Invariants of Finite Groups II

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 G be an algebraic group over S and let it act on a scheme X/S by \sigma. A pair (Y,\phi) with Y a scheme and \phi:X\to Y an S-morphism is called a categorical quotient of X by G if \begin{array}{ccc}G\times_S X&\stackrel{\sigma}{\to} &X\\p_2\downarrow&&\phi\downarrow\\X&\stackrel{\phi}{\to}&Y\end{array} commutes and if given any pair (Z,\psi), such that \psi\circ \sigma=\psi\circ p_2 (that is, satisfying the above), then there exists a unique S-morphism \chi:Y\to Z such that \psi=\chi\circ\phi.

Definition: The pair (Y,\phi) is called a geometric quotient if

  1. \phi\circ \sigma=\phi\circ p_2 as before
  2. \phi is surjective, and the image of \Psi=(\sigma,p_2):G\times_S X\to X\times_S X is the entirety of X\times_S X. This condition is equivalent to the geometric fibers of \phi being the orbits of geometric points of X, for geometric points over an algebraically closed field of sufficiently high transcendence degree. (If G and X are of finite type over S and Y, then any alg closed field works.)
  3. \phi is submersive, that is, a subset U\subset Y is open iff \phi^{-1}(U) is open.
  4. The structure sheaf \mathscr{O}_Y is the subsheaf of \phi_*\mathscr{O}_X consisting of invariant functions.

So the conditions, in turn, roughly say that it is compatible with the action, has the orbits as fibers, that Y has the quotient topology, and that the functions on Y are the ones which were invariant along orbits in X.
Note that categorical quotients are unique up to unique isomorphism due to the universal property.

Proposition: Let \sigma be an action of G/S on X/S, and let (Y,\phi) be a geometric quotient of X by G. Then (Y,\phi) is a categorical quotient of X by G.

Proof:Let \psi:X\to Z be any S-morphism with \psi\circ \sigma=\psi\circ p_2. Let \{V_i\} be an open affine cover of Z. For each i, \psi^{-1}(V_i) is an invariant open subset of X, so \psi^{-1}(V_i)=\phi^{-1}(U_i) for some subset U_i in Y. As \phi is submersive, U_i is open.

As \phi is surjective, \{U_i\} is an open cover for Y. Thus, any morphism \chi:Y\to Z with \psi=\chi\circ\phi must have \chi(U_i)\subset V_i. Therefore, it is given by h_i making the following diagram commute:

\begin{array}{ccc}\mathscr{O}_Z(V_i)&\stackrel{h_i}{\to}&\mathscr{O}_Y(U_i)\\  \psi^*\downarrow &&\swarrow\phi^*\\\mathscr{O}_X(\psi^{-1}(V_i))&&\end{array}

As \phi^* is injective by the fourth condition on being a geometric quotient, h_i is uniquely determined, if it exists. Thus, at most one \chi exists. Now for any g\in \mathscr{O}_X(V_i), we can see that \psi^*(g) is an invaraint element of \mathscr{O}_X(\phi^{-1}(U_i)), hence it is in the subring \phi^* \mathscr{O}_Y(U_i), and so the h_i exist.

h_i defines h_i:U_i\to V_i. As \chi_i=\chi_j on U_i\cap U_j, we can glue together to get \chi:Y\to Z. QED.

We call a quotient universal if for all Y'\to Y and X'=X\times_Y Y', \phi':X'\to Y' is a quotient of X' by G. If this only works for flat base extension, we call it uniform.

Definition: We call an action \sigma closed if for all geometric points x\in X, the orbit in \bar{X}=X\times_S Spec\Omega for \Omega algebraically closed is closed.

Proposition: If a geometric quotient exists, then the action is closed.

Proof:Let x be a geometric point of X over \Omega. The orbit of x is \bar{\phi^{-1}}\bar{\phi}(x). Thus, the orbit is closed, because \bar{\phi} is continuous, and x is a closed point. QED

Proposition: (Y,\phi) is a categorical quotient if \phi\circ \sigma=\phi\circ p_2, \mathscr{O}_Y is the subsheaf of invariants of \phi_*\mathscr{O}_X, and if W is an invariant closed subset of X, then \phi(W) is closed in Y, and \phi(\cap W_i)=\cap \phi(W_i) for arbitrary collections of invariant closed sets. In fact, if these conditions hold, \phi is submersive.

However, this last proposition one doesn’t QUITE imply the existence of a geometric quotient.

Theorem:Let k be a field of characteristic zero. Let X be an affine scheme over k and G be a reductive algebraic group. Let \sigma:G\times X\to X be an action. Then a universal categorical quotient (Y,\phi) of X by G exists, \phi is universally submersive, and Y is an affine scheme. Moreover, if X is algebraic, then Y is algebraic over k. Moreover, X noetherian implies Y noetherian.

Before we begin we will need a pair of algebraic lemmas

Lemma:Let R be a ring and R_0 the ring of invariants of R. Then if S_0 is an R_0-algebra, it is the ring of invariants of R\otimes_{R_0}S_0.

Lemma: If \{I_\alpha\}_{\alpha\in J} is a set of invariant ideals in R, then (\sum I_\alpha)\cap R_0=\sum(I_\alpha\cap R_0).

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 R=\Gamma(X,\mathscr{O}_X). Then we get an induced action of G on R. Let R_0\subset R be the ring of invariants, and Y=Spec R_0. Let \phi:X\to Y be the morphism induced by the inclusion R_0\subset R.

The first lemma implies that for any open affine U\subset Y, \mathscr{O}_Y(U) is the ring of invariants of \mathscr{O}_X(\phi^{-1}(U)), and therefore \mathscr{O}_Y\subset \phi_*\mathscr{O}_X is the subsheaf of invariants.

Now we rephrase the second lemma geometrically: let W_\alpha\subset X be the closed subset defined by I_\alpha. Then the second lemma says that Closure(\cap_\alpha W_\alpha)=\cap_\alpha Closure(\phi(W_\alpha)). Apply this case to W_1, an invariant closed subset of X, W_2=\phi^{-1}(y), with y\in Y closed, then \phi(W_1) is closed. And such, \phi(\cap W_\alpha)=\cap \phi(W_\alpha), and so (Y,\phi) is a categorical quotient.

To show universality, let Y'\to Y. We only need to worry about Y' affine, and we must show that Y' is a categorical quotient of X\times_Y Y' by G. By the first lemma, \Gamma(Y',\mathscr{O}_{Y}) is still the ring of invariants of \Gamma(X,\times_Y Y',\mathscr{O}_{X\times_Y Y'}), and so we are done.

Now let X be noetherian. If I\subset R_0 is any ideal, then lemma 1 applied to S_0=R_0/I gives us that IR\cap R_0=I. So the poset of ideals in R_0 is a subposet of the ideals in R. Thus the ACC descends, and R_0 is noetherian, and so Y is.

Now suppose that X is of finite type over k. We’ll first worry about the case where R is graded over k and G preserves the grading. Then R_0 is a subgraded algebra, and so it is finitely generated over k, 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 (Y,\phi) is a geometric quotient of X by G iff the action of G on X is closed.

Proof: We already saw one direction, so we will just assume that \sigma is a closed action.

SUppose that \Psi(G\times X) is a proper subset of X\times_Y X. Then for some algebraically closed field \Omega, there exist x_1,x_2\in X geometric points with \phi(x_1)=\phi(x_2), but with disjoint orbits O(x_1) and O(x_2). By assumption, these are closed invariant disjoint subsets of \bar{X}. Thus, there exists an invariant f\in \Gamma(X,\mathscr{O}_X)\otimes_k \Omega with f(x_1)=0 and f(x_2)=1. Since the ring of invariants is for \Gamma(\bar{X},\mathscr{O}_X) is R_0\otimes_k \Omega with R_0 the invariants of \Gamma(X,\mathscr{O}_X), there exists an invariant such that f(x_1)\neq fx_2). Thus \phi(x_1)\neq \phi(x_2), a contradiction. QED

Finally, to connect this back to finite groups, let V be an affine variety over k=\bar{k} of characteristic zero, and G a finite group acting on V. Then the orbits of G are finite, and therefore closed, so the action is closed. By the theorem, a universal quotient V/G exists, and it is an affine scheme of finite type over k. By the corollary, it is in fact a geometric quotient. To see that V/G is a variety merely requires that we check that R_0 is reduced. This follows from the fact that R is reduced (as V is a variety) and R_0\subset R must then be. Thus, affine varieties have quotients by finite groups which are affine varieties.

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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