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.

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.

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?

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.

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.