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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 mod- ules. • 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; " : 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][x1, Theorem 3.5]. 0 0 Lemma 1.1. Let k ;A 2 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 2 Mk. Let B be an A-algebra, that is: B 2 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 2 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 2 Mk. 2 1 APPLICATION: SEMIDIRECT PRODUCT OF (FINITE) GROUP SCHEMES A linear map ' : V −! W of two G-modules V and W is said to be G-linear provided each ' ⊗ idA : V ⊗ A −! W ⊗ A is G(A)-linear for all A 2 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 2 AutR(V ⊗k R) be an invertible R-linear transformation. Then µ := (idV ⊗ λ) ◦ : 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)( ): 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 = fa 2 A j There is b 2 A such that ab = 1 = bag 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 2 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 2 Mk and g 2 G(A). Let B 2 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 2 Mk. In this way, each g 2 G(A) gives rise to an automorphism h : XA −! XA; x 7! gB:x 8x 2 X(B);B 2 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) 8f 2 k[X] ⊗k A; x 2 X(B);B 2 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 ' : G −! H of k- functors is called a homomorphism of group schemes provided each 'A : G(A) −! H(A) is a homomorphism of abstract groups for all A 2 Mk. This is the case if and only if the comorphism '∗ : k[H] −! k[G] is a Hopf algebra homomorphism. Let ' : G!H be a morphism of group schemes. Then we define the kernel of ' via ker('): Mk ! Groups;A 7! ker('A) It is not hard to see, that ker(') is represented by the Hopf algebra k[G]=k[G]'∗(k[H]y), i.e. ker(') ⊆ G is a closed subgroup. Defining the image of ' is not so easy, as the naive approach A 7! im('A) might not be representable. Instead, one makes this functor larger by - roughly speaking - allowing also h 2 H(A) which only appear in the image after some extension of A in the following sense Definition 1.6. Let A; B 2 Mk. We call a ring homomorphism : 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 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 , more precisely: B 2 MA is an A-algebra). If 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(')(A) := fh 2 H(A) j There is an fppf-A-algebra B and g 2 G(B) such that hB = 'B(g)g Remark. Being precise, im(') is then a so-called k-group faisceau (examples of faisceau are also affine schemes, so im(') ⊆ H is a subgroup-faisceau). As A itself is an fppf-A- algebra, the functor A 7! im('A) is always a k-subgroup functor of im('). 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 2 H(A);A 2 Mk, then we say H acts on N via automorphisms.
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