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3
votes
2answers
335 views

What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve?

Lately I am studying the bend-and-break, and I follow the proof in the following note written by Olivier Debarre: http://www.math.ens.fr/~debarre/M2.pdf There is a detail that I just cannot go ...
8
votes
0answers
480 views

Eilenberg-Steenrod axioms of sheaf cohomology

Cohomology of a space is often defined axiomatically: a cohomology theory is a functor from pairs of spaces to abelian groups satisfying the Eilenberg-Steenrod axioms. Is there a similar ...
8
votes
0answers
369 views

Deducing properness from H^i(X,F) finitely generated over \Gamma(O_X)

Suppose that $X$ is a quasi-projective variety over a field $k$ and that we further know that for every coherent sheaf $\mathcal{F}$, $H^i(X,\mathcal{F})$ is finitely generated over $\Gamma(O_X)$. Is ...
8
votes
0answers
288 views

Dualizing complex of the product of two locally compact spaces

Hello! In the setting of locally compact Hausdorff spaces, I would like to understand the relation between the exterior product ${\mathbb D}_X\boxtimes{\mathbb D}_Y$ of the dualizing complexes of two ...
4
votes
0answers
93 views

$A_\infty$ structure on sum of twists of structure sheaf

Fix $n$ and let $P^n$ be projective $n$-space. Let $S = k[x_0, \dots, x_n]$. Set $A^0 = \bigoplus_{d \ge 0} H^0(P^n, \mathcal{O}(d))$ and $A^n = \bigoplus_{d < -n} H^n(P^n, \mathcal{O}(d))$. I ...
4
votes
0answers
335 views

Cohomology of a sheaf with only one stalk

Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish). I believe that ...
3
votes
0answers
93 views

Deformation of vector bundle on projective space with same Hilbert polynomial as multiple of structure sheaf

Let $E$ be a vector bundle on projective space ${\bf P}^n$ whose Hilbert polynomial is the same as $\mathcal{O}^{{\rm rank}(E)}$. Does there exist a vector bundle over ${\bf P}^n \times {\rm ...
3
votes
0answers
154 views

Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?

It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e. $$ ...
3
votes
0answers
52 views

Convolution of DQ-Modules

On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push ...
2
votes
0answers
171 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 ...
2
votes
0answers
120 views

$\delta$-functor and commutativity of pull-back with right derivation

Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, ...
2
votes
0answers
61 views

Cohomology and quotients for the canonical topology

Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example ...
2
votes
0answers
221 views

Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...
2
votes
0answers
302 views

Weil Kostant Integrality Result as Stated by Brylisnki

I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued ...
2
votes
0answers
127 views

Intersections of components of 'simple' ('local") Zariski coverings

I would like to study the ordered Cech cohomology with respect to a Zariski covering of a variety. I can pass to the limit with respect to refinements; the components of the 'limit covering' will be ...
2
votes
0answers
133 views

orientations on a stratified space

For this question let $k$ be any field of characteristic not equal to $2$ and $X$ a stratified space whose strata are topological manifolds. I'm not sure what the definition of "stratified space" ...
2
votes
0answers
307 views

Sheaf Cohomology on Zariski-Riemann Spaces

Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
1
vote
0answers
96 views

Help understanding the proof of a theorem about Cohomology of vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states: Let $\mathcal{E}$ be a vector bundle on ...
1
vote
0answers
182 views

Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
1
vote
0answers
68 views

How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?

For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to ...
1
vote
0answers
196 views

why is the local field Q_l not an etale sheaf over a scheme X?

I would like to know the reason why the local field Q_l is not an etale sheaf over a scheme X while its ring of integers Z_l can be regarded as a constant etale sheaf?
1
vote
0answers
199 views

On inverse images with respect to Zariski-etale topology.

For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
1
vote
0answers
205 views

Non-realizable CR structures?

Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$, $v = ...
0
votes
0answers
106 views

Cohomology of pushforward under the double cover

Given a double cover $\pi: C \to \mathbb P^1$, where $C$ is a genus $g$ curve over algebraically closed field, I want to compute the group $\mathrm H^1(\mathbb P^1, \pi_*\mathbb G_m)$ in flat ...
0
votes
0answers
129 views

Cohomology group vs sheaf of cohomology group

Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf ...
0
votes
0answers
90 views

Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that: $m^*(L) \simeq L \boxtimes L $. Then why is $L$ an equivariant sheaf on $G$ with the action the ...
0
votes
0answers
291 views

Global sections of twisting ideal sheaf of a smooth closed point on a projective space

Let $X = \mathbb{P}^n_k$ be a projective space over an algebraically closed field $k$ and $x$ be a closed point. Given an integer $m$ and a positive integer $r$. What are the global sections of ...
0
votes
0answers
166 views

Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?

Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...