1 Application: Semidirect Product of (Finite) Group Schemes

1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

General assumptions and notations:

• k denotes an algebraically closed ﬁeld.

• All k-algebras are assumed to be associative.

• If A is a k-algebra, we denote by mod(A) the category of ﬁnite-dimensional modules.

• Given a commutative ring R, we denote by MR the category of commutative R- algebras.

1 Application: Semidirect product of (Finite) Group Schemes

Assume from now on that char(k) = p > 0. A k-functor is a functor from Mk to the category of Sets. Let G be an aﬃne group scheme over k. By deﬁnition,

G : Mk −→ Groups

∼ is a functor such that there is some k-algebra A and a natural equivalence G = Speck(A) of k-functors (G is representable via A). Then k[G] := A is the coordinate ring of G.

The multiplication G × G −→ G, the unit ek −→ G and the inversion G −→ G are natural transformations. Hence, Yonedas Lemma gives rise to algebra homomorphisms

∆ : A −→ A ⊗k A, ε : A −→ k, S : A −→ A. In this way, A obtains the structure of a commutative Hopf algebra (cf. [2] for some more details).

1.1 Base Change

The following lemma can be proven as [1][§1, Theorem 3.5].

0 0 Lemma 1.1. Let k ,A ∈ Mk. Then the tensor product A ⊗k k of k-algebras obtains the structure of a k0-algebra via α0.a⊗β0 := a⊗α0β0. Moreover, there is a natural equivalence 0 ∼ k0 Speck0 (A ⊗k k ) = Speck(A) ◦ Resk of functors.

Let now G be a group scheme with coordinate ring k[G] and ﬁx some algebra A ∈ Mk.

Let B be an A-algebra, that is: B ∈ MA. Composition of structure maps k → A → B

1 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES shows that we can view B automatically as a k-algebra. Moreover, if λ : R → S is a homomorphism of A-algebras, then λ is automatically also a homomorphism of k- algebras. Hence, deﬁning GA(B) = G(B), GA(λ) = G(λ), we obtain a functor

GA : MA −→ Groups

By deﬁnition, the elements of GA(B) correspond to k-algebra homomorphisms k[G] → B.

By the above lemma, those correspond to A-algebra homomorphisms k[G] ⊗k A → ∼ B. Hence GA = SpecA(k[G] ⊗k A), so that GA is an aﬃne group scheme over A with coordinate ring k[G] ⊗k A.

1.2 Actions

Definition 1.2. Let G be a group scheme and X be an affine scheme. A (left)-action of G on X is a morphism G × X −→ X of affine schemes such that each G(A) × X(A) → X(A) is an action of G(A) on X(A) for all A ∈ Mk.

Recall that a k-vector space V gives rise to a k-functor

Va = V ⊗k − : Mk −→ Sets,Va(A) = V ⊗k A, Va(λ) = idV ⊗ λ

If V is ﬁnite-dimensional, then Va is aﬃne algebraic:

Lemma 1.3. Let V be a ﬁnite-dimensional k-vector space. Then the functor Va is an aﬃne algebraic scheme with coordinate ring S(V ∗).

Proof. According to the ﬁnite-dimensionality of V , there are natural equivalences (the last one being the universal property of the symmetric algebra over V ∗)

∼ ∗ ∗ ∼ ∗ ∼ ∗ V ⊗k − = (V ) ⊗k − = Homk(V , −) = Speck(S(V ))

As any basis of V ∗ generates S(V ∗) as an algebra, the claim follows.

G-modules and their homomorphisms can be described in the following way:

Deﬁnition 1.4. Let G be a group scheme and V a k-vector space, then V is a G-module provided there is an action G × Va −→ Va such that each G(A) 3 g : V ⊗k A −→ V ⊗k A is A-linear for all A ∈ Mk.

2 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

A linear map ϕ : V −→ W of two G-modules V and W is said to be G-linear provided each ϕ ⊗ idA : V ⊗ A −→ W ⊗ A is G(A)-linear for all A ∈ Mk.

If V is a G-module, then V corresponds to a homomorphism ρ : G → GL(V ), g 7→ (v ⊗ a 7→ g.v ⊗ a) of group schemes. Here

GL(V ): Mk → Groups,A 7→ AutA(V ⊗k A) is the general linear group. If V is n-dimensional, then GL(V ) ∼= GL(n).

Remark. If λ : R → S is a morphism in Mk, then GL(V, λ) : AutR(V ⊗k R) → AutS(V ⊗k S) is deﬁned as follows:

Let ψ ∈ AutR(V ⊗k R) be an invertible R-linear transformation. Then µ := (idV ⊗

λ) ◦ ψ : V ⊗k R → V ⊗k S is R-linear, where we consider S as an R-module via pullback along λ. Hence V → V ⊗k S, v 7→ µ(v ⊗ 1) is k-linear and corresponds to GL(V, λ)(ψ): V ⊗ S 7→ V ⊗ S, v ⊗ s 7→ s.µ(v) S-linear. In particular, one-dimensional G-modules correspond to homomorphisms G → GL(1) =

Gm, the characters of G. Here Gm denotes the multiplicative group:

× Gm : Mk → Groups,A 7→ A = {a ∈ A | There is b ∈ A such that ab = 1 = ba}

The set of all characters X(G) forms an abelian group with pointwise multiplication. We writes this group additively and it is easy to see that (identifying characters with their corresponding one-dimensional G-module) λ⊗k µ = λ+µ for λ, µ ∈ X(G). Consequently, the group X(G) acts on mod(G) with auto-equivalences, each λ via V 7→ V ⊗k λ.

Another way, in which G-modules arise is from actions on aﬃne schemes X. Fix

A ∈ Mk and g ∈ G(A). Let B ∈ MA, then the structure map i : A → B is k-linear. We put gB := G(i)(g). Remark. If A → B is injective, then G(A) → G(B) enjoys the same property (take a left inverse and use functoriality). In particular, G(k) is mapped injectively into G(A) for any A ∈ Mk. In this way, each g ∈ G(A) gives rise to an automorphism

h : XA −→ XA, x 7→ gB.x ∀x ∈ X(B),B ∈ MA

3 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

of the aﬃne scheme XA which gives rise to a comorphism λ(h): k[X]⊗k A −→ k[X]⊗k A. Letting each g act via λ(g−1), k[X] obtains the structure of a G-module. Recalling that 1 k[X] ⊗k A = Mor(XA, AA) is the algebra of natural transformations from XA into the 1 aﬃne line AA, the action is given via (cf. [3][p. 26])

−1 (g.f)(x) = f(gB .x) ∀f ∈ k[X] ⊗k A, x ∈ X(B),B ∈ MA

1.3 Semidirect Products of Group Schemes

By Yonedas Lemma, morphisms of aﬃne schemes correspond to algebra homomorphisms between their coordinate rings (with arrows turned around). If we consider morphisms of group schemes, then those algebra maps must be Hopf algebra maps:

Lemma 1.5. Let G, H be group schemes. A natural transformation ϕ : G −→ H of k- functors is called a homomorphism of group schemes provided each ϕA : G(A) −→ H(A) is a homomorphism of abstract groups for all A ∈ Mk. This is the case if and only if the comorphism ϕ∗ : k[H] −→ k[G] is a Hopf algebra homomorphism.

Let ϕ : G → H be a morphism of group schemes. Then we deﬁne the kernel of ϕ via

ker(ϕ): Mk → Groups,A 7→ ker(ϕA)

It is not hard to see, that ker(ϕ) is represented by the Hopf algebra k[G]/k[G]ϕ∗(k[H]†), i.e. ker(ϕ) ⊆ G is a closed subgroup.

Deﬁning the image of ϕ is not so easy, as the naive approach A 7→ im(ϕA) might not be representable. Instead, one makes this functor larger by - roughly speaking - allowing also h ∈ H(A) which only appear in the image after some extension of A in the following sense

Deﬁnition 1.6. Let A, B ∈ Mk. We call a ring homomorphism ψ : A → B

• ﬂat, provided if M → N is an injective homomorphism of A-modules, then M ⊗A

B → M ⊗A B is also injective.

• faithfully ﬂat, provided ψ is ﬂat and additionally if M is an A-module such that

M ⊗A B = 0, then M = 0.

4 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

(B always considered as an A-module via pullback along ψ, more precisely: B ∈ MA is an A-algebra). If ψ is faithfully ﬂat, then B is also called an fppf-A-algebra.

Remark. Taking M = A in [2][13.1 (3)], we see that each faithfully ﬂat homomorphism A → B must be injective. Here is now the deﬁnition (cf. [2][15.5] or [3][p.70]) of the image:

im(ϕ)(A) := {h ∈ H(A) | There is an fppf-A-algebra B and g ∈ G(B) such that hB = ϕB(g)}

Remark. Being precise, im(ϕ) is then a so-called k-group faisceau (examples of faisceau are also aﬃne schemes, so im(ϕ) ⊆ H is a subgroup-faisceau). As A itself is an fppf-A- algebra, the functor A 7→ im(ϕA) is always a k-subgroup functor of im(ϕ).

Let now H and N be group schemes. Assume that H acts on the k-functor N , i.e. there is a morphism τ : H × N −→ N of aﬃne schemes. If each τA(h, −): N (A) −→

N (A) is a group automorphism for all h ∈ H(A),A ∈ Mk, then we say H acts on N via automorphisms. If this is the case, then we obtain a group scheme

Mk −→ Groups,A 7→ N (A) o H(A), λ 7→ N (λ) × H(λ) the semidirect product of N and H, denoted N oH. Note that the underlying k-functor is the direct product of N and H. To give another characterization of semidirect products, we require the following deﬁnition. We denote by ek : Mk → Groups,A 7→ {1} the trivial group scheme.

Deﬁnition 1.7. A sequence of group schemes

ι π δ : ek −→ N −→ G −→ H −→ ek

is called exact, provided ker(ι) = ek, ker(π) = im(ι), im(π) = H. Moreover, we call δ split exact provided there is a morphism ϕ : H → G such that π ◦ ϕ = idH.

Remark. In view of [2][15.3] ι is a closed embedding (a monomorphism in the category of algebraic group schemes). The same must hold for ϕ (if it exists) as each ϕA has - considering πA - a left inverse. Hence ϕA is injective for all A ∈ Mk.

Lemma 1.8. Split extensions correspond to semidirect products.

5 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

Proof. Let ι π ek −→ N −→ G −→ H −→ ek be a split extension. By deﬁnition, there is ϕ : H → G such that π ◦ ϕ = idH. We let g ∈ G(A), then

−1 g = (g · ϕA(πA(g)) ) · ϕA(πA(g)) ∈ ker(πA) · im(ϕ)(A)

Consequently, the image of the homomorphism

ψ : ker(π) o im(ϕ) −→ G, (n, h) 7→ n · h

given by multiplication is all of G. Let g ∈ ker(πA) ∩ im(ϕ)(A). Then gB = ϕB(h) for some fppf-A-algebra B and h ∈ H(B). Thus

h = πB ◦ ϕB(h) = πB ◦ gB = πB ◦ G(ιA)(g) = H(ιA) ◦ πA(g) = H(ιA)(e) = e

Hence gB = ϕB(e) = e, so that g = e by injectivity. Thus, ker(π) ∩ im(ϕ) = ek. As the kernel of ψ is clearly isomorphic to ker(π) ∩ im(ϕ) under x 7→ (x, x−1), we conclude that ∼ ψ is an isomorphism. Hence G = ker(π) o im(ϕ) = N o H. If conversely G = N oτ H, then the setting ι(n) = (n, e), π(n, h) = h, ϕ(h) = (e, h) constitutes (obviously) a split exact sequence.

As an application, one immediately obtains the following result concerning the structure of Lie algebras of semidirect products:

Lemma 1.9. If G = N o H is a semidirect product of algebraic group schemes, then (denoting their Lie algebras with gothic letters) g = n o h.

Proof. Consider the corresponding exact sequence

π ek −→ N −→ G −→ H −→ ek

By deﬁnition, there is ϕ : H → G such that π ◦ ϕ = idH. As taking Lie algebras is a functor, we obtain

d(π ◦ ϕ) = d(π) ◦ d(ϕ) = d(idH) = idh

In particular, d(π) is surjective (it has a right inverse). Combining this with the left

6 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES exactness of the Lie functor (as shown in previous talks), we obtain a split exact sequence

d(π) 0 −→ n −→ g −→ h −→ 0 of restricted Lie algebras. Quite similar to the proof of the lemma before, one now shows that g = ker(d(π)) ⊕ im(d(ϕ)). As ker(d(π)) ⊆ g is a p-ideal and im(d(ϕ)) ⊆ g a p- ∼ subalgebra, it follows that g = ker(d(π)) o im(d(ϕ)) = n o h is a semidirect product.

1.4 Hopf algebras of Semidirect Products

Consider again N o H. As mentioned before k[N ] is an H-module. As H acts via automorphisms on N , each λ(h) is an automorphism of the A-Hopf algebra k[N ] ⊗k A by Lemma 3.5. We also say, that H acts on k[N ] via automorphisms of Hopf algebras. A consequence is the following: All the structure maps

ε : k[N ] → k, η : k → k[N ], m : k[N ] ⊗k k[N ] → k[N ]

∆ : k[N ] → k[N ] ⊗k k[N ],S : k[N ] → k[N ] are maps of H-modules, where we consider k as the trivial H-module. (For elementary module operations cf. [3]). Recall that an algebraic group scheme G is called ﬁnite, provided its coordinate ring is a ﬁnite-dimensional k-algebra. We then call kG := k[G]∗ the Hopf algebra of G.

Theorem 1.10. The assignment G 7→ kG is an equivalence of categories

{ﬁnite group schemes over k} −→ {ﬁnite-dimensional cocommutative Hopf algebras over k}

Moreover, the categories mod(G) and mod(kG) are also equivalent. Elementary module constructions correspond to each other (the trivial module correspods to the trivial module, tensor products to tensor products and so on).

Let now N and H be ﬁnite, in particular N o H is also ﬁnite as its coordinate ring is (as a k-algebra (!)) given by the tensor product k[N ] ⊗k k[H]. Dualizing all the abovementioned structure maps, we still get maps of H-modules. Hence H also acts on the Hopf algebra kN via automorphisms. In particular, kN is a kH-module bialgebra.

7 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

Hence we can form the smash product kN #kH which has the structure of a Hopf algebra by Theorem 2.6 of my ﬁrst talk.

Theorem 1.11. The Hopf algebras kG and kN #kH are isomorphic.

Proof. We consider the natural emdeddings i := d(ιN ): kN → kG, j := d(ιH): kH → kG. The group H acts on G and N via conjugation. As discussed before, this yields actions on k[G], k[N ]. Moreover, by [3][I, 7.18] the induced action of H on kG is given by restriction of the adjoint representation of the Hopf algebra kG to kH. The comorphism k[G] −→ k[N ] of the inclusion N → G is given by the restriction of functions. This clearly deﬁnes an H-linear map. Consequently, its dual i : kN → kG enjoys the same property. As an upshot of the above, we obtain i ∈ AlgkH(kN , (kG)j). The universal property of the smash product thus provides a homomorphism

ϕ : kN #kH → kG, u#v 7→ i(u) · j(v)

By deﬁnition of the semidirect product, the multiplication N ×H → G is an isomorphism of schemes and thus induces an isomorphism kN ⊗k kH → kG of vector spaces given by multiplication. Consequently, ϕ is surjective, hence an isomorphism of k-algebras for dimension reasons. We are left to show that ϕ is a Hopf algebra map. Examplary, we will show that ϕ respects the comultiplication, that is, (ϕ ⊗ ϕ) ◦ ∆kN #kH = ∆kG ◦ ϕ. In the following computation, we will suppress the embeddings i, j for notational reasons.

X (ϕ ⊗ ϕ) ◦ ∆kN #kH(u#v) = u(1)v(1) ⊗ u(2)v(2) (u),(v) X X = u(1) ⊗ u(2) · v(1) ⊗ v(2) (multiplication inside kG ⊗k kG) (u) (v)

= ∆kG(u) · ∆kG(v)

= ∆kG(uv) = ∆kG(ϕ(u#v))

As extensions of ﬁnite-dimensional Hopf algebras are known to be free, an application of ?? yields

8 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES

Corollary 1.12. Let G = N o H be a semidirect product of ﬁnite group schemes. Then G-modules correspond to coherent kN -kH-modules. Moreover, a) kN is a G-module in a natural way via the left regular representation of kN and the given kH-module structure. b) If M is a G-module and N an H-module, then M ⊗k N is a G-module in a natural way. The kH-action is given by the tensor product and kN acts via u.m ⊗ n := u.m ⊗ n. c) If N is an H-module, then kN ⊗k N is a G-module in a natural way. Moreover,

we have a natural equivalence τ : kN ⊗k − −→ kG ⊗kH − given by τN (a ⊗ n) =

d(ιN )(a) ⊗ n, when the functors are considered to have values in ﬁnite-dimensional module categories.

d) Let M,N be G-modules. If N is commutative, then M ⊗kN N has a natural structure

of a G-module. The H-module structure is given by a quotient of M ⊗k N and kN acts via a.m ⊗ n := a.m ⊗ n.

9 References

References

[1] R.Farnsteiner: Lecture Notes on Homological algebra https://www.dropbox.com/s/kzm63mxoltkjds4/main.pdf?dl=0

[2] W.Waterhouse: Introduction to Aﬃne Group Schemes

[3] J. Jantzen : Representations of Algebraic Groups, Second Edition

[4] Molnar, Richard K. : Semi-Direct Products of Hopf-Algebras, Journal of Algebra 47, 29-51 (1977)

[5] Massey, W., Peterson, F. : The Cohomology Structure of certain Fibre Spaces-I , Topology Vol. 4, pp. 47-65. Pergmon Press. (1965)

[6] R.Farnsteiner.: Polyhedral Groups, McKay quivers, and the ﬁnite algebraic groups with tame principal blocks, Invent. math. 166 (2006)