Questions tagged [sheaf-cohomology]
The sheaf-cohomology tag has no usage guidance.
342
questions
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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 $\...
- 721
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votes
1
answer
132
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Pullback morphism of a hyperplane inclusion is zero in the derived category
Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
3
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0
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176
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Is pullback map on sheaf cohomology injective for surjective morphisms?
Consider a surjective map $f\colon X\to Y$ of smooth projective varieties. It is well known (see e.g. Voisin's Hodge theory I, Lemma 7.28) that the map $H^i(Y,\mathbb Q)\to H^i(X,\mathbb Q)$ is ...
- 2,235
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75
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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 ...
- 721
1
vote
1
answer
225
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Does anyone have a good example of an injective resolution?
I'm learning about injective resolutions and derived functor sheaf cohomology, and it seems that every source on injective resolutions gives no examples. I feel like just one good example would make ...
- 127
0
votes
0
answers
87
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Serre duality for non-compact Riemann surfaces
Suppose $X$ is a Riemann surface. If $X$ is compact, then Serre duality tells us that we have an isomorphism in sheaf cohomology
$$ H^1(X,E) \cong H^0(X,\Omega\otimes E^\ast)^\ast $$
Can we say ...
- 488
2
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1
answer
159
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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
- 53
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0
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53
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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}...
- 6,000
2
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0
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128
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Cohomology of equivariant toric vector bundles using Klyachko's filtration
I am trying to understand Klyachko's following description of the cohomology groups of locally free (hopefully more generally of reflexive) sheaves on toric varieties.
Whereas detailed literature ...
- 455
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1
answer
295
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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 ...
- 3,432
4
votes
1
answer
328
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Reference for isomorphism between group cohomology and singular cohomology
Let $G$ be a (discrete) group, $X$ a topological space that works as a classifying space for $G$, and $\mathcal{L}$ a local system on $X$ with stalk $L$. It is a fairly standard result that
$$ H^i(G, ...
- 488
2
votes
0
answers
232
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Action of algebraic group in cohomology of equivariant algebraic vector bundle
Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
- 21.1k
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A question about cohomology with local coefficient
Let's consider the next theorem.
Theorem
[The cohomology Leray-Serre Spectral sequence] Let $R$ be a
commutative ring with unit. Given a fibration $F\hookrightarrow E\overset{p}{%
\rightarrow }B$, ...
- 1,161
2
votes
1
answer
261
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Commutative group scheme cohomology on generic point
Setup:
Let $k$ be an algebraically closed field.
Let $C$ be a smooth connected projective curve over $k$.
Let $J$ be a smooth commutative group scheme over $C$ with connected fibers.
Let $j:\eta\to C$ ...
- 123
1
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0
answers
161
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Computing the first sheaf cohomology
I am looking for some examples of computing the dimension of the first sheaf cohomology for smooth projective surfaces. To be more precisely, let $X$ be a smooth, projective surface. Let $D$ be an ...
- 451
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votes
1
answer
151
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In what sense is the complex $\mathscr{L}^\bullet$ unique?
This is in Section III.12. Algebraic Geometry by Hartshorne. Assume $X\to\mathrm{Spec}(A)$ is a projective morphism of Noetherian schemes. Let $\mathscr{F}$ be coherent over $X$, flat over $A$.
...
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1
answer
261
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If $\mathrm{Ext}^i(E,F)$ commutes with base change, then is $\mathrm{Ext}^{i+1}(E,F)$ representable?
Consider a projective morphism of Noetherian schemes $p:X\to \mathrm{Spec}(A)$. Let $\mathcal{E},\mathcal{F}$ be coherent $\mathcal{O}_X$-modules flat over $A$. For every (Noetherian) ring map $A\to B$...
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votes
1
answer
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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}...
- 373
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0
answers
74
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How to estimate the locus of non-zero cohomology for a equivariant toric reflexive sheaf, with a Klyachko description
I am trying to analyze Macaulay2 package "ToricVectorBundles". The package deals with equivariant reflexive sheaves on complete toric varieties. Such a sheaf is described by a set of ...
4
votes
4
answers
546
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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
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
- 21.1k
3
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0
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59
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On the exactness of the restriction to small etale site in the case of differentiable stacks
Let $X$ be a manifold considered as a stack on the big etale site of manifolds. Consider the functor $$ F \mapsto F_X$$ which sends a sheaf on the stack $X$ to the sheaf on the manifold $X$.
In Lemma ...
- 313
25
votes
1
answer
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When (or why) is a six-functor formalism enough?
The six functor formalism in a given cohomology theory consists of for each space a derived category of sheaves and six different ways to construct functors between those categories (four involving a ...
- 137k
3
votes
1
answer
235
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Čech-like cohomology with the “other nerve”
Let $X$ be a space and $\mathcal U$ a cover of $X$. Instead of Čech cohomology, I would like to take the following construction:
let
$$I= \{ \text{finite nonempty intersections of elements of }\,\...
- 1,702
1
vote
0
answers
198
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Exterior product of Euler Exact Sequence
Consider the Euler exact sequence:
$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $
This ...
- 131
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vote
0
answers
129
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Sheaf cohomology definition
I have seen multiple definitions for sheaf cohomology and wanted to ask for the reason.
One goes through injective resolutions and the other through flasque resolutions.
For paracompact bases it holds ...
- 11
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0
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85
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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 ...
- 531
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vote
1
answer
384
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Sheaf cohomology in number theory
I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
- 176
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0
answers
444
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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 ...
- 884
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0
answers
191
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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 ...
- 373
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1
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448
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Unbounded acyclic resolutions
Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
- 27.3k
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1
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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 ...
- 23.2k
3
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Long exact sequence in Borel-Moore homology
The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence
$$\cdots \to H^{...
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0
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Homotopy invariant Bloch-Ogus cohomologies with a vanishing property
I am looking for examples (in any characteristic) of homotopy invariant Bloch-Ogus cohomology theories given by Zariski sheaves $\Gamma(n)$, such that $\Gamma(0) = \mathbb{Z}$ is the constant sheaf. ...
- 5,851
2
votes
1
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201
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Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)
Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering
$$ \dotsb \...
- 463
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0
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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 ...
- 373
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1
answer
239
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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 ...
- 7,161
2
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0
answers
60
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Isomorphism of cohomology bundles for smooth homotopic fibrations
Let $M,N$ be closed smooth manifolds. Let $f_0,f_1\colon M\to N$ be two smooth fibrations which are homotopic to each other in the class of smooth (equivalently, continuous) maps. Let $E^i_0, E^i_1$ ...
- 21.1k
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1
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434
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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
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208
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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 ...
- 10.3k
6
votes
1
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428
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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, ...
- 1,093
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votes
1
answer
178
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On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 553
3
votes
1
answer
332
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Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 553
1
vote
1
answer
220
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A question involving the three-dimensional Kloosterman sum
Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
- 553
5
votes
0
answers
246
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Cohomology of coherent sheaves on Deligne Mumford stacks
Suppose that $\cal X$ is tame Deligne Mumford stack with generic trivial inertia. Let $X$ be its muduli space and $f:{\cal X}\to X$ the projection.
Let $\cal F$ be a coherent sheaf on $\cal X$.
Is it ...
- 163
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0
answers
264
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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 ...
- 1,111
3
votes
0
answers
171
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Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology
Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
- 817
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0
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181
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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
178
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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_\...
- 1,084
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0
answers
120
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Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?
Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be).
Let $\mathcal{E}$ be a locally free sheaf on $X$, and
$$\nabla:\mathcal{E} \to \...
- 1,338
1
vote
0
answers
326
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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