All Questions
Tagged with sheaf-theory sheaf-cohomology
123 questions
4
votes
0
answers
48
views
Resolution of constant sheaf by $L^2$ function sheaves
Let $X$ be a compact Hausdorff space equipped with a Radon measure of full support.
Then $U\mapsto L^2(U)$ is a fine sheaf, hence can be taken for a first step in an acyclic resolution of the constant ...
- 460
2
votes
1
answer
232
views
existence of a coherent sheaf
I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
- 613
6
votes
0
answers
223
views
Under what generality are the compactly supported singular and sheaf cohomologies equal?
Edit: I have since resolved my question.
If X is locally compact Hausdorff in addition to being cohomologically locally contractible with coefficients in $A$ - eg it is a manifold or an open subset of ...
- 1,021
3
votes
1
answer
240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 178
0
votes
0
answers
116
views
How can I calculate $\chi(\mathscr{O}(P))$
Let be X a reduced and irreducible curve over a field $L_0$. Let $L$ an extension of $L_0$ and set
\begin{gather*}
\overline{X}=L \otimes X.
\end{gather*}
Assume $\overline{X}$ also irreducible. Now, ...
- 69
1
vote
0
answers
141
views
Homeomorphic endomorphism of schemes inducing equivalence of sheaves
Let $F: X \to X$ to be an endomorphism of scheme $X$, which is additionally assumed to induce an universal homeomorphism on the underlying topological space $| X|$. Then it is known that this induces ...
- 6,028
8
votes
0
answers
644
views
Trying to understand "Shtukas"
I'm studying Goss' Basic structures of function field Arithmetic, chapter 6 about Shtukas. I'm trying to understand some details about some concepts. This chapter is based on a Mumford's paper An ...
- 69
7
votes
1
answer
607
views
Converses to Cartan's Theorem B
Here is a phrasing of some Cartan Theorem B statements:
Consider the following conditions:
$X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...
- 1,109
1
vote
0
answers
126
views
Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
- 613
2
votes
0
answers
148
views
Equivalence of cohomology with compact support
Let $𝑋$ be a connected CW complex, $𝜌:\pi_1(𝑋)→\mathrm{Aut}(G)$ a representation, and $G_\rho$ the associated sheaf. It follows from here, that the following two cohomologies are isomorphic.
(1)The ...
- 241
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
27
votes
0
answers
469
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,021
6
votes
0
answers
103
views
Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?
In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
- 1,021
0
votes
0
answers
57
views
Lifting of quadrics containing hyperplane section for projectively normal curves
Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
- 6,028
4
votes
1
answer
541
views
Clarification on smooth de Rham theorem
I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology:
Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex
$$\mathbb{R}...
- 511
4
votes
4
answers
696
views
Canonical product in sheaf cohomology
EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
- 21.8k
0
votes
0
answers
91
views
Why does the associated sheaf vanish?
I am learning local cohomology from Hartshorne’s book Local Cohomology.
My question is about understanding a line in the proof of proposition 1.11 in this book.
The set-up for proposition 1.11 is that ...
- 639
9
votes
0
answers
475
views
Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
- 898
0
votes
0
answers
215
views
Singular cohomology to cohomology of quasi-coherent sheaf
Let $X$ be a projective nonsingular variety (integral Noetherian scheme of finite type that is proper over a field $k=\overline{k}$ such that $\Omega^1_X$ is locally free). Suppose one knows the ...
- 511
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
0
votes
0
answers
247
views
Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?
I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off?
Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...
- 511
16
votes
1
answer
448
views
Zorn's lemma for Grothendieck sites
In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
- 163
6
votes
0
answers
226
views
Is the right adjoint to presheaf direct image interesting?
Let $X\overset{f}{\to}Y$
be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...
- 10.5k
6
votes
1
answer
441
views
Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
- 1,153
6
votes
0
answers
322
views
What's the point of fine sheaves? (As opposed to soft ones)
Why do people care to define fine sheaves? What useful property do they have for which softness is not sufficient?
some observations (because I feel guilty about a the one-line question):
The point ...
- 1,141
1
vote
0
answers
215
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 $\...
3
votes
0
answers
194
views
Hypercovers with sieves
Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
- 1,094
1
vote
0
answers
355
views
Global section of pullback of an ideal sheaf
For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
- 119
6
votes
1
answer
327
views
Topology on cohomology of a sheaf of topological groups
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
Is there a natural way to introduce topology on $H^i(X,...
- 1,170
2
votes
1
answer
501
views
Higher direct image with compact support of a constant sheaf
Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...
6
votes
2
answers
631
views
Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
- 1,343
3
votes
1
answer
258
views
Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
- 1,805
3
votes
1
answer
331
views
Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism
If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
- 1,805
5
votes
1
answer
209
views
Cohomology of doubly pinched torus via spectral sequences
Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
- 191
1
vote
0
answers
103
views
$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality
This is a question arising when I am reading
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
...
- 1,168
1
vote
1
answer
521
views
Connecting homomorphism in Cech cohomology
Let $M$ be a smooth manifold and $\mathcal{U}$ be a good open cover of $M$. If I have an exact sequence of sheaves
$$0 \longrightarrow A \stackrel{f}\longrightarrow B \stackrel{g}\longrightarrow C \...
- 19
10
votes
0
answers
958
views
intuition about perverse sheaves
firstly, I would know if my very basic intuition on perverse sheaves is correct .
secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .
my intuition ...
- 546
1
vote
1
answer
278
views
Relation between characteristic cycle and singular support of constructible sheaf
Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $...
- 21.8k
4
votes
1
answer
289
views
Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$
I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\...
- 6,028
4
votes
2
answers
1k
views
Sheaf cohomology commutes with colimits of sheaves
Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...
- 6,028
13
votes
1
answer
649
views
Sheaves in combinatorics and discrete geometry
I am looking for examples for the application of sheaves, sheaf-like constructions or the (co)homology of sheaves to problems in combinatorics and discrete geometry.
For example given a poset $(P,\...
- 430
4
votes
0
answers
120
views
Understanding a step in proof of sheaf version Verdier duality
Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss.
So all proofs I can find factors through a particular statement, which goes ...
- 1,856
3
votes
1
answer
225
views
Subspace inclusion with non-vanishing higher direct images
I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
- 10.5k
1
vote
1
answer
2k
views
Pullback map on global sections surjective
Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ ...
- 6,028
6
votes
1
answer
760
views
The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
- 294
2
votes
1
answer
600
views
Pushforward in Compactly Supported Cohomology
Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
- 1,761
5
votes
0
answers
268
views
Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
- 59
6
votes
1
answer
221
views
When is derived category of ringed space perfectly generated?
Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...
We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. ...
- 501
5
votes
2
answers
382
views
Obstructions to abelian sheaf being quasi-coherent
Let $X$ be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $X$ to be quasi-coherent with respect to some affine scheme structure on $X$ is that its higher ...
user137767
2
votes
1
answer
1k
views
Cohomology of tangent sheaf of a hypersurface
Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
user125056