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Group Schemes Over Fields and Abelian Varieties

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Group Schemes Over Fields and Abelian Varieties Group schemes overfields and Abelian varieties Daniele Agostini May 5, 2015 We will work with schemes over afield k, of arbitrary characteristic and not necessarily algebraically closed. An algebraic variety will be a geometrically integral and separated scheme, offinite type overk. 1 Group schemes over Speck Definition 1.1 (Group scheme). A group scheme over k is a scheme G Speck that is also −→ group object in the category of schemes overk. This means that there are three morphisms the multiplication mapm:G G G. • ×k −→ the identity elemente : Speck G. • −→ the inversion mapι:G G. • −→ that satisfy the usual axioms of abstract groups. def 1 Example 1.1 (Additive group). The additive group is defined to be Ga,k = Ak = Speck [T],with the group law given by addition: m : Speck [T] Speck [T] Speck [T] T T 1 + 1 T × k −→ �→ ⊗ ⊗ e : Speck Speck [T] T 0 −→ �→ ι : Speck [T] Speck [T] T T −→ �→ − def 1 Example 1.2 (Multiplicative group). The multiplicative group is defined to be Gm,k = Ak 0 = 1 \{ } Speck [T,T − ] with the group law given by multiplication: 1 1 1 m : Speck [T,T − ] Speck [T,T − ] Speck [T,T − ] T T T × k −→ �→ ⊗ 1 e : Speck Speck [T,T − ] T 1 −→ �→ 1 1 1 ι : Speck [T,T − ] Speck [T,T − ] T T− −→ �→ Remark 1.1. If G is a group scheme, then the underlying topological space is not an abstract group in general (the usual translations are defined just for the rational points), however, for every scheme S, the set of morphisms G(S) = Hom(S, G) does in fact inherit the structure of an abstract group. For example, in the case of the additive groupG a we see that G (S) = Hom (S, Speck [T]) = ( S) a,k k O with the group structure induced by addition. Conversely, using Yoneda’s Lemma, one can prove that a group scheme is precisely a scheme G such that for every schemeS we have a group structure onG (S): m :G (S) G(S) G(S) e G(S) ι :G (S) G(S) S × −→ S ∈ S −→ that is functorial in S. We can rephrase this by saying that the functor of points h : Sch/k Sets G −→ factors through the forgetful functor Groups Sets. −→ 2 Definition 1.2. LetG Speck be a group scheme. ThenG is said to be −→ 1.finite ifG Speck isfinite. −→ 2. ´etale ifG Speck is etale.´ −→ 3. local if it isfinite and connected. Definition 1.3 (Rank of afinite group scheme) . Let G be afinite group scheme over afield k. The rank ofG is the integer dim (G). k O Example 1.3 (Constant group schemes). Let G be an abstractfinite group. Then we can define a group scheme (G) in this way: as a scheme we set (G) = g GSpeck � ∈ We define the group structure via Yoneda: for every scheme S we see that (G)(S) is the set of continuous functions S G, where G has the discrete topology. It is clear that the group −→ structure onG induce a group structure on (G)(S) that gives in turn a group scheme structure. 1.1 Subgroup schemes, kernels Definition 1.4 (Subgroup schemes). Let G be a group scheme. A (normal) subgroup scheme of G is a closed subscheme H G such that H(S) is a (normal) subgroup of G(S) for every scheme S. ⊆ ThenH becomes a group scheme in its own right, with a group law inherited from that onG. Remark 1.2. Equivalently, a subgroup scheme ofG is a closed subschemej:H � G such that −→ 1. the restriction of the multiplication mapm:H H G factors throughH. ×k −→ 2. the restriction of the inversion mapιH G factors throughH. −→ 3. the identitye : Speck G factors throughH. −→ def Example 1.4 (Roots of unity). Let n N be a positive integer, then we can define µ = ∈ n,k Speck [T]/(Tn 1). For every scheme S we have µ (S) = f (S) f n = 1 and this makes − n,k { ∈O S | } µn,k into a subgroup scheme ofG m,k. Moreover µn,k is afinite group scheme and it is always etale´ if k has characteristic zero. In characteristicp > 0 instead,µ n,k is etale´ if and only ifp � n. Example 1.5 (Roots of zero). Suppose that k has characteristic p > 0. Then for every m N ∈ def pm positive integer we define αpm = Speck [T]/(T ). For every scheme S we have αpm (S) = pm f (S) f = 0 and this makes α m into a subgroup scheme of G . It is clear that α m is a { ∈O S | } p a p finite local group scheme. Example 1.6 (Reduced group scheme). Suppose that k is a perfectfield and let G be a group scheme offinite type overk. ThenG red is a subgroup scheme ofG. Indeed, since k is perfect, we see that Gred is geometrically reduced [TS15, Lemma 32.4.3] so that G G is again geometrically reduced [TS15, Lemma 32.4.7]. Now consider the red ×k red restriction of the multiplication map m:G G G: since the domain is reduced, this red ×k red −→ must factor through Gred. It is clear that the inversion map and that the identity map factors throughH, and then we are done. As usual, many examples of subgroup schemes arise as kernel of group homomorphisms: Definition 1.5 (Homomorphism of group schemes). Let G, H be group schemes over k.A homomorphism of group schemes is a k-morphism of schemes f:G H that respects the multi- −→ plication in the usual way. Equivalently, the induced map f :G (S) H(S) is a homomorphism S −→ of abstract groups for everyk-schemeS. 3 Definition 1.6 (Kernels). Let f:G H be an homomorphism of group schemes. Then we −→ define Kerf as the scheme-theoreticfiber of f over e : Speck H. For every scheme S one has H −→ (Kerf )(S) = Ker ( f :G (S) H(S)) S −→ and then it is clear that Kerf is a subgroup scheme ofG. Example 1.7. Fix ann N positive integer and consider the morphism ∈ m :G G induced byT Tn n m,k −→ m,k �→ then it is clear that this is a homomorphism of group schemes and thatµ n,k = Kerm n,k. As another example, suppose that k has positive characteristic p,fix a positive integer m and consider the morphism m F :G G induced byT Tp m,k a,k −→ a,k �→ then this is a homomorphism of group schemes andα m,k = KerF m,k. 1.1.1 Frobenius Speaking of homomorphisms, we want to introduce an useful tool and rich source of examples: the Frobenius morphism. In this section, wefix afieldk of positive characteristicp > 0. Definition 1.7 (Absolute Frobenius). LetX be a scheme overk. Then we define a morphism Frob :X X X −→ that on every open affine subscheme SpecA X acts as −→ A A a ap −→ �→ In general, the absolute Frobenius is not a morphism of k-schemes. Tofix this problem one can introduce other notions of Frobenius: Definition 1.8 (Arithmetic Frobenius). Let X be a scheme over k. Then we denote by X( p) the fiber product X(p) X Frobk Speck Speck The morphismX (p) X is called the arithmetic Frobenius. −→ (p) (p) (p) Example 1.8. If X = Speck [X1, ... , Xn]/( f1, ... , fr), then X = Speck [X1, ... , Xn]/( f1 , ... , fn ) (p) wheref i is obtained fromf i by replacing the coefficients with theirp-th power. Moreover, the arithmetic Frobenius in this case is given by ( ) ( ) k[X , ,X ]/( f , ,f ) k[X ,...,X ]/( f p ,...,f p ) X X a ap 1 −→ n 1 −→ n −→ 1 n 1 n i �→ i �→ In particular this is still not a morphism ofk-schemes. Definition 1.9 (Relative Frobenius). Let X be a scheme over k. Then by the universal property of thefiber product, the absolute Frobenius Frob and the structure morphism X Speck give X −→ rise to a morphism ofk-schemes F(p) :X X(p) −→ called the relative Frobenius ofX. 4 Example 1.9. We continue with the previous example of X = Speck [X1, ... , Xn]/( f1, ... , fn). In this case the relative Frobenius is induced by the map ofk-algebras ( ) ( ) k[X ,...,X ]/( f p ,...,f p ) k[X ,...,X ]/( f ,...,f ) X Xp 1 n 1 n −→ 1 n 1 n i �→ i Lemma 1.1. Let G be a group scheme over k. Then G( p) is a group scheme over k as well and the relative Frobenius F(p) :G G(p) is a group homomorphism. −→ Proof. Basically one pulls back the operations on G to G(p) and checks that everything works. 1.2 Quotients byfinite group schemes One of thefirst result that we learn in abstract algebra is the Isomorphism Theorem: namely that if we have a surjective homomorphism of abstract groups f:G H then we have an induced −→ isomorphism G/Kerf ∼ H. Can we say something similar also for group schemes? Can we −→ define the quotient for the action of a group in general? Definition 1.10 (Actions of group schemes). An action of a k-group scheme G on a k-scheme X is a map µ:G X X ×k −→ that satisfies the usual compatibilities. This is the same as saying that for every k-scheme S we have an action µ :G (S) X(S) X(S) S × −→ that is functorial inS. We say that the action is free if the map (µ,pr ) :G X X X 2 ×k −→ ×k is a closed embedding.
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