which coincides with Q as a group.
We give a few facts about the set-up so far.
be a short exact sequence of G-groups, with the image of R in Q contained in the centre Z(Q) of Q and with σ a section of π (σ not necessarily a homomorphism). Additionally, fix γ ∈ Z 1 (G, Q) and denote the induced element in Z 1 (G, S) also by γ. Then the following hold: (i) There is an exact sequence of cohomology (ii) Moreover, the association of the short exact sequence to the longer is functorial: Suppose we have another short exact sequence of G-groups with the image of R ′ in Z(Q ′ ) and with σ ′ a section of π ′ ; and a commutative diagram where the vertical arrows are G-equivariant homomorphisms. Then there is a 2 More generally, R can be a further right G-set with a right action on Q which commutes with the action of G on Q, i.e. (q r ) g = q gr g . Then one can define q * g = q gγ(g) for q ∈ Q, g ∈ G and γ ∈ Z 1 (G, R).
commutative diagram of the two sequences arising from (i) above.
(iii) There is another short exact sequence of G-groups with the image of R contained in the centre of Q γ . Moreover in the situation of (ii), there is a commutative diagram where γ ′ indicates the cocycle η • γ.
(iv) Thus we get by (i) a new exact sequence of cohomology: which is also functorial in the manner of (ii).
(v) The map where δγ denotes the map g → γ(g)δ(g), is a well-defined bijection, taking the trivial class in H 1 (G, Q γ ) to the class of γ in H 1 (G, Q).
(vii) If β ∈ Z 1 (B, Q) denotes the restriction of γ to B, then we have the following commutative diagram commutes: on elements we have (iv) is clear from (i) and (iii).
(vi) is easy to check from the definitions of the maps concerned. For instance, Putting together the components of the last proposition: where rightward arrows are part of four exact sequences running through the central vertical square.
(ii) With the hypotheses of Proposition 2.2.1(ii), we have the following commutative partial cuboid Proof. For the first partial cuboid, the front and back faces are subdiagrams of Proposition 2.2.1(vi),(viii); the top and bottom faces commute by (ix); the vertical squares commute by (x).
For the second partial cuboid, the front and back faces commute by Proposition 2.2.1(ii) and (iv). The vertical squares commute by (ix).

Versions of the five lemma
the following hold: (i) If h 2 and h 4 are surjective and h 5 is injective, then h 3 is surjective.
(ii) If h 2 and h 4 are injective and the restriction maps S G γ → S B β are surjective for any γ ∈ Z 1 (G, S) with γ| G B = β, then h 3 is injective. (iii) If the hypotheses of (i) and (ii) hold, then h 3 is an isomorphism.
Proof. Assume the hypotheses of (i) and take γ ∈ H 1 (B, Q). We need to produce a pre-image β, say, such that h 3 (β) = γ. We get started by the diagram chase used to prove the five lemma for diagrams of abelian groups.
The picture is now as follows: where we would like to assert the existence of an element replacing '?' and mapping to γ.
In the case of abelian groups, one would continue the proof of the five lemma by taking the difference γ − θ; observing that this maps under π B to 1 and continuing the diagram chase. Since we cannot do this in the case of pointed sets we use twisting to reduce to the case θ = 1. For this we use Proposition 2.2.2(i), replacing the back face with the middle of the diagram above. * Here we are using the fact that the bijection in Proposition 2.2.1(v) takes the trivial element in H 1 (G, Q η ) to the element η in H 1 (G, Q) and the fact that the partial cuboid is commutative. Now in the front bottom row, as γ ′ ∈ ker π ′ B , we have γ ′ ∈ im ι ′ B . Thus we may replace ? 1 with an element κ mapping to γ ′ . Since h 2 is a surjection, we may replace ? 2 with an element λ mapping to κ. Then we replace ? 3 with µ = ι ′ G (λ) and by the fact that the front left square commutes, h ′ 3 (µ) = γ ′ . Finally if we replace ? with ν := θ η (µ) the commutativity of the central vertical square gives us our preimage of γ.
For (ii) the picture in the partial cuboid is as follows: ) and hence is equal to 1 since the composition of these two maps is trivial. Thus γ = δ since θ γ is a bijection.
(iii) is now obvious.
An identical proof using Proposition 2.2.2(ii) shows: (iii) If the hypotheses of (i) and (ii) hold, then h 3 is an isomorphism.

Proposition 3.1.3. Assume the hypotheses of 2.2.1(ii) and let B be a subgroup of G.
Then in each of the following diagrams: and (i) If h 2 and h 4 are surjective and h 5 is injective, then h 3 is surjective.
(ii) If h 2 and h 4 are injective then h 3 is injective.
(iii) If h 2 and h 4 are isomorphisms and h 5 is an injection then h 3 is an isomorphism.
Proof. The usual proof of the five lemma [Rot09, 2.72] goes through in these cases. Where one would take the 'difference' of two elements g and h in an abelian group one uses the element gh −1 . The proof then works as normal.

Corollary 3.1.4. Suppose Q is a G-group admitting a filtration by G-groups Q
(a) If B is a subgroup of G and for all 1 ≤ j ≤ n we have then H 1 (G, Q) H 1 (B, Q).

(b) Suppose Q ′ is another G-group admitting another such filtration by G-groups
Proof. (a) This is a simple induction using the previous results. Firstly the fact that Q j is normalised by both Q and G mean that that (Q γ ) j := (Q j ) γ gives a filtration of Q γ ; we wish to show that the hypotheses (i) and (ii) give us: (*) For Suppose (*) is known to hold for all G-groups with filtrations up to n − 1, then we must just check that Q G ։ Q B . So set R = Q n = (Q γ ) n , and S = Q γ /(Q γ ) n . The hypotheses of Proposition 3.1.3(i) now hold and we have Q G ։ Q B as required; proving (*).
For the conclusion of (a) note that the case n = 1 is immediate from the hypotheses. If (a) is known to hold for G-groups admitting a filtration of length up to n − 1 then set R = Q n , and S = Q/Q n , then by (*) the hypotheses of of Theorem 3.1.1(iii) hold. We conclude H 1 (G, Q) H 1 (B, Q).
Remark 3.1.5. The results in this section involving B exist in a more general form which we do not need for the purposes of this paper. It is not necessary for B to be a subgroup of G-merely that there is some homomorphism ρ : B → G.

Algebraic G-groups
Let G be an algebraic group defined over an algebraically closed field k of characteristic p ≥ 0. Let Q be another algebraic group defined over k. Then we say Q is an algebraic G-group if Q is a G-variety (i.e. the map G×Q → Q; (g, q) → q g is a morphism of varieties), and G acts as group automorphisms of Q. In this case, following [Hum75,8.4] one can form a semidirect product GQ with multiplication defined by g 1 q 1 .g 2 q 2 = g 1 g 2 q g 2 1 q 2 . One checks that the results and proofs of the previous sections apply to the category of algebraic groups provided one is clear on the nature of this category: Firstly, one insists that maps are morphisms of varieties, i.e. continuous in the Zariski topology. Thus by definition Z i (G, Q) is the set of regular maps γ : i 1 G → Q satisfying the appropriate cocycle condition. This moreover ensures that G acts regularly on Q γ . 3 With this change note that where we have made cocycle-coboundary definitions for H i (G, Q), these coincide with the usual rational (Hochschild) cohomology definitions of cohomology for algebraic groups in the case where Q is a (rational) G-module, such as those used in [CPSvdK77]. This is all worked out carefully in [McN10,4.1]. This means in particular that the results and proofs quoted in Proposition 2.2.1 go over to our situation; this applies especially to results quoted from [Ser94].
Secondly, by a subgroup we will mean always a closed subgroup; i.e. the image of an injection of algebraic groups. For instance, since a cocycle γ in Z 1 (G, Q) is a morphism γ : G → Q, the subgroup {gγ(g) : g ∈ G} of the semidirect product GQ is now a closed subvariety of GQ. By a complement to Q in GQ, we use the definition in [McN10,4  Note that a sectioned extension is in particular a strongly exact sequence; that is to say that the map dπ : Lie Q → Lie S is onto. If Q is a sectioned extension of algebraic groups R and S with R abelian, it follows that Q corresponds to a (regular) 2-cocycle class in H 2 (S, R): e.g. [Ser88, VII.4(a)]. (Obviously in the category of abstract groups, all extensions will be 'sectioned', since they need simply to be defined set-wise.) We record the obvious fact:

Lemma 3.2.5. For any algebraic group G and any G-group Q, the extension of Ggroups
Remark 3.2.6. Much of the cohomology theory for algebraic groups predicated on regular maps fails if one considers non-sectioned extensions of algebraic groups. We thank Steve Donkin for the following example. Let R = F p , Q = G a and S = Q/F p , where G a is the 1-dimensional additive group. (Then S is isomorphic to G a too.) Imagine there were a morphism of varieties σ : S → Q with σ a section of the quotient map Q → S. Then q + F p = πσ(q + F p ) = σ(q + F p ) + F p . So σ(q + F p ) − q ∈ F p . Since S is connected, F p is discrete and σ is continuous, the image of this map must be constant, hence 0. So σ(q + F p ) is constantly q. If 0 r ∈ F p then σ(q + r + F p ) = q = q + r. So r = 0; a contradiction.
Indeed the extension 1 → R → Q → S → 1 cannot correspond to a 2-cocycle φ : S × S → R since such regular cocycles from a continuous to a discrete group must again be trivial.
Worse, letting G S, no such exact sequence of Proposition 2.2.1(i) can exist using rational cohomology: if it did, one would have with H i (G, R) = 0 for i = 1, 2, as continuous maps γ : G → R or γ : G × G → R must be constant (and so cocycles are zero maps). Thus we would have H 1 (G, Q) H 1 (G, S). The latter contains the identity map, but the above argument also shows there can be no preimage; a contradiction.
Happily, sectioned extensions exist in some general settings as in the lemma below. We need some further definitions: Definition 3.2.7. Let G be a group and Q a G-group. We say that a (finite) filtration ; we say that it is a filtration by G-groups if each Q i is normalised by G; we say it is a filtration by G-modules if it is a filtration by G-groups and for each i, Q i /Q i+1 has the structure of a (rational, linear) G-module; finally, we say that it is sectioned if for each i there is a map Q/Q i → Q.
The following is a result of Rosenlicht.

Lemma 3.2.8 (c.f. [Spr98, 14.2.6]). Let Q be a connected unipotent algebraic subgroup of an algebraic group G. Then (i) there exists a (regular) section to the morphism G → G/Q of varieties;
(ii) G is isomorphic (as a variety) to G/Q × Q.
In particular, any filtration of Q by unipotent subgroups is sectioned.

Unipotent algebraic G-groups
Here we prove that any connected unipotent algebraic G-group Q admits a (sectioned) central filtration by G-modules. This allows us to use results like 3.1.4. First we need four technical lemmas. Proof. Since V is a G-module, there is a G-equivarient isomorphism between V and T 0 (V), its tangent space at the identity. Under this map, M is sent to a Recall that a vector group V is an algebraic group isomorphic to a direct product of n copies of G a , the additive group of the field.

Lemma 3.3.2. Let U be a connected unipotent algebraic group over a field k of characteristic p > 0, V a vector group over k and let π : U → V be an isogeny. Then U is a vector group, hence isomorphic to V.
Proof. If u ∈ U, π(u p ) = pπ(u) = 0, so the map u → u p is a continuous map from U to the kernel of π. Such a map must be constant as U is connected, thus identically 0. Hence u p = 0. By [Ser88,Proposition 11], U is thus a vector group; dimensions imply that U V. Proof. Let dim V = n. Without loss of generality we may enlarge F to be a subgroup F q × · · · × F q n times ≤ V for some q = p r . Let v ∈ V and take λ ∈ k with λ F q .
Then for each g ∈ G we have Substituting (1) into (2) we have Thus we have λ f 1 = f 3 − f 2 ∈ F. Since λ F q , we must have f 1 = 0. We conclude g(λv) = λg(v) for any λ F q . Now fix a finite subfield F s of k with F q F s and take λ a generator of the multiplicative group F × s . Then for any 0 µ ∈ F q , µ = λ t for some t ∈ N. But for any g ∈ G and v ∈ V, Thus g acts linearly on V.

Lemma 3.3.4. Let Q be a connected unipotent algebraic G-group with a (central) filtration {Q i } such that successive quotients have the structure of G-modules or finite groups. Then Q has a (central) filtration by G-modules.
Proof. Let m := |{i : Q i /Q i+1 is finite}|. Take j minimum such that A := Q j /Q j+1 is finite. Then C := Q j−1 /Q j is a G-module. Letting B = Q j−1 /Q j+1 , we have the following complex induced from the short exact sequence 1 → A → B → C → 1: Clearly ι is an injection; the image of B • under π is a connected subgroup of C of the same dimension as C hence is equal to C, showing that π is surjective. If b ∈ ker π then we obviously have b ∈ A ∩ B • ; thus the complex is a short exact sequence.
Now, B • is a connected unipotent group isogenous to the vector group C. By i } is a filtration of Q by G-modules or finite groups with j − 1 the minimum value of i such that Q i /Q i+1 is finite. By induction we get a filtration {R i } of Q with Q/R 2 finite. This is the image of a connected group and is hence trivial. Thus {R i } i≥2 is a (central) filtration of Q by G-modules and finite groups with m − 1 successive quotients being finite.
Repeating this process, we get a central filtration Q by G-modules with no finite quotients.
We are now in a position to prove our second main result.

Theorem 3.3.5. Let G be an algebraic group over an algebraically closed field k and Q a connected unipotent G-group. Then Q has a (sectioned) central filtration by G-modules.
Proof. Form the semidirect product H = GQ and embed it as a closed subgroup of GL n (k), in such a way that Q is represented by strictly upper triangular matrices. Write M n (k) for the algebra of n × n matrices over k and regard M n (k) as a G-module via matrix conjugation. Let A be the subalgebra of M n (k) generated by Q ≤ GL n (k) ⊂ M n (k). Then every element in Q, and therefore A, is λ.I + x n for x n nilpotent; thus A is a local algebra with radical J, say; in fact, J is the ideal given by all the nilpotent elements of A. Now G acts on A by algebra automorphisms; thus G also acts as a group of automorphisms of each power J i of J since each is a characteristic ideal. Equally, G acts on each of the multiplicative groups 1 + J i , again by matrix conjugation. Note that for j 1 ∈ J i , and j 2 ∈ J we have (1 + j 1 )(1 + j 2 ) = (1 + j 2 )(1 + j 1 ) modulo J i+1 and so (1 + J i )/(1 + J i+1 ) ≤ Z ((1 + J)/(1 + J i+1 )). Moreover, there is a G-equivarient automorphism from the additive G-group J i /J i+1 to the multiplicative G-group (1 + J i )/(1 + J i+1 ). So R = 1 + J has a central filtration by G-modules. We may refine this to a filtration R = R 0 > R 1 > R 2 > · · · > 1 with consecutive quotients being simple G-modules. Now, since Q was unipotent, Q − 1 ⊂ A consists of nilpotent matrices and therefore contained in J. So Q ⊆ R = 1 + J. Thus, letting Q i = Q ∩ R i , we have a filtration of Q with successive quotients being either simple G-modules, or finite G-groups by 3.3.1. By Lemma 3.3.4, there is now a central filtration of Q by G-modules.
Recall that such a filtration is automatically sectioned by Lemma 3.2.8.

Non-abelian cohomology of a group extension
The following is a generalisation (by a small amount) of the five-term exact sequence from the Lyndon-Hochschild-Serre spectral sequence to our nonabelian situation. Here our groups can be algebraic or abstract, bearing in mind the impositions made in §3.2 Assuming the hypotheses for (i), take δ and γ in Z 1 (G, Q) with δ(u) = q −u γ(u)q for each u ∈ U. We wish to show δ ∼ γ. First, replacing γ by the cocycle g → q −g γ(g)q we have δ(u) = γ(u). Now consider the exact sequence The image of δ under the map θ −1 γ given in Proposition 2.2.1(v) is now in the kernel of res. Thus θ −1 γ (δ) = 1 by exactness of this sequence. So δ = θ γ (1) = γ. Statement (ii) is similar. Note that the definition given there is in the situation of abstract groups-it is possible that the definition given is not even valid for algebraic groups, since a choice of a (possibly non-regular) map is called for.
The lemma has the following consequences for algebraic groups. For the second part, since Z(G) • is a torus, we also have H 1 (Z(G) • , Q) = 0 for any unipotent algebraic group Q. Thus the statement follows from Lemma 3.4.1(ii).

Application: 1-cohomology of reductive G with coefficients in a unipotent group
In this section, G will be a connected, reductive group defined over an algebraically closed field k and B a Borel subgroup of G. Recall that for any parabolic subgroup P of G, there is a short exact sequence where R u (P) is the unipotent radical of P, and L is a complement to R u (P) in P, i.e. a Levi subgroup. We have a splitting homomorphism ι : L → P such that π • ι is the identity map on L.
In [CPSvdK77], Cline, Parshall, Scott and van der Kallen proved a result which has the following generalisation Using the results of the previous sections we will generalise this result in the case n = 1, by replacing V by an arbitrary unipotent G-group Q. Remark 3.5.3. A more general result when n = 0 is clear: for any affine G-variety X we may choose a G-equivarient embedding of X into a G-module W with fixed points To prove our second generalisation of results in [CPSvdK77], let us recall (and extend) the notion of a Frobenius twist-if G is defined over a field k of characteristic p one has Frobenius morphisms F r = F r 1 : G → G (r) induced by the field automorphism σ q : k → k by x → x p r (see [Jan03,I.9.2]). Where G is defined over the prime field k 0 ≤ k we have F r (G) G. In this case, if Q is any algebraic Ggroup (for instance a G-module) then one can define a new G-group Q [r] which is identical to Q as a group, but where we twist the G-action by precomposing with F r . If both Q and the action map G × Q → Q are also defined over k 0 , one has a G-equivariant map of algebraic groups given by Q → Q [r] ; call this the twist map. In the case Q is a G-module, it is also k-semilinear. Denote the fixed points G(k) F r of the group of k-points G(k) by G(q). If G is connected reductive, G(q) is called a finite reductive group.
The following is an easy improvement of the main result [CPSvdK77, 6.6] to the case where G is connected and reductive. We will only need it in the case that m = 1. ) for all n ≥ 1, the former equality since |Z(q)| is coprime to p, the latter as Z is connected. Now, Lemma 3.5.4 asserts the existence, for each G-module Q j /Q j+1 an integer r j such that for r ≥ r j restriction induces isomorphisms H i (G, Q j /Q j+1 ) → H i (G(p r ), Q j /Q j+1 ) for i = 0, 1, and an injection H 2 (G, Q j /Q j+1 ) → H 2 (G(p r ), Q j /Q j+1 ). Take r 0 = max j r j . Then for any r ≥ r 0 , take B = G(p r ); we see the hypotheses of 3.1.4(a) hold and we conclude H 1 (G, Q) H 1 (G(p r ), Q). Now we deal with the case where Q is not connected; note again that Q and Q/Q • are defined over k 0 . We have H 1 (G, Q) H 1 (G, Q • ) as G is connected.
We claim there is also an isomorphism H 1 (G(q), Q • ) H 1 (G(q), Q) for all sufficiently large q. Clearly this will now finish the proof in the non-exceptional case of the lemma. To prove the claim, from the short exact sequence we deduce that in the exact sequence of Proposition 2.2.1(i) the map (Q/Q • ) G = Q/Q • → H 1 (G, Q • ) is trivial. We have shown that there exists an integer r 0 with H 1 (G, Q • ) → H 1 (G(q), Q • ) if q = p r and r ≥ r 0 . Thus using the commutative diagram of of 2.2.1(vi) with G(q) taking the role of B, we see the map (Q/Q • ) G → H 1 (G, Q • ) is also trivial. We now wish to show H 1 (G(q), Q/Q • ) = {1}. Since G acts trivially on Q/Q • , so does G(q). As Q/Q • is a p-group, we can take a filtration of it by subgroups isomorphic to F p under addition. Now q is already big enough so that H 1 (G(q), k) = 0 from Lemma 3.5.4. Since H 1 (G(q), F p ) → H 1 (G(q), k) = 0 is clearly an injection, we have H 1 (G(q), Q/Q • ) = {1} by applying the long exact sequence of Proposition 2.2.1(i) inductively through this filtration of Q/Q • . One last application of Proposition 2.2.1(i) applied to 1 → Q • → Q → Q/Q • → 1 yields the exact sequence 1 → H 1 (G(q), Q • ) → H 1 (G(q), Q) → 1.
An easy twisting argument now shows that H 1 (G(q), Q • ) → H 1 (G(q), Q) is an isomorphism of pointed sets, thereby proving the claim.
In the exceptional case when p = 2, given in the theorem exactly the same proof shows that H 1 (G, Q [1] ) H 1 (G, Q [2] ), using the exceptional case from [CPSvdK77, 7.1] and the fact that for a simple algebraic group G, H 1 (G, V) H 1 (G, V [1] ) for a simple module V unless p = 2, G = Sp 2n and V = V(ω 1 ). 5 Definition 3.5.7. Following [CPSvdK77, 6.7(b)], we denote by H 1 gen (G, Q) the stable value of H 1 (G(p r ), Q).
Remark 3.5.8. If one knows the weights of G on a filtration of Q, one can establish an explicit value of r 0 in the theorem above using the proof of 3.5.4 and the explicit constants given for e 0 and f 0 in [CPSvdK77].

Subgroup Structure
Finally, we give a corollary on subgroup structure. For that we need the following lemma, which pins down when abstract complements in semidirect products of connected, reductive groups with unipotent groups are complements also as algebraic groups. Thus there is a bijection between the set of conjugacy classes of closed, connected, reductive subgroups H of GQ withH = G and the set H 1 (G, Q [1] ). J → K and induces an isomorphism on Lie algebras, π| J is an isomorphism by [Spr98,5.3.3]. Now we can define a map σ : G → H. Let σ i be the other well-known special isogeny Sp 2n i → SO 2n i +1 for 1 ≤ i ≤ m, and τ : K → J the inverse of π| J . Then we define σ : G → H by setting σ| H i = σ i for 1 ≤ i ≤ n and σ| K = τ.
Then the composition of maps G σ → H π → G induces the first Frobenius endomorphism F : H i → H i for 1 ≤ i ≤ m, and an isomorphism on K. So under this morphism we have g → h → hγ(h) −1 = F(g) for g ∈ H i . Defining δ = γ • σ we have h = F(g)δ(g). Now one checks that δ defines a cocycle in Z 1 (G, Q [1] ): for g 1 , g 2 ∈ G i with 1 ≤ i ≤ m, we have F(g 1 )δ(g 1 )F(g 2 )δ(g 2 ) = F(g 1 )F(g 2 )δ(g 1 g 2 ) by the fact that σ i is a homomorphism, so that δ(g 1 g 2 ) = δ(g 1 ) F(g 2 ) δ(g 2 ), showing that δ| G i defines a cocycle in H 1 (G i , Q [1] ). For g 1 , g 2 ∈ K, δ| K defines a cocycle in Z 1 (K, Q), so that under the injection Z 1 (K, Q) → Z 1 (K, Q [1] ), δ| K defines a cocycle in Z 1 (K, Q [1] ). Hence δ defines a cocycle in Z 1 (G, Q [1]  Part (iv) follows from (b) noting that we must be in the exceptional case of Theorem 3.5.6. Thus the Q-conjugacy classes of reductive subgroups H withH = G is in bijection with the Q Z -conjugacy classes of semisimple subgroups K of (G/Z)Q Z withK = G/Z, whereK denotes the image of K under the quotient morphism (G/Z)Q Z → G/Z. We have shown already that this is in bijection with H 1 (G/Z, (Q Z ) [1]  We recall Serre's notion of G-complete reducibility from [Ser98]. A subgroup H of G is said to be G-completely reducible (or G-cr) if whenever H is contained in a parabolic subgroup P of G, it is contained in some Levi subgroup of that parabolic.
Using the above theorem we can show that a closed reductive subgroup H of G is G-cr if and only if whenever H is in a parabolic subgroup P of G, each parabolic subgroup P 1 of H is in some Levi subgroup of that parabolic; in other words Corollary 3.6.2. Let H be a closed, connected, reductive subgroup of G contained in