Questions tagged [sheaf-cohomology]

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4 votes
0 answers
202 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 ...
3 votes
1 answer
497 views

Elementary way to compute Hodge numbers of Grassmanian

I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf ...
10 votes
0 answers
825 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 ...
1 vote
0 answers
160 views

Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
4 votes
0 answers
100 views

Serre vanishing on one-point blow-ups

This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry. Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
1 vote
1 answer
235 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 $...
1 vote
0 answers
103 views

Compute Cech cohomology with two open sets

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...
4 votes
1 answer
281 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 votes
1 answer
282 views

Formal character of local cohomology groups with support in Schubert cells

Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$. ...
8 votes
1 answer
480 views

Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?

Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...
1 vote
0 answers
354 views

Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)

I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2 & Example 12.9.2): Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point $(0:...:1)...
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 ...
1 vote
0 answers
133 views

Surjectivity of multiplicative map (in more specific case)

(I have asked the question Surjectivity of multiplicative map. I ask here the more specific case.) Let $S$ be a smooth complex algebraic surface, and $D$ be a divisor on $S$ such that $D^2>0$ and $...
6 votes
0 answers
559 views

Calculation in prismatic cohomology

In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper ...
14 votes
1 answer
559 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,\...
1 vote
1 answer
157 views

Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes. Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$. We can define the subsheaf $\...
4 votes
0 answers
119 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 ...
3 votes
1 answer
198 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 ...
1 vote
0 answers
56 views

local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$. Suppose that $K$ is locally ...
2 votes
1 answer
777 views

Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves

I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$. ...
2 votes
0 answers
130 views

A infinity structure on Yoneda Ext group

I am currently trying to control an $A_\infty$-algebra of the form $\mathrm{Ext}_X(F\oplus F'[2n-2],F\oplus F'[2n-2])$ where $X$ is a nice enough scheme and $F,F'$ are sheaves that are NOT locally ...
1 vote
1 answer
927 views

Injectivity of the cohomology map associated to the pullback of line bundles

Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...
2 votes
0 answers
606 views

Čech-Alexander complex in computing (crystalline/prismatic) cohomology

I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology). They seemed to be introduced as a method ...
3 votes
0 answers
168 views

Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology

There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
1 vote
1 answer
1k 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 $\mathcal{L}$ ...
1 vote
0 answers
185 views

Group cohomology of sheaves under closed immersion

Suppose $X$ is a scheme over Spec $\mathbb{Z}$, and $p$ is a non-zero prime in $\mathbb{Z}$. Then we have a closed immersion from the special fibre $i_p: X_p \rightarrow X$. If $\mathscr{F}$ is a ...
6 votes
1 answer
675 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 ...
9 votes
1 answer
676 views

Leray-Hirsch theorem for Dolbeault cohomology

In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this: Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...
2 votes
1 answer
370 views

Very weak Riemann-Roch on curves (by J. Kollar)

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14): 1.13 (Very weak Riemann-Roch on curves)...
6 votes
1 answer
327 views

Naive question on local cohomology

Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims: $$...
3 votes
1 answer
438 views

Does local cohomology commute with pullback?

Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open. For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
8 votes
0 answers
253 views

Global functions on a product of schemes over artinian ring

For a morphism of schemes $f:X\to S$ with $S=\text{Spec}(R)$ affine, let's write $A(X)=H^0(X,\mathcal{O}_X)$. I'm interested in the morphism of $R$-algebras $$ c:A(X)\otimes_R A(Y)\to A(X\times_SY) $$ ...
3 votes
0 answers
182 views

Sheaf cohomology of the complement of a schubert variety

Let $k$ be a field, $d,n \in \mathbb{N}$ and denote by $Gr(d,n)$ the Grassmannian, which parameterizes the $d$-dimensional linear subspaces of $n$-dimensional $k$-vector space, considered as a ...
12 votes
3 answers
2k views

How to compute the cohomology of a local system?

Suppose we have a reasonable topological space $X$ (i.e. a complex algebraic variety or a manifold) whose integral singular cohomology and fundamental group we understand well. Suppose that we are ...
2 votes
1 answer
535 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 vote
1 answer
374 views

Multiplicative structure for sheaf cohomology of flag varieties

Let $F,F'$ be two locally free sheaves on a smooth complex algebraic variety. There is a cup-product $H^i(X, F) \otimes H^j(X,F') \to H^{i+j}(X,F \otimes F')$. In particular if $F$ is the sheaf of ...
8 votes
1 answer
571 views

Can one determine the trace map for a nonsingular projective variety explicitly?

I've never understood how one would actually go about computing a trace map associated with the canonical sheaf on a smooth projective variety, if it's even possible. Hartshorne proves that the ...
5 votes
0 answers
245 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 ...
4 votes
1 answer
242 views

Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$

Let $\pi:X\rightarrow S$ be a smooth scheme over $S$ and let's assume for simplicity that $S=\mathrm{Spec} R$ is affine, Noetherian and regular. Let $\mathcal F$ be a coherent sheaf on $X$ and let's ...
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 ...
4 votes
0 answers
232 views

Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence

Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$. In the beginning he states that there exisits ...
6 votes
1 answer
214 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 $\mathcal{D}(\mathcal{O}\text{-}Mod)$. ...
1 vote
0 answers
164 views

Compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ for a semi-abelian scheme $A$

How can I compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ (where $A$ denotes a semi-abelian scheme over $S$, $\mathbb G_m$ denotes the multiplicative group over $S$ and $\underline{\text{Hom}}$ ...
1 vote
0 answers
111 views

Iitaka dimension of a $\mathbb{Q}$-Cartier Prime divisor

Let $X$ be a normal projective variety and $D$ a prime divisor such that $mD$ is Cartier for some integer $m>0$. Suppose $H^1(X,\mathcal{O}_X)=0$ and $mD|_D\sim 0$. My questions are the following: ...
1 vote
0 answers
165 views

Cohomological criterion for being projectively normal

Let $X$ be a smooth projective variety over some algebraically closed field $K$ and let $\mathcal{L}$ be a line bundle that is generated by global sections. I want to know whether the ring $\sum_{n\in\...
5 votes
2 answers
373 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 ...
1 vote
0 answers
830 views

Cohomology of the pullback of a coherent sheaf (considered as an abelian sheaf)

Let $X$ be an affine Noetherian scheme, $Y$ a separated Noetherian scheme, $f:X\rightarrow Y$ a morphism of schemes inducing a homeomorphism on the underlying topological spaces. Let $F$ be a ...
1 vote
0 answers
75 views

Identities for beta functions and twisted cohomology

This is a question about notation, I apologize if it is too basic. In the paper Cho, Koji; Matsumoto, Keiji, Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I, ...
9 votes
1 answer
508 views

Deducing properness from $H^i(X, \mathcal{F})$ 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 ...
1 vote
0 answers
76 views

Sections of nodal curves

We work over an algebraically closed field. Suppose $X\subset \mathbf{P}^n$ is an integral projective curve and $\pi:X\to Y$ is a linear projection that identifies two distinct points $p,q\in X$ to a ...

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