# Milnor excision for algebraic stacks

Recall that a commutative square of commutative rings

$$\begin{matrix} A&\to&B\\ \downarrow &&\downarrow\\ A^\prime&\to&B^\prime\end{matrix}$$

is called a Milnor square if the vertical maps are surjective and the square is both a pullback and pushout of rings.

It has been shown that Milnor squares give rise to diagrams that are still both pullbacks and pushouts in the category of all schemes after applying $$\operatorname{Spec}$$. Since it is standard that the $$\operatorname{Spec}$$ functor sends pushouts to pullbacks, perhaps a more interesting way to state this fact is that a functor of points representable by a scheme $$S$$ satisfies Milnor excision:

$$S(A)\simeq S(A^\prime)\times_{S(B^\prime)}S(B).$$

Question: Is it true that Deligne-Mumford stacks or Artin stacks also satisfy Milnor excision? Is there a reference?

Note: Lurie shows that spectral Deligne-Mumford stacks satisfy a weaker condition in SAG chapter 16 called cohesion, which gives excision as above when all of the maps of rings in the square are surjective. I'm primarily interested in the spectral DM case, but a proof in the non-derived case should be enough to suss out what's going on.

Edit: It turns out that this is quite a hard problem in complete generality, but the case I care about is, specifically, quasicompact quasiseparated Deligne Mumford stacks (but no affine/quasiaffine diagonal!). Not sure if that makes things any easier.

• The standard proof, as far as I'm aware, that schemes satisfy Milnor excision builds an explicit pushout of affine schemes and shows that it is a locally ringed space and also an affine scheme, which then by categorical generalities implies the result. These pushouts are not preserved by the Yoneda embedding into the huge fppf site, so it's a bit tricky. Aug 27 '20 at 19:57

In upcoming joint work with Jarod Alper, Jack Hall and Daniel Halpern-Leistner:

Artin algebraization for pairs and applications to the local structure of stacks and Ferrand pushouts

we prove more generally the existence of pushouts of affine morphisms along closed immersions in the category of (quasi-separated) algebraic stacks. This in particular implies that Milnor squares are pushouts in the category of (quasi-separated) algebraic stacks. Let me sketch how this is proved:

Let $$X=\operatorname{Spec} B$$ and $$Y=\operatorname{Spec} A$$ and similarly for the primes so we have a cartesian square: $$\require{AMScd}$$ $$\begin{CD} X' @>f'>> Y'\\ @V g' V V @VV g V\\ X @>>f> Y \end{CD}$$ with $$g$$, $$g'$$ closed immersions. By assumption, this is co-cartesian in the category of affine schemes. To show that this is co-cartesian in the category of algebraic stacks, let $$Z$$ be an algebraic stack together with maps $$u\colon X\to Z$$ and $$v\colon Y'\to Z$$ and a $$2$$-isomorphism $$ug'\cong vf'$$. We can replace $$Z$$ with an open quasi-compact neighborhood of the images of $$u$$ and $$v$$ and assume that $$Z$$ is quasi-compact.

Let $$p\colon Z_1\to Z$$ be an affine smooth presentation. Consider the pull-backs along $$u$$, $$ug'\cong vf'$$ and $$v$$ and call these $$X_1\to X$$, $$X'_1\to X'$$ and $$Y'_1\to Y'$$. The easiest case is if $$Z$$ has affine diagonal. Then $$p$$ is affine and $$X_1$$, $$X'_1$$, $$Y'_1$$ are also affine. Then we can take the pushout of the three affine schemes resulting in $$Y_1\to Y$$. This gives us a map $$Y_1\to Z_1\to Z$$. One then observes that $$Y_1\to Y$$ is smooth (flatness is [Fer, Thm 2.2 (iv)] and finite presentation can be proven similarly and smoothness then follows by considering fibers). Then take $$X_2=X_1\times_X X_1$$ etc. We obtain two maps $$Y_2\rightrightarrows Y_1\to Z_1\to Z$$. Since $$Y_2$$ also is a pushout in the category of affine schemes (they are stable under flat base change by [Fer, Thm 2.2 (iv)]) these two maps coincide (*). By descent, we obtain a map $$Y\to Z$$.

(*) It remains to show that any two maps $$Y\to Z$$ fitting in the diagram are isomorphic up to unique 2-isomorphism. For this, one takes two maps and pull-back the diagonal of $$Z$$. This is then turned into an existence question. Again, if the diagonal is affine, it is immediate.

When the diagonal is not affine, then the $$X_1$$, $$X'_1$$ and $$Y'_1$$ above are merely algebraic spaces. One can take an étale affine presentation of $$X_1$$ and pull this back to $$X'_1$$. The crucial step is then to extend this to an étale presentation of $$Y'_1$$. This is where the Artin algebraization alluded to in the title comes in. It is also needed when you want to construct the pushout $$Y$$ of a diagram $$X\leftarrow X'\rightarrow Y'$$ of algebraic stacks (affine / closed immersion).

Edit: In [TT], the case where $$\Delta_Z$$ is (ind-)quasi-affine is handled. The crucial result is [TT, Thm 5.7/5.8] which in the setup above proves that $$Y_1$$ exists when $$X_1$$ is (ind-)quasi-affine. This settles the case when $$Z$$ is an algebraic space or a Deligne–Mumford stack with separated diagonal. The case where $$f$$ is finite/integral is easier and treated in [Fer] and [R, Thm. A.4]. Also see MO question Ferrand pushouts for algebraic stacks.

[Fer] Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), no. 4, 553–585.
[R] David Rydh, Compactification of tame Deligne–Mumford stacks, preprint, https://people.kth.se/~dary/tamecompactification20110517.pdf
[TT] Michael Temkin and Ilya Tyomkin, Ferrand pushouts for algebraic spaces, Eur. J. Math. 2 (2016), no. 4, 960–983.

• This is great, thanks! Aug 28 '20 at 14:36
• Oh, one minor question: Is the quasiseparated hypothesis known to be necessary, in the sense that there exists a counterexample that doesn't satisfy Milnor excision? If there is a counterexample, it would be interesting to know. Thanks again =). Aug 28 '20 at 18:16
• Milnor excision, as you stated it in the question, holds at least for stacks with quasi-separated diagonal (that is, the double-diagonal is quasi-compact). In particular, it holds for algebraic spaces and algebraic stacks with separated diagonals. I think we can tweak our proof to get it to work in general. Aug 30 '20 at 12:56
• Well, the tweaking is probably not so easy after all... Aug 30 '20 at 13:25
• Yeah, this whole question seems quite tricky. I'm trying to see if I can get it working using locally ringed topoi for higher DM stacks until your paper comes out. Trying to see which pushouts are preserved along the wrong adjoints is very counterintuitive! Anyway, excited to see the paper whenever it comes out! Aug 30 '20 at 19:53