All Questions
25 questions
4
votes
1
answer
445
views
Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
- 3,472
2
votes
1
answer
265
views
Formula for the Euler characteristic of a local system on $\mathbb{P}^1$
Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
- 7,547
1
vote
0
answers
217
views
Artin-Winters proof of semi-stable reduction theorem: details
I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...
1
vote
0
answers
135
views
Base change of cohomology when the cohomology is a torsion
Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...
- 51
5
votes
1
answer
654
views
First Chern class of torsion sheaves
Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there ...
- 113
8
votes
1
answer
2k
views
Cohomology of Grothendieck topology
My naïve cartoon picture of the construction of étale cohomology is this:
start with a scheme, associate to it a Grothendieck topology (making a site).
A functor from the Grothendieck topology to ...
- 2,050
4
votes
0
answers
211
views
Does cohomology and base change hold if supported at a point?
I have a flat, quasicompact, and separated map $p : X \to \mathbb{A}^1$ and I know that $R^i p_* \mathcal{O}_X$ vanishes everywhere except possibly $0 \in \mathbb{A}^1$.
Q1: Does "cohomology and ...
- 1,104
1
vote
0
answers
152
views
What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?
Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.
What is the ...
- 435
9
votes
1
answer
1k
views
Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?
Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$.
Do the cohomology groups $H^i(Y,\mathcal{O}...
- 371
4
votes
0
answers
343
views
Does hypercohomology of the Koszul complex compute sheaf cohomology?
Let $i:X \to \mathbb{P}^n$ be a smooth projective variety defined by the vanishing locus of polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ which have degrees $>0$ and are pairwise coprime, ...
- 1,716
12
votes
1
answer
2k
views
difference between the small and big étale/flat/... site
What is the difference between the small and the big étale (or flat or syntomic or ...) site? How does the cohomology vary? When should I use which one? Up to now, I have always used the small sites.
user19475
9
votes
4
answers
3k
views
Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...
- 5,929
3
votes
0
answers
130
views
Action of automorphisms on cohomology with supports
Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
- 525
1
vote
1
answer
695
views
Axioms for sheaf cohomology
Let $R$ be a commutative ring and $X$ a topological space. Define a sheafy cohomology theory (see here) to be a collection of functors $\mathrm{H}^q:\mathrm{Sh}(X;R\mathrm{Mod})\to R\mathrm{Mod}$ such ...
user62675
4
votes
1
answer
1k
views
Leray's theorem up to some degree
I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology.
The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such ...
- 440
0
votes
1
answer
235
views
cohomology of an intermediate extension of a local system
Let $V$ be affine $n$-space over a field $k$; and $j \colon U \to V$ an open subscheme of $V$. Let $L$ be an $\ell$-adic local system on $U$.
Suppose the cohomology of $H^{\bullet}(U,L)$ does not ...
- 3,472
2
votes
0
answers
347
views
l-adic cohomology and perverse sheaves
Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...
- 3,472
2
votes
1
answer
333
views
A functorial property of higher right derived functors
Let $f:X \to Y$ be a projective morphism of complex Noetherian schemes. Assume $Y$ is smooth and for all $y \in Y$, $f^{-1}(y)$ is of pure dimension $1$. Let $\mathcal{F}_1, \mathcal{F}_2$ and $\...
- 833
2
votes
2
answers
1k
views
Cohomology of sheaf extended by zero
Let $X$ be a projective scheme of pure dimension $1$. Let $U$ be a open subscheme and $j:U \to X$ the open immersion. Let $\mathcal{F}$ be a coherent sheaf on $U$.
Denote by $j_!(\mathcal{F})$ the ...
- 2,323
0
votes
1
answer
156
views
The Existence of Pure Resolutions, Given a Degree Sequence?
I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
user24421
0
votes
1
answer
660
views
Example of non-vanishing of first cohomology of a torsion coherent sheaf on a curve
By a curve we mean a projective scheme of pure dimension one. Can some one give an example of a curve $C$ and a torsion coherent sheaf on $C$ such that its first cohomology group does not vanish?
...
- 2,032
3
votes
1
answer
1k
views
Compare global sections of restriction and pullback of sheaves
Let $X$ be a projective scheme and $X \subset \mathbb{P}^n$ for some positive integer $n$. Let $j:Z \hookrightarrow X$ be a closed subscheme. Is it true that $H^0(j^*\mathcal{N}_{X|\mathbb{P}^n}) \...
- 2,323
4
votes
1
answer
2k
views
Is the higher direct image sheaf of a locally free sheaf over $\mathbb{P}^1$ locally free?
Let $f:X \to \mathbb{P}^1$ be a projective flat morphism, $X$ is a projective scheme. Let $\mathcal{F}$ be a locally free sheaf on $X$. Are the higher direct image sheaves $R^if_*\mathcal{F}$ locally ...
- 2,032
2
votes
2
answers
645
views
Vanishing of Ext group
Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. Then we get the short exact sequence $$0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$$
for some sheaf $...
- 1,070
6
votes
2
answers
385
views
cohomology and $j_!$
I have a projective variety $X$ and an open immersion $j : U \to X$.
Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between $H^i(X,...
- 758