Ok, here’s my second talk. This one went a bit heavier on the technical stuff, and is mostly out of Geometric Invariant Theory by Mumford and Fogarty (I have access to the second edition). At some point in my Algebraic Geometry from the Beginning series, I’ll try to get to explaining all of the terms used here. Also, I’ll be getting back to that next week (hopefully) now that a big pile of my commitments have been resolved. Also, this is technically the title of the talk, as you will shortly notice, I didn’t really stick to the topic I was intending to.
We start by defining what we mean by a quotient.
Definition: Let be an algebraic group over and let it act on a scheme by . A pair with a scheme and an -morphism is called a categorical quotient of by if commutes and if given any pair , such that (that is, satisfying the above), then there exists a unique -morphism such that .
Definition: The pair is called a geometric quotient if
- as before
- is surjective, and the image of is the entirety of . This condition is equivalent to the geometric fibers of being the orbits of geometric points of , for geometric points over an algebraically closed field of sufficiently high transcendence degree. (If and are of finite type over and , then any alg closed field works.)
- is submersive, that is, a subset is open iff is open.
- The structure sheaf is the subsheaf of consisting of invariant functions.
So the conditions, in turn, roughly say that it is compatible with the action, has the orbits as fibers, that has the quotient topology, and that the functions on are the ones which were invariant along orbits in .
Note that categorical quotients are unique up to unique isomorphism due to the universal property.
Proposition: Let be an action of on , and let be a geometric quotient of by . Then is a categorical quotient of by .
Proof:Let be any -morphism with . Let be an open affine cover of . For each , is an invariant open subset of , so for some subset in . As is submersive, is open.
As is surjective, is an open cover for . Thus, any morphism with must have . Therefore, it is given by making the following diagram commute:
As is injective by the fourth condition on being a geometric quotient, is uniquely determined, if it exists. Thus, at most one exists. Now for any , we can see that is an invaraint element of , hence it is in the subring , and so the exist.
defines . As on , we can glue together to get . QED.
We call a quotient universal if for all and , is a quotient of by . If this only works for flat base extension, we call it uniform.
Definition: We call an action closed if for all geometric points , the orbit in for algebraically closed is closed.
Proposition: If a geometric quotient exists, then the action is closed.
Proof:Let be a geometric point of over . The orbit of is . Thus, the orbit is closed, because is continuous, and is a closed point. QED
Proposition: is a categorical quotient if , is the subsheaf of invariants of , and if is an invariant closed subset of , then is closed in , and for arbitrary collections of invariant closed sets. In fact, if these conditions hold, is submersive.
However, this last proposition one doesn’t QUITE imply the existence of a geometric quotient.
Theorem:Let be a field of characteristic zero. Let be an affine scheme over and be a reductive algebraic group. Let be an action. Then a universal categorical quotient of by exists, is universally submersive, and is an affine scheme. Moreover, if is algebraic, then is algebraic over . Moreover, noetherian implies noetherian.
Before we begin we will need a pair of algebraic lemmas
Lemma:Let be a ring and the ring of invariants of . Then if is an -algebra, it is the ring of invariants of .
Lemma: If is a set of invariant ideals in , then .
These two lemmas appear as statements in the proof of Theorem 1.1 in GIT (page 28-29) which is the theorem above.
With these in hand, we can prove the theorem:
Proof: Let . Then we get an induced action of on . Let be the ring of invariants, and . Let be the morphism induced by the inclusion .
The first lemma implies that for any open affine , is the ring of invariants of , and therefore is the subsheaf of invariants.
Now we rephrase the second lemma geometrically: let be the closed subset defined by . Then the second lemma says that . Apply this case to , an invariant closed subset of , , with closed, then is closed. And such, , and so is a categorical quotient.
To show universality, let . We only need to worry about affine, and we must show that is a categorical quotient of by . By the first lemma, is still the ring of invariants of , and so we are done.
Now let be noetherian. If is any ideal, then lemma 1 applied to gives us that . So the poset of ideals in is a subposet of the ideals in . Thus the ACC descends, and is noetherian, and so is.
Now suppose that is of finite type over . We’ll first worry about the case where is graded over and preserves the grading. Then is a subgraded algebra, and so it is finitely generated over , because it is noetherian, so it is of finite type.
The general case can be obtained from the graded case without too much difficulty. QED
Extension: In the situation above, then is a geometric quotient of by iff the action of on is closed.
Proof: We already saw one direction, so we will just assume that is a closed action.
SUppose that is a proper subset of . Then for some algebraically closed field , there exist geometric points with , but with disjoint orbits and . By assumption, these are closed invariant disjoint subsets of . Thus, there exists an invariant with and . Since the ring of invariants is for is with the invariants of , there exists an invariant such that . Thus , a contradiction. QED
Finally, to connect this back to finite groups, let be an affine variety over of characteristic zero, and a finite group acting on . Then the orbits of are finite, and therefore closed, so the action is closed. By the theorem, a universal quotient exists, and it is an affine scheme of finite type over . By the corollary, it is in fact a geometric quotient. To see that is a variety merely requires that we check that is reduced. This follows from the fact that is reduced (as is a variety) and must then be. Thus, affine varieties have quotients by finite groups which are affine varieties.