35
votes
Accepted
Does every sheaf embed into a quasicoherent sheaf?
That already fails for $X$ equal to $\text{Spec}\ R$, where $R$ is a DVR with generic point $\eta = \text{Spec}\ K$. Since there are only two nonempty open subsets of $X$, namely all of $X$ and $\{\...
29
votes
Accepted
When (or why) is a six-functor formalism enough?
When defining a homotopy-coherent structure, you have to strike the correct balance between supplying enough data (so that all isomorphisms (between isomorphisms, ...) that you need later are actually ...
- 21.3k
21
votes
Accepted
Is there a complex which computes Cech cohomology?
This is the kind of thing sieves are good for. For an open cover, let $S$ denote the sieve it generates, so $S$ is poset of open subsets $V$ such that $V$ is contained in some element of the cover. ...
- 9,273
16
votes
Poincare duality on the level of complexes
One way of finding a "fully derived" version of Poincaré duality is Atiyah duality. This says that for any closed manifold $M$ there is an equivalence of spectra (in the sense of algebraic topology)
$$...
- 16.5k
14
votes
Is there a complex which computes Cech cohomology?
While I'm sure a hocolim solution also exists (and works in general), one can avoid it, at least for compact topological spaces, by using open covers which are indexed by the points of your space $X$ (...
- 18.7k
14
votes
Accepted
Different definition of sheaf cohomology
Taking global sections is the same thing as computing the hom from $O_X$. In other words, there is an isomorphism of functors $\Gamma(X,-)\cong\hom(O_X,-)$, so both functors have the same derived ...
- 47.3k
14
votes
Accepted
Zorn's lemma for Grothendieck sites
The statement you formulate is not true in this generality.
The idea of the following counterexample is to exploit the fact that the assumption "all restriction maps along morphisms of the site ...
- 1,110
13
votes
How to compute the cohomology of a local system?
Suppose $X$ is a connected CW complex with fundamental group $\pi:=\pi_1(X,x)$. Then the cellular chain complex $C_*(\widetilde{X})$ of the universal cover is a chain complex of free $\mathbb{Z}\pi$-...
- 35.9k
12
votes
Accepted
Cohomology of Grothendieck topology
Artin, M. Grothendieck topologies. (English) Zbl 0208.48701 Cambridge, Mass.: Harvard University. 133 p. (1962). (pdf copy)
These notes seem to fit your description precisely. They are concise, start ...
10
votes
Accepted
Algebraic groups without torsors
I am just posting my comment as an answer. If you want to produce a nontrivial torsor, it suffices to produce a nontrivial torsor after base change to the algebraic closure of a residue field of the ...
10
votes
Accepted
First cohomology of tangent sheaf of rational curve
Let $C$ be the union of 5 lines in general position in $\mathbb{P}^2$ (hence with 10 pairwise intersection points $P_{ij}$, $1 \le i < j \le 5$) and let $F$ be the equation of $C$. We have the ...
- 39.3k
10
votes
Accepted
Clarification on smooth de Rham theorem
The following statements are true:
The complex $0 \to \mathbb{R}_X \to \Omega^0_X \to \Omega^1_X \to \dots$ is an acyclic complex (of sheaves), i.e. it is exact at each step. This is the content of ...
- 37k
10
votes
Converses to Cartan's Theorem B
For (4), there are finite CW complexes $X$ which are not contractible, but such that $H^q(X, \mathcal{F})$ vanishes for all (finite rank) locally constant sheaves $\mathcal{F}$. Specifically, take $X$ ...
- 156k
9
votes
Accepted
Sheaf cohomology of a complement of finitely many points
For what follows, I recommend SGA 2 (available on Arxiv). There is an exact sequence
$$0\rightarrow H^1(X,\mathcal{O}_X)\rightarrow H^1(X\smallsetminus p,\mathcal{O}_X)\rightarrow H^2_{p}(X,\mathcal{O}...
- 38k
9
votes
difference between the small and big étale/flat/... site
You can always compute the cohomology as the limit over hypercoverings of the final object. But the hypercoverings of the final object are the same in both categories, so it seems that the cohomology ...
- 16.1k
9
votes
Accepted
Elementary way to compute Hodge numbers of Grassmanian
One of the ways is to use the projective bundle theorem that says that if $X = \mathbb{P}_Y(E)$ is a projectivization of a rank $r$ vector bundle $E$ over $Y$ then
$$
H^\bullet(X) = H^\bullet(Y) \...
- 39.3k
9
votes
Estimates for certain double-Kloosterman sums
I can do it for $q$ prime, and thus $q$ squarefree, by a long but elementary manipulation followed by the Weil bound. Until the end, the elementary manipulations will work for $q$ arbitrary.
Taking ...
- 148k
8
votes
Accepted
Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Edit. This follows from the Elkik-Fujita Vanishing Theorem. There is a more general vanishing theorem due to Elkik and Fujita. One version of this theorem (where I read the theorem) is Theorem 1.3.1 ...
8
votes
How to compute the cohomology of a local system?
In general you should have $H^1(\pi_1(X,x),V)\cong H^1(X,L)\,,$ where on the left we have the group cohomology of $\pi_1(X,x)$ acting on $V$ according to the representation. You will also have an ...
- 1,198
8
votes
Accepted
$H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles
Yes, this is called non-abelian sheaf cohomology. If $X$ is a topological space and $\mathcal{G}$ is a sheaf of groups, then $H^0(X, \mathcal{G})$ is the global sections of $\mathcal{G}$, and there is ...
- 156k
8
votes
Reference request: Kleiman's proof of Snapper's Lemma
Note that Nitsure's paper is part of the book FGA Explained. There is a proof of Snapper's lemma in Theorem B.7 of Appendix B ("Basic intersection theory" by Kleiman) in the same book. ...
- 28.7k
8
votes
Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Let $x_0,\ldots, x_n$ be homogeneous coordinates, then a section of
$H^0(\mathbb{P}^n,\mathcal{T}_{\mathbb{P}^n}(d))$ can be expanded as a sum
$$\sum_i f_i(x_0,\ldots, x_n) \frac{\partial}{\partial ...
- 35.2k
7
votes
Computation of cohomology of ideal sheaves
The computation of the restriction morphism $H^i(Y,O_Y) \to H^i(X,O_X)$ is usually non-trivial (unless you know for instance an explicit locally free resolution of the ideal sheaf of $X$ in $Y$).
On ...
- 39.3k
7
votes
Accepted
Can one determine the trace map for a nonsingular projective variety explicitly?
This is a good question. It's one that my colleague Joe Lipman spent a lot of time thinking about. You can look at some of his papers for a more explicit answer for computing the trace. Probably you ...
- 35.2k
7
votes
Divisors whose restriction is big
Take $Y=\mathbb{P}^1$ and let $X=\mathbb{F}_n=\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-n))$ be the Hirzebruch surface with a section $C_0$ such that $C_0^2=-n$.
Let $H$ be an ample divisor on $\...
- 66.3k
7
votes
Accepted
Cohomology of divisors on Hirzebruch surfaces
Sure. To compute $H^0$ one can first pushforward to the base $\mathbb{P}^1$. If $a \ge 0$ one obtains
$$
p_*\mathcal{O}(a\Gamma + bF) \cong
p_*\mathcal{O}(a\Gamma) \otimes \mathcal{O}(b) \cong
S^a(\...
- 39.3k
7
votes
Accepted
First Chern class of torsion sheaves
The coefficient $r$ is equal to the length of $\mathcal{T}$ at the generic point of $Z$, so it is positive.
- 39.3k
7
votes
On the upper-bound for a type of quintuple Kloosterman sums
The Newton polyhedron method for your sum is as follows:
We first replace the quadratic character with an additive character and a simpler quadratic character using a Gauss sum, then introduce a ...
- 148k
7
votes
Accepted
Smooth analogue of Cartan's Theorem B
It seems like Gro-Tsen's comment pretty much answers your main question, doesn't it? If "simple" = $C^\infty$ manifold, and "nice" = sheaf of modules over $C^\infty_X$. Then such a ...
- 35.2k
7
votes
Accepted
Is the square of a special line bundle also special?
In general, the answer is no.
If $C$ has genus $g \geq 3$, take any effective divisor $D$ of degree $d$ such that $g - 1 < d < 2g-2$ and the support of $D$ is contained in an effective canonical ...
- 66.3k
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