Stratification by gerbes

David Rydh and I have been dscussing residual gerbes and stratifcations by gerbes. We have now shown that if X is an algebraic stack whose inertia I_X is quasi-compact over X, then X has a canonical stratification by locally closed algebraic substacks which are gerbes… but this stratification is indexed by a possibly infinite well ordered set. This is the stratification of type (a) of Lemma Tag 06RF.

As the example in this post shows we cannot always expect to find a finite stratification. To me an intriguing question is what possible order types one can obtain from the canonical stratification of these algebraic stacks. My first guess is that the index should in any case always be countable (but I do not even have a heuristic argument for this).

The result above relies on a very general “generic flatness” result which also allows one to prove the existence of residual gerbes at any point of an algebraic stack whose inertia is quasi-compact.

My next goal is to revise the chapter on formal deformation theory.


Gerbes have now found their way into the stacks project. I am still working on this, but some basic material is now there.

It was a little bit more complicated than the discussion here. The reason is that you want to say what it means for one algebraic stack to be a gerbe over another. I decided not to give a geometric characterization as the definition, but rather to define what it means for one stack in groupoids over a site to be a gerbe over another. This is done in Stacks, Section Tag 06NY. We then discuss what this really means for a morphism of algebraic stacks in Morphisms of Stacks, Section Tag 06QB.

It turns out that the “topological/stack-theoretic” definition does entail that if an algebraic stack X is a gerbe over the algebraic stack Y then X —> Y is flat and locally of finite presentation and is fppf locally over Y of the form [Y/G] for some flat and locally finitely presented group algebraic space G. In fact this characterizes gerbes — i.e., we could have defined them this way. See Lemma Tag 06QH and Lemma Tag 06QI.

We say that an algebraic stack X is a gerbe if it is a gerbe over some algebraic space. Similarly to the above it turns out that this happens if and only if the inertia of X is flat and locally of finite presentation over X, see Proposition Tag 06QJ.

Monomorphisms of Algebraic Spaces

Let f : X —> Y be a monomorphism of algebraic spaces. Is f representable (by schemes)? After hitting this with a bunch of standard arguments I was led to the following commutative algebra question:

Question: Let A —> B be a local homomorphism of local rings such that the two maps B —> B ⊗_A B are essentially etale and such that A is their equalizer. Then is the map A —> B essentially etale?

This is a first approximation; I have been unable to find an exact translation of the problem on monomorphisms into algebra. The answer to the question is (I think) yes if A is a local ring of dimension 0, or if A —> B is flat (descent of \’etale ring maps).

By the way, I should mention that the statement on monomorphisms of algebraic spaces is true when the morphism is locally of finite type. Namely, any separated, locally quasi-finite morphism of algebraic spaces is representable (by schemes), see Lemma Tag 0418.

Slicing presentations

In the stacks project a Deligne-Mumford stack is an algebraic stack X such that there exists a scheme U and a surjective etale morphism U —> X. An algebraic stack X is said to be DM if the diagonal Δ : X —> X x X is unramified. In fact Theorem Tag 06N3 says:

X is DM if and only if X is Deligne-Mumford.

An algebraic stack X is said to be quasi-DM if the diagonal Δ : X —> X x X is locally quasi-finite. The analogue of the theorem above is Theorem Tag 06MF which says:

X is quasi-DM if and only if there exists a scheme U and a surjective, flat, locally finitely presented, and locally quasi-finite morphism U —> X.

The proofs of these theorems are completely parallel. Assume X is DM (resp. quasi-DM). We try to construct etale (resp. loc fp + flat + loc quasi-finite) maps from schemes toward X. In both cases the strategy is the following:

  1. Pick a smooth morphism U —> X,
  2. choose a suitable point x of X,
  3. let F be the fibre of U over x, and
  4. “slice” U, i.e., find a complete intersection V(f_1, …, f_d) ⊂ U such that f_1, …, f_d form a regular system of parameters (resp. regular sequence) at some point of F

In both cases the proof shows that, after possibly shrinking U, the morphism V(f_1, …, f_d) —> X is flat, locally finitely presented, and unramified (resp. locally quasi-finite). A bit of care is needed in choosing the point x on X. I decided to use “finite type points”; in both cases one then has to do a bit of work to show that the “fibre F” has desirable properties: in the DM case one need to produce x such that F —> U is unramified and in the quasi-DM case such that F —> U is locally quasi-finite.

The reasoning above is completely standard. However, there is a way to deduce the first theorem from the second. I decided against arguing like this in the stacks project as it is perhaps a little nonstandard. Here is the argument. Let X be DM. By the second theorem we can find U —> X which is surjective, flat, locally of finite presentation, and locally quasi-finite. Let H_{d, lci}(U/X) be the LCI locus in the relative degree d Hilbert stack of U over X (see Section Tag 06CJ). Then H_{d, lci}(U/X) —> X is smooth (this is explained in the proof of Theorem Tag 06DC). But of course it is clear that H_{d, lci}(U/X) —> X has relative dimension 0, hence it is etale. This doesn’t quite finish the proof because H_{d, lci}(U/X) is (as defined in the stacks project) an algebraic stack and not an algebraic space; but a straightforward argument shows (because X is DM) that the disjoint union for varying d of the open substacks of H_{d, lci}(U/X) having trivial inertia surjects onto X.

Residual gerbes

This morning I introduced a notion of residual gerbe for a point x on an algebraic stack X. See the (currently) last section in the chapter Properties of Algebraic Stacks. I decided that the residual gerbe of X at x should be a reduced, locally Noetherian algebraic stack Z whose underlying space |Z| is a singleton which comes with a monomorphism Z —> X such that the unique point of Z maps to x.

In the generality of the stacks project I cannot show that residual gerbes always exist. If a residual gerbe Z does exist, then it is unique. In fact, it turns out that there exists a field and a surjective, flat, locally finitely presented morphism z : Spec(k) —> Z (which is a very convenient property to have because we work in the fppf topology). For any algebraic stacks there are alway points where the residual gerbe does exist, namely the points of finite type.

In Appendix B of the preprint “Etale devissage, descent and push-outs of algebraic stacks” David Rydh has shown (I think) that residual gerbes (as defined above) exist for any point of a quasi-separated algebraic stack (his results are actually stronger). This implies that the definition above does not conflict with the definition in the book by Laumon and Moret-Bailley.

One curiosity is that we haven’t yet defined gerbes in the stacks project. The fact that a residual gerbe “is” a gerbe (over something) isn’t yet documented in the stacks project.

Let me know if you have any comments or suggestions.

Formal deformation theory

Alex Perry wrote a chapter on formal deformation theory for the stacks project following Schlessinger and Rim. Please read the introduction of that chapter for more information.

I intend to work on this chapter a little bit more in the near future in order to allow for finite residue field extensions (i.e., work with Λ —> k of finite type). The way the chapter is written however, I believe only minor changes will have to be made.

Once this is done we intend to use this material to study the formal local structure of algebraic stacks and to explain Artin’s criteria for Algebraic Stacks. One big obstruction looming in the future is the general Neron desingularization (Popescu). I’m not yet sure how to deal with this.

More immediately what we really need now is a couple of examples where the theory applies directly as written up. Alex and I listed a few obvious examples at the end of the chapter. If you feel like writing one of these up (should not be more than a few pages) using the framework we have in place please email me (so we don’t do double work).