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8
votes
0answers
555 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
376 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
297 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 ...
6
votes
0answers
158 views

“Flat” and “Affine” morphisms of smooth manifolds

Let $f: X \to Y$ be a qcqs morphism of qcqs schemes. We have the following characterizations: $f$ is flat $\iff$ $f^*:QCoh(Y) \to QCoh(X)$ is exact. $f$ is affine $\iff$ $f_*: QCoh(X) \to QCoh(Y)$ ...
5
votes
0answers
128 views

Push forward of the constant sheaf for a Serre's fibration

Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
4
votes
0answers
49 views

Reference request: local cohomology in disjoint union

I have a topological space $X$ and two disjoint, closed subspaces $Y$ and $Z$ of $X$. I believe that in this situation, for any abelian sheaf $\mathcal{F}$ on $X$ and any $p \in \mathbb{N}$, there is ...
4
votes
0answers
137 views

Proper base change for non-quasicoherent sheaves

For a proper flat map $f: X \to Y$ (of reasonable schemes) and a closed embedding $i: Y' \to Y$, we know by EGA the base change quasi-isomorphism, where $f'$ and $i'$ are the pullbacks of $f$ and $i$: ...
4
votes
0answers
108 views

Cohomology of $Sym^m Q \otimes Sym^k Q \otimes L^p$

Let $V$ be a complex vector space. Let $L=\mathcal{O}(-1)$ and $Q=V/L$ be the quotient bundle over $\mathbb{P}V$. I'm trying to compute the cohomologies with coefficients in $Sym^m Q \otimes Sym^k Q ...
4
votes
0answers
115 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
355 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 ...
4
votes
0answers
326 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 ...
3
votes
0answers
121 views

First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$

Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, ...
3
votes
0answers
68 views

$H^{1}(C, N_{C/X}(-m)) = 0$, for $C$ a irreducible curve on $X$ through $m$ general points

I was reading A. de Jong and J. Starr's paper "Low degree complete intersections are rationally simply connected", which can be found at http://www.math.stonybrook.edu/~jstarr/papers/nk1006g.pdf, and ...
3
votes
0answers
95 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 ...
3
votes
0answers
135 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
216 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
62 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 ...
3
votes
0answers
134 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
227 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
158 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
68 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
237 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
309 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
134 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 ...
1
vote
0answers
153 views

Importance and intuition of global sections in sheaf cohomology

I am trying to understand why global sections of a sheaf are "important" or interesting objects of study. Perhaps I have too weak of a background to appreciate it (and that is certainly an acceptable ...
1
vote
0answers
61 views

Picard sequence for sujective morphisms

Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...
1
vote
0answers
107 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
207 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
72 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
202 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
213 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
215 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
52 views

Cohomology adding the infinite point

Let $F$ be a closed subset of $\mathbb{C}$. (We assume $F \neq \emptyset, F \neq \mathbb{C},$ and $0 \notin F$.) Of course $F$ is not a closed subset of $\overline{\mathbb{C}}$ in general but it ...
0
votes
0answers
116 views

Canonicity of Cech cohomology

For a topological space X, consider the Leray covering U_λ(i.e. ∩U_λ is sufficiently fine, e.g. affine for Zariski topology) of X. For a sheaf F on X, the cohomology H^i(X,F) is calculated via Cech ...
0
votes
0answers
50 views

constructibility for pushforward

Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
0
votes
0answers
57 views

Applicability of Scheja's theorem

First, let me begin by recalling Scheja's (cohomology extension) theorem. Let $X$ be a complex manifold of dimension $n$ and $Z\subset X$ a complex submanifold of dimension $d.$ Let $\mathcal{F}$ be ...
0
votes
0answers
131 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
145 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
127 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
428 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
167 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 $H^\ast (Z,i^{-1}S)$ ...