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Results tagged with algebraic-stacks
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user 40
11
votes
Accepted
Ferrand pushouts for algebraic stacks
Yes, this is exactly Theorem A.4 in my old preprint Compactification of tame Deligne–Mumford stacks which is long overdue to appear on the arXiv. The proof is rather terse but fairly standard (compare …
- 5,039
1
vote
Residual gerbe and field of moduli
(1): There is a natural injection $K\to L$, indeed we have $\operatorname{Spec} L\to \mathcal{G}\to \operatorname{Spec} K$.
(2) and (3): No, it can happen that there are no objects defined on any str …
- 5,039
9
votes
Accepted
Milnor excision for algebraic stacks
In upcoming (now on arXiv:2205.08623) joint work with Jarod Alper, Jack Hall and Daniel Halpern-Leistner:
Artin algebraization for pairs with applications to the local structure of stacks and Ferrand …
- 5,039
7
votes
universal property of blow up for stacks?
Yes, this can, for example, be checked using fppf-descent.
Pick a presentation $p:U\to X$ and pull-back everything along $p$. Since blowing-up commutes with flat base change, you may then use the uni …
- 5,039
10
votes
Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?
I don't know the answer to your question in general (on what I would call "fpqc-stacks with affine diagonal") but the answer is at least true for many algebraic stacks. In particular, in Perfect compl …
- 5,039
5
votes
Accepted
Universal homeomorphism of stacks and etale sites
(1) is by definition. The standard proof of (2) is via descent of étale morphisms along universal submersions (see SGA1, Exp IX, Thm 4.10, or http://stacks.math.columbia.edu/tag/04DY, or my paper "Sub …
- 5,039