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Results tagged with stacks
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user 11682
In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
4
votes
Accepted
Descending a monomorphism of stacks
A morphism of stacks is an iso iff it is a mono and an epi
F. $F'$ is an epi iff $F$ is
And this is enough :
First case : assume $F$ is representable, i.e. $\Delta_F$ is a mono. …
- 4,008
9
votes
Accepted
stackification commutes with finite limits?
And fortunately there is an excellent reference online: Tag04Y1 in the Stacks Project. I quote:
Lemma 8.4. Let $C$ be a site. …
- 4,008
4
votes
Algebraic stacks as (étale) groupoid algebraic spaces/schemes
The notion of "torsor under a groupoid in a topos" already appears in
Breen, Lawrence Tannakian categories. Motives (Seattle, WA, 1991), 337–376, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc …
- 4,008
4
votes
Does every morphism BG-->BH come from a homomorphism G-->H?
As a complement to the answers above : it is kind of well-known (at least I thought it was) that the natural morphism
$$\operatorname{\mathbf{Hom}}_{gr} (G,H) \to \operatorname{\mathbf{Hom}}(BG,BH)$ …
- 4,008
2
votes
Good introductory references on algebraic stacks?
Besides the references already given, I like Dan Edidin's
Notes on the construction of the moduli space of curves
https://arxiv.org/abs/math/9805101
I quote :
"In section 3 we return to curves and …
- 4,008
6
votes
When quotient stacks (for nonsmooth group) are algebraic and related questions
About 1. : no, smoothness isn't essential.
"Flat is enough" : De Jong's slogan to express this result due to M.Artin.
https://www.math.columbia.edu/~dejong/wordpress/?p=1584
I quote :
"Given a flat, f …
- 4,008
3
votes
Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
If $G$ is an affine groupe scheme over some base $S$, you can consider the groupoid $G\rightrightarrows S$. The corresponding prestack $[G\rightrightarrows S]^{pre}$ is (equivalent to) the prestack of …
- 4,008
4
votes
Finite etale atlas for Deligne-Mumford stacks
over an algebraically closed field of characteristic zero say (but would work in characteristic p as well by defining precisely X as a stack of roots in the sense of Vistoli - see Charles Cadman, Using stacks … Also, you may want to consider the following closely related notion, taken from
Fundamental Groups of Algebraic Stacks Behrang Noohi http://arxiv.org/abs/math/0201021
"An algebraic stack being uniformizable …
- 4,008
4
votes
Representability of morphism of stacks
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic
stacks over $S$. …
- 4,008
18
votes
Phenomena of gerbes
You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the str …
- 4,008
4
votes
Phenomena of gerbes
A typical example from deformation theory : fix $i:X_0\to X$ of first order thickening defined by a square zero ideal $\mathcal I$, and let $\mathcal E_0$ be a locally free sheaf of finite rank on $X …
- 4,008