Geometric quotients for finite groupoids

In this comment David Rydh formulates the conjecture that for a finite affine groupoid (U, R, s, t, c) the spectrum of the ring of invariants may be a geometric quotient for the stack [U/R]. In fact, the same question came up in a recent conversation with Jarod Alper here in the department.

I have an idea for generating invariant functions, which sounds so familiar to me that I am sure it is in the literature (let me know if you have a reference), or maybe I have already tried using it in the past. First, recall that if s,t are finite locally free flat so B is finite locally free over A then for any element x of A gives rise to an invariant element y by taking y = Nm_s(t(x)). In words y is the norm of t(x) with respect to the finite locally free ring map s : A —> B. Thus, in the general case where s, t are finite we try to find an element y in A which behaves like the norm of t(x) with respect to s. Maybe a falsifiable version of the conjecture above would be to conjecture the existence of a y in A such that for every prime p of A the value of y in k(p) is a power of the Nm of t(x) restricted to B \otimes_{s, A} k(p)?

My idea is to try to do the following. Take a finite free extension phi : A —> B’ and a surjection pi : B —> B’ such that pi o phi = s. (It may be convenient for later arguments to allow only certain types of ring maps A —> B, such as my personal favorite: finite flat relative complete intersections.) Now for any element x of A we can let y = Nm_phi(x’) where x’ in B’ is any element with pi(x’) = t(x). It is clear that y will NOT be R-invariant in general, simply because we have put too little restrictions on B’. But on the other hand, I am pretty confident that the ideal generated by all y of the form Nm_phi(x’) will be R-invariant. Namely, it should just cut out the set of points which are R-equivalent to a zero of the function x.

However, if A is an Artinian ring, then we can choose B’ so that B’ and B have the same maximal ideals. In this case if A has positive residue characteristics then it is quite easy to show that y^{p^n} for large n is independent of the choice of x’ and presumably is an invariant element of A (I haven’t checked this completely). This could then be the start of a kind of induction argument in the Noetherian case. But in characteristic zero I do not even know how to produce enough invariant functions in the Artinian case.

4 thoughts on “Geometric quotients for finite groupoids

  1. The idea of finding invariants using the norm is of course an old trick (FGA, SGA3, etc). In the non-flat case it can also be used if U is normal and R=>U is finite and equidimensional, i.e., every irreducible component of R dominates a component of U (then the quotient can be constructed using Chow schemes) but this is a very restrictive case.

  2. OK, let me continue the discussion in the post a bit. Let s, t : A —> B and c : B —> B \otimes_{s, A, t} B define a finite affine groupoid scheme (U, R, s, t, c).

    Remark 1. If J is an ideal of A, then the restriction of the groupoid to A’ = A/J replaces the ring B by B’ = B/(s(J)B + t(J)B). This will automatically be a finite groupoid again.

    Remark 2. If A —> A’ is finite free then the restriction of the groupoid to A’ replaces B by B’ = A’ \otimes_A B \otimes_A A’. This will automatically be a finite groupoid again.

    Remark 3. If A is Artinian the general case reduces to the case where the groupoid is connected, meaning that for any u_1, u_2 of U there exists a point r of R with t(r) = u_1 and t(r) = u_2.

    Remark 4. If A is Artinian, then combining remarks 2 and 3 we reduce to the case where R is connected and for any point r of R the residue field of r agrees with the residue field of t(r) and with the residue field of s(r).

    Remark 5. Situation as in Remark 4. Then we can find a subfield k of A such that A is finite over k and such that s(x) – t(x) is in the radical of B for all x in k. This follows on applying the result for the finite flat case.

    Now I have the following question: Let k be a field and consider the category whose objects are finite k-algebras A (hence Artinian) and maps are maps A —> B which are not k-linear but are k-linear modulo the radicals of A and B. Is there any kind of descent theory for such gadgets?

  3. Continuing the previous comment. Let k be a field and let A, B be finite k-algebras. Let f : A —> B be a ring map which is a k-algebra map when dividing by the radicals of A and B. Let M be a finite A-module, and let N be a finite B-module. Let g : M —> N be an f-linear module map. Then I claim that g is a differential operator of finite order over k. Namely, for any c in k consider the map g_c : M —> N which maps x to cg(x) – g(cx). By assumption the map g_c is zero modulo the radicals. Hence g_c(M) is contained in a strict sub module of N. By induction on the length of N we conclude that g is a finite order differential operator. The order is bounded by the dimension of N over k.

  4. Continuing the previous comments. Suppose (A, B, s, t, c) defines an affine groupoid scheme where A, B are local Artinian with the same residue field of characteristic zero and with maximal ideals of square zero. Then the invariant subring of A surjects onto the residue field of A. This follows by a slightly annoying computation using only what it means to be a groupoid and the fact that a group scheme in characteristic zero is reduced (it also involves dealing with first order differential operators, i.e., derivations).

    Note: In order for the conjecture of the post to be true it is necessary that the invariant subring of A surjects onto the residue field whenever given an affine groupoid scheme where A, B are local Artinian rings with the same residue field of characteristic zero. So this is a good test case to look at.

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