All Questions
Tagged with stacks etale-cohomology
10 questions
3
votes
0
answers
152
views
Descent of classifying stack
Let $X$ be a variety over $k$ and $G$ be a finite abelian group. Then we know that $H_{fppf}^{2}(X,G)$ is in bijective correspondence with isomorphism classes of $G$-banded gerbes.
Now we consider a ...
- 253
25
votes
1
answer
3k
views
Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...
- 711
6
votes
1
answer
504
views
Irreducible components of an algebraic stack
Let $\mathcal{X}$ be an algebraic stack of finite type over a (separably closed) field $ k$. Let's say that $\mathcal{X}$ has finite dimension $d \in \mathbb{Z}$. Is it still true that the number of ...
- 1,061
5
votes
1
answer
511
views
What is the relationship between the $\ell$-adic cohomology of a DM stack and that of its coarse moduli space?
Let $\mathscr{X}$ be a smooth proper DM stack over a field $k$ (perhaps assumed to be separably closed and/or of char. $0$) and let $\pi \colon \mathscr{X} \rightarrow X$ be its coarse moduli space.
...
- 3,058
9
votes
1
answer
891
views
Universal homeomorphism of stacks and etale sites
A morphism between schemes is a universal homeomorphism if it is integral, surjective, universally injective. For morphism between algebraic stacks, this notion also make sense.
It is well know that ...
- 313
3
votes
1
answer
956
views
Reference for etale cohomology on stacks
Is there good reference for general theory.of etale cohomology on stacks and more advanced topics?
Thanks
- 781
4
votes
0
answers
477
views
Cohomology of BG, algebraically
Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient ...
- 805
4
votes
1
answer
337
views
family of gerbes over smooth and proper algebraic varieties
Let $X$ be a smooth and proper variety over $\mathbb{C}$. Let $F$ be an $\mathbb{A}^1$ family of $\mathbb{G}_m$ gerbes over $X$. Suppose the fibers over every point away from 0 in $\mathbb{A}^1$ are ...
- 3,758
3
votes
0
answers
2k
views
When is the base change morphism an isomorphism?
This is a rewrite of a previous question, which was in turn a follow up question to Leray-Hirsch principle for étale cohomology The motivation is to clarify some points in Torsten Ekedahl's ...
- 23.5k
13
votes
2
answers
840
views
When can cohomology be calculated on the coarse moduli space?
Suppose $\cal{X}$ is a DM-stack, and X its coarse moduli space. Let F be a sheaf on $\cal{X}$, and $\pi : \mathcal{X} \to X$ the projection. In all examples I have seen, it has been true that
$H^i(\...
- 40.2k