12 VICTORIA HOSKINS 3. Algebraic group actions and quotients In this section we consider group actions on algebraic varieties and also describe what type of quotients we would like to have for such group actions. 3.1. Affine Algebraic groups. An algebraic group (over k) is a group object in the category of schees (over k). By a Theore of Chevalley, every algebraic group is an extension of an abelian variety (that is, a sooth connected projective algebraic group) by an affine algebraic group (whose underlying schee is affine) [22, Theore 10.25]. In this course, we only work with affine algebraic groups and cover the results which are ost iportant for our purposes. A good reference for affine algebraic group schees is the book (in preparation) of Milne [23]. For those who are interested in discovering ore about algebraic groups, see [3, 22, 11]. Definition 3.1. An algebraic group over k is a schee over k with orphiss e : Spec k (identity eleent), : (group law) and i : (group inversion) such that we have coutative diagras id Spec k e id id e Spec k id = = Spec k (i,id) e (id,i) Spec k. e We say is an affine algebraic group if the underlying schee is affine. We say is a group variety if the underlying schee is a variety (recall in our conventions, varieties are not necessarily irreducible). A hooorphis of algebraic groups and H is a orphis of schees f : H such that the following square coutes f f H H H f H. An algebraic subgroup of is a closed subschee H such that the iersion H is a hooorphis of algebraic groups. We say an algebraic group is an algebraic quotient of if there is a hooorphis of algebraic groups f : which is flat and surjective. Reark 3.2. (1) The functor of points h of an algebraic group has a natural factorisation through the category of (abstract) groups, i.e, for every schee X the operations, e, i equip Ho(X, ) with a group structure and with this group structure, every ap h (f) : Ho(X, ) Ho(Y, ) for f : Y X is a orphis of groups. In fact, one can show using the Yoneda lea that there is an equivalence of categories between the category of algebraic groups and the category of functors F : Sch rp such that the coposition Sch F rp Set is representable. When restricting to the category of affine k-schees, this can give a very concrete description of an algebraic group, as we will see in the exaples below. (2) Let O() := O () denote the k-algebra of regular functions on. Then the above orphiss of affine varieties correspond to k-algebra hooorphiss : O() O() O() (coultiplication) and i : O() O() (coinversion) and the identity eleent corresponds to e : O() k (counit). These operations define a Hopf algebra
MODULI PROBLEMS AND EOMETRIC INVARIANT THEORY 13 structure on the k-algebra O(). Furtherore, there is a bijection between finitely generated Hopf algebras over k and affine algebraic groups (see [23] II Theore 5.1). (3) By a Theore of Cartier, every affine algebraic group over a field k of characteristic zero is sooth (see [23] VI Theore 9.3). Moreover, in Exercise sheet 3, we see that every algebraic group is separated. Hence, in characteristic zero, the notion of affine algebraic group and affine group variety coincide. (4) In the definition of hooorphiss, we only require a copatibility with the group law ; it turns out that the copatibility for the identity and group inversion is then autoatic. This is well known in the case of hooorphiss of abstract groups, and the algebraic case can then be deduced by applying the Yoneda lea. (5) For the definition of a quotient group, the condition that the hooorphis is flat is only needed in positive characteristic, as in characteristic zero this orphis is already sooth (this follows fro the Theore of Cartier entioned above and the fact that the kernel of a hooorphis of sooth group schees is sooth; see [22] Proposition 1.48) Exaple 3.3. Many of the groups that we are already failiar with are affine algebraic groups. (1) The additive group a = Spec k[t] over k is the algebraic group whose underlying variety is the affine line A 1 over k and whose group structure is given by addition: (t) = t 1 + 1 t and i (t) = t. Let us indicate how to show these operations satisfy the group axios. We only prove the associativity, the other axios being siilar and easier. We have to show that ( id) = (id ) : k[t] k[t] k[t] k[t]. This is a ap of k-algebras, so it is enough to check it for t. We have (( id) )(t) = ( id)(t 1 + 1 t) = t 1 1 + 1 t 1 + 1 1 t and siilarly ((id ) )(t) = t 1 1 + 1 t 1 + 1 t 1 which copletes the proof. For a k-algebra R, we have a (R) = (R, +); this justifies the nae of the additive group. (2) The ultiplicative group = Spec k[t, t 1 ] over k is the algebraic group whose underlying variety is the A 1 {0} and whose group action is given by ultiplication: (t) = t t and i (t) = t 1. For a k-algebra R, we have (R) = (R, ); hence, the nae of the ultiplicative group. (3) The general linear group L n over k is an open subvariety of A n2 cut out by the condition that the deterinant is non-zero. It is an affine variety with coordinate ring k[x ij : 1 i, j n] det(xij ). The co-group operations are defined by: (x ij ) = n k=1 x ik x kj and i (x ij ) = (x ij ) 1 ij where (x ij ) 1 ij is the regular function on L n given by taking the (i, j)-th entry of the inverse of a atrix. For a k-algebra R, the group L n (R) is the group of invertible n n atrices with coefficients in R, with the usual atrix ultiplication. (4) More generally, if V is a finite-diensional vector space over k, there is an affine algebraic group L(V ) which is (non-canonically) isoorphic to L di(v ). For a k-algebra R, we have L(V )(R) = Aut R (V k R). (5) Let be a finite (abstract) group. Then can be naturally seen as an algebraic group k over k as follows. The group operations on ake the group algebra k[] into a Hopf algebra over k, and k := Spec(k[]) is a 0-diensional variety whose points are naturally identified with eleents of.
14 VICTORIA HOSKINS (6) Let n 1. Put µ n := Spec k[t, t 1 ]/(t n 1), the subschee of n-roots of unity. Write I for the ideal (t n 1) of R := k[t, t 1 ]. Then (t n 1) = t n t n 1 1 = (t n 1) t n + 1 (t n 1) I R + R I which iplies that µ n is an algebraic subgroup of. If n is different fro char(k), the polynoial X n 1 is separable and there are n distinct roots in k. Then the choice of a priitive n-th root of unity in k deterines an isoorphis µ n Z/nZ k. If n = char(k), however, we have X n 1 = (X 1) n in k[x], which iplies that the schee µ n is non-reduced (with 1 as only closed point). This is the siplest exaple of a non-reduced algebraic group. A linear algebraic group is by definition a subgroup of L n which is defined by polynoial equations; for a detailed introduction to linear algebraic groups, see [1, 15, 40]. For instance, the special linear group is a linear algebraic group. In particular, any linear algebraic group is an affine algebraic group. In fact, the converse stateent is also true: any affine algebraic group is a linear algebraic group (see Theore 3.9 below). An affine algebraic group over k deterines a group-valued functor on the category of finitely generated k-algebras given by R (R). Siilarly, for a vector space V over k, we have a group valued functor L(V ) given by R Aut R (V k R), the group of R-linear autoorphiss. If V is finite diensional, then L(V ) is an affine algebraic group. Definition 3.4. A linear representation of an algebraic group on a vector space V over k is a hooorphis of group valued functors ρ : L(V ). If V is finite diensional, this is equivalent to a hooorphis of algebraic groups ρ : L(V ), which we call a finite diensional linear representation of. If is affine, we can describe a linear representation ρ : L(V ) ore concretely in ters of its associated co-odule as follows. The natural inclusion L(V ) End(V ) and ρ : L(V ) deterine a functor End(V ), such that the universal eleent in (O()) given by the identity orphis corresponds to an O()-linear endoorphis of V k O(), which by the universality of the tensor product is uniquely deterined by its restriction to a k-linear hooorphis ρ : V V k O(); this is the associated co-odule. If V is finite diensional, we can even ore concretely describe the associated co-odule by considering the group hooorphis End(V ) and its corresponding hooorphis of k-algebras O(V k V ) O(), which is deterined by a k-linear hooorphis V k V O() or equivalently by the co-odule ρ : V V k O(). In particular, a linear representation of an affine algebraic group on a vector space V is equivalent to a co-odule structure on V (for the full definition of a co-odule structure, see [23] Chapter 4). 3.2. roup actions. Definition 3.5. An (algebraic) action of an affine algebraic group on a schee X is a orphis of schees σ : X X such that the following diagras coute Spec k X e id X X X id σ X = σ id X X X σ σ X. Suppose we have actions σ X : X X and σ Y : Y Y of an affine algebraic group on schees X and Y. Then a orphis f : X Y is -equivariant if the following diagra coutes X id f Y σ X X f σ Y Y.
MODULI PROBLEMS AND EOMETRIC INVARIANT THEORY 15 If Y is given the trivial action σ Y = π Y : Y Y, then we refer to a -equivariant orphis f : X Y as a -invariant orphis. Reark 3.6. If X is an affine schee over k and O(X) denotes its algebra of regular functions, then an action of on X gives rise to a coaction hooorphis of k-algebras: σ : O(X) O( X) = O() k O(X) f hi f i. This gives rise to a hooorphis Aut(O(X)) where the k-algebra autoorphis of O(X) corresponding to g is given by for f O(X) with σ (f) = h i f i. f h i (g)f i O(X) Definition 3.7. An action of an affine algebraic group on a k-vector space V (resp. k-algebra A) is given by, for each k-algebra R, an action of (R) on V k R (resp. on A k R) σ R : (R) (V k R) V k R (resp. σ R : (R) (A k R) A k R) such that σ R (g, ) is a orphis of R-odules (resp. R-algebras) and these actions are functorial in R. We say that an action of on a k-algebra A is rational if every eleent of A is contained in a finite diensional -invariant linear subspace of A. Lea 3.8. Let be an affine algebraic group acting on an affine schee X. Then any f O(X) is contained in a finite diensional -invariant subspace of O(X). Furtherore, for any finite diensional vector subspace W of O(), there is a finite diensional -invariant vector subspace V of O(X) containing W. Proof. Let σ : O(X) O() O(X) denote the coaction hooorphis. Then we can write σ (f) = n i=1 h i f i, for h i O() and f i O(X). Then g f = i h i(g)f i and so the vector space spanned by f 1,..., f n is a -invariant subspace containing f. The second stateent follows by applying the sae arguent to a given basis of W. In particular, the action of on the k-algebra O(X) is rational (that is, every f O(X) is contained in a finite diensional -invariant linear subspace of O(X)). One of the ost natural actions is the action of on itself by left (or right) ultiplication. This induces a rational action σ : Aut(O()). Theore 3.9. Any affine algebraic group over k is a linear algebraic group. Proof. As is an affine schee (of finite type over k), the ring of regular functions O() is a finitely generated k-algebra. Therefore the vector space W spanned by a choice of generators for O() as a k-algebra is finite diensional. By Lea 3.8, there is a finite diensional subspace V of O() which is preserved by the -action and contains W. Let : O() O() O() denote the coultiplication; then for a basis f 1,..., f n of V, we have (f i ) O() V, hence we can write n (f i ) = a ij f j j=1 for functions a ij O(). In ters of the action σ : Aut(O()), we have that σ(g, f i ) = j a ij(g)f j. This defines a k-algebra hooorphis ρ : O(Mat n n ) O() x ij a ij. To show that the corresponding orphis of affine schees ρ : Mat n n is a closed ebedding, we need to show ρ is surjective. Note that V is contained in the iage of ρ as n n f i = (Id O() e ) (f i ) = (Id O() e ) a ij f j = e (f j )a ij. j=1 j=1
16 VICTORIA HOSKINS Since V generates O() as a k-algebra, it follows that ρ is surjective. Hence ρ is a closed iersion. Finally, we clai that ρ : Mat n n is a hooorphis of seigroups (recall that a seigroup is a group without inversion, such as atrices under ultiplication) i.e. we want to show on the level of k-algebras that we have a coutative square O(Mat n n ) Mat O(Mat n n ) O(Mat n n ) ρ ρ O() ρ O() O(); that is, we want to show for the generators x ij O(Mat n n ), we have ( ) (a ij ) = (ρ (x ij )) = (ρ ρ )( Mat(x ij )) = (ρ ρ ) x ik x kj = k k a ik a kj. To prove this, we consider the associativity identity (id ) = ( id) and apply this on the k-algebra level to f i O() to obtain a ik a kj f j = (a ij ) f j j k,j as desired. Furtherore, as is a group rather than just a seigroup, we can conclude that the iage of ρ is contained in the group L n of invertible eleents in the seigroup Mat n n. Tori are a basic class of algebraic group which are used extensively to study the structure of ore coplicated algebraic groups (generalising the use of diagonal atrices to study atrix groups through eigenvalues and the Jordan noral for). Definition 3.10. Let be an affine algebraic group schee over k. (1) is an (algebraic) torus if = n for soe n > 0. (2) A torus of is a subgroup schee of which is a torus. (3) A axial torus of is a torus T which is not contained in any other torus. For a torus T, we have coutative groups X (T ) := Ho(T, ) X (T ) := Ho(, T ) called the character group and cocharacter group respectively, where the orphiss are hooorphiss of linear algebraic groups. Let us copute X ( ). Lea 3.11. The ap is an isoorphis of groups. θ : Z X ( ) n (t t n ) Proof. Let us first show that this is well defined. Write for the coultiplication on O( ). Then (t n ) = (t t) n = t n t n shows that θ(n) : is a orphis of algebraic groups. Since t a t b = t a+b, θ itself is a orphis of groups. It is clearly injective, so it reains to show surjectivity. Let φ be an endoorphis of. Write φ (t) k[t, t 1 ] as i < a it i. We have (φ (t)) = φ (t) φ (t), which translates into a i t i t i = a i a j t i t j. i i,j Fro this, we deduce that at ost one a i is non-zero, say a n. Looking at the copatibility of φ with the unit, we see that necessarily a n = 1. This shows that φ = θ(n), copleting the proof.
MODULI PROBLEMS AND EOMETRIC INVARIANT THEORY 17 For a general torus T, we deduce fro the Lea that the (co)character groups are finite free Z-odules of rank di T. There is a perfect pairing between these lattices given by coposition <, >: X (T ) X (T ) Z where < χ, λ >:= χ λ. An iportant fact about tori is that their linear representations are copletely reducible. We will often use this result to diagonalise a torus action (i.e. choose a basis of eigenvectors for the T -action so that the action is diagonal with respect to this basis). Proposition 3.12. For a finite diensional linear representation of a torus ρ : T L(V ), there is a weight space decoposition V = χ X (T ) where V χ = {v V : t v = χ(t)v t T } are called the weight spaces and {χ : V χ 0} are called the weights of the action. Proof. To keep the notation siple, we give the proof for T =, where X (T ) = Z; the general case can be obtained either by adapting the proof (with further notation) or by induction on the diension of T. The representation ρ has an associated co-odule and the diagra V χ ρ : V V k O( ) = V k[t, t 1 ]. ρ V V k[t, t 1 ] ρ id ρ V k[t, t 1 ] id V k[t, t 1 ] k[t, t 1 ] coutes. Fro this, it follows easily that, for each integer, the space V = {v V : ρ (v) = v t } is a subrepresentation of V. For v V, we have ρ (v) = Z f (v) t where f : V V is a linear ap, and because of the copatibility with the identity eleent, we find that v = Z f (v). If ρ (v) = Z f (v) t, then we clai that f (v) V. Fro the diagra above f (v) t t Z ρ (f (v)) t = (ρ Id k[t,t 1 ])(ρ (v)) = (Id V )(ρ (v)) = Z and as {t } Z are linearly independent in k[t, t 1 ], the clai follows. Let us show that in fact, the f for a collection of orthogonal projectors onto the subspaces V. Using the coutative diagra again, we get f (v) t t = f (f n (v)) t t n, Z,n Z which again by linear independence of the {t } shows that f f n vanishes if n and is equal to f n otherwise; this proves that they are orthogonal idepotents. Hence, the V are linearly independent and this copletes the proof. This result can be phrased as follows: there is an equivalence between the category of linear representations of T and X (T )-graded k-vector spaces. We note that there are only finitely any weights of the T -action, for reasons of diension.
18 VICTORIA HOSKINS 3.3. Orbits and stabilisers. Definition 3.13. Let be an affine algebraic group acting on a schee X by σ : X X and let x be a k-point of X. i) The orbit x of x to be the (set-theoretic) iage of the orphis σ x = σ(, x) : (k) X(k) given by g g x. ii) The stabiliser x of x to be the fibre product of σ x : X and x : Spec k X. The stabiliser x of x is a closed subschee of (as it is the preiage of a closed subschee of X under σ x : X). Furtherore, it is a subgroup of. Exercise 3.14. Using the sae notation as above, consider the presheaf on Sch whose S-points are the set {g h (S) : g (x S ) = x S } where x S : S X is the coposition S Spec k X of the structure orphis of S with the inclusion of the point x. Describe the presheaf structure and show that this functor is representable by the stabiliser x. The situation for orbits is clarified by the following result. Proposition 3.15. Let be an affine algebraic group acting on a schee X. The orbits of closed points are locally closed subsets of X, hence can be identified with the corresponding reduced locally closed subschees. Moreover, the boundary of an orbit x x is a union of orbits of strictly saller diension. In particular, each orbit closure contains a closed orbit (of inial diension). Proof. Let x X(k). The orbit x is the set-theoretic iage of the orphis σ x, hence by a theore of Chevalley ([14] II Exercise 3.19), it is constructible, i.e., there exists a dense open subset U of x with U x x. Because acts transitively on x through σ x, this iplies that every point of x is contained in a translate of U. This shows that x is open in x, which precisely eans that x is locally closed. With the corresponding reduced schee structure of x, there is an action of red on x which is transitive on k-points. In particular, it akes sense to talk about its diension (which is the sae at every point because of the transitive action of red ). The boundary of an orbit x is invariant under the action of and so is a union of -orbits. Since x is locally closed, the boundary x x, being the copleent of a dense open set, is closed and of strictly lower diension than x. This iplies that orbits of iniu diension are closed and so each orbit closure contains a closed orbit. Definition 3.16. An action of an affine algebraic group on a schee X is closed if all -orbits in X are closed.