Finally we’re ready to discuss what sorts of quotients exist when group schemes act on (some) other schemes. Recall that for simplicity, every scheme/variety/morphism in sight is assumed to be over the spectrum of a fixed algebraically closed field, . For further simplicity, assume all schemes are of finite type. In this post, we’ll begin the discussion with the simplest case, that is, the main results involving taking quotients of reductive group schemes acting on affine schemes. This will be the main motivation for generalizing results to the non-affine case which we also begin to do here. This theory is well developed and sometimes a bit technical though I’ve tried to avoid anything too difficult below and hopefully the examples illustrate some of what’s going on. (The main reference is Mumford’s GIT, and proofs of unproven facts can be found there.)
We need one more definition before getting to the main results, the notion of a slightly weaker notion than a geometric quotient, namely:
Definition: Suppose a group scheme acts on a scheme . (Recall this means that there is a morphism such that ….). A categorical quotient for this setup will be a morphism such that
i) The two maps are equal.
ii) And is the “smallest” scheme with this property. In other words, if satisfies the same condition as in i), then there is a unique map such that .
A formal exercise shows that geometric quotients (see GSM (I)) are already categorical ones. The converse is certainly not true, the main difference in the definitions being that orbits need not necessarily be separated in the categorical quotient case. We will see examples of this below.
The main theorem in the affine case have already been stated and proved by Charles in his talks on Invariants. I’ll restate it here but certainly won’t reprove it.
Theorem: If is an affine scheme and is a reductive group acting on , then a categorical quotient exists and is equal to , the spectrum of the ring of invariants. In fact, inherits many properties from (algebraic, noetherian, connected, irreducible, …). The scheme is a geometric quotient if and only if the action is closed (that is to say, each orbit is).
As an example, recall one of the examples from the GSM(I), the action of on given by . Here there were three types of orbits, the origin, the hyperbolas and the two axes minus the origin. Notice that the ring of invariants is isomorphic to , and the quotient map is given by which sends . Our theorem asserts this is a categorical quotient and it is clearly not a geometric quotient, the origin and the two axes are collapsed to the origin in the target.
For another example, consider the action of on itself by the adjoint action, that is . For any matrix , we can compute the characteristic polynomial . Using this, we can define a equivariant map (in fact an actual morphism) which sends . This is actually a categorical quotient. To verify this, we need to verify that . This is not too hard, I’ll outline the proof. 1) Since any matrix is equivalent to one in Jordan normal form, an invariant function is determined by its values on Jordan matrices. 2) The set of matrices with diagonal normal form is Zariski dense in the set of all invertible matrices, so an invariant function is determined by its values on diagonal matrices. 3) Let be the ring of functions on the diagonal matrices. Since diagonal matrices with permuted elements are equivalent under the action of the group, invariant functions must be symmetric in the x’s. By the Fundamental Theorem of Symmetric Functions, these are polynomials in the elementary symmetric functions in the x’s. These functions though agree with the functions when restricted to the diagonal matrices.
However, we see in the above example that the quotient is not a geometric one, as there are matrices with different Jordan normal forms that have the same characteristic polynomial.
In fact, there are examples of non-projective algebraic varieties acted on by finite groups where the categorical quotient does not even exist in the category of algebraic varieties. As usual at this point, we’ll simply say the constructions are a little involved (the interested can look at an example Hironaka which is explained in GIT as well as in the Appendix of Hartshorne’s Algebraic Geometry).
The search for geometric quotients continues, but we see from the above examples that even in some simple cases we’ll be out of luck. Suppose now that acts on (maybe no longer affine). The idea will be to find and characterize open sets where geometric quotients do exist, but of course, we get to wade through a little more theory first!
To motivate the theory, we return to the affine case and consider the following Lemma:
Lemma: Suppose that is acted on by an affine algebraic group . Then there exists an equivariant embedding of into such that acts on through a linear representation.
Sketch of Proof: Suppose that generate the k algebra . The Lemma follows by looking at the collection of -translates of the ‘s and taking the linear space they span. One need only remember that if the action is given (dually) by , if we write then the invariant subspace spanned by the -translates of is finite dimensional and is contained in the span of the finitely many ‘s.
We attempt to follow a similar path when is quasi-projective. Recall that a map to projective space is given by a line bundle and a collection of n + 1 global sections of . We define a lifting of an action of on to an action of on :
Defintion: Suppose is an action of an algebraic group scheme on and is an invertible sheaf on . We abuse notation and identify with its total space, that is, the line bundle with its natural projection . Then a linearization of is an action such and such that the zero section of is -equivariant.
Alternatively, one can rephrase this definition to say that there exists an isomorphism of invertible sheaves which “satisfy the cocycle condition”. Since we won’t use this formulation, we’ll leave it to the reader to work this out (or see Mumford’s GIT 1.3).
So as not to go too far out of the way, we’ll state some important facts about G-linearizations:
Fact 1: Suppose that is a connected affine algebraic group, and is a normal algebraic variety, then there is some positive integer n such that admits a linearization.
Fact 2: Suppose that is an irreducible affine algebraic group acting on a quasi projective . Then there exists a -equivariant embedding where acts on through a linear representation .
Suppose that acts on in the natural way (call the action ). The group sits inside where as the complement of the hypersurface given by the vanishing of the determinant in the variables . The action is the restriction of the rational map given by matrix multiplication. This map is not defined at any point such that . The restriction of the first projection of to this set of points, call them , maps onto . Since has codimension 2 or more in , it must be the case that is the restriction of a line bundle on (exercise). The formula for the action though, shows then that this bundle must be . If the bundle admits a linearization, then and so would be trivial. This can’t be the case though because and restrict to the same bundle over . The bundle they restrict to though is a generater of which is isomorphic to (exercise). Thus we see that does not admite a linearizaiton for this action. The same argument though shows that does admit a linearization!
When admits an action by a reductive group , the idea will be to cover by affine invariant open sets , take the quotient and glue together the results. Of course, such a naive cover won’t always exist! Instead, we’ll find such a cover of an open set of , and this really is the best we can hope for. The construction of that open subset will depend upon a parameter, namely the choice of a -linearizized line bundle . But that will come in the next installment!