Questions tagged [sheaf-cohomology]

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. (Def: http://en.m.wikipedia.org/wiki/Sheaf_cohomology)

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First higher direct image of a constant sheaf

Let $X$, $F$ be manifolds and $f:X\times F\to X$ be a natural projection. Then is the first higher direct image of a constant sheaf $R^1f_*(\mathbb{Z}_{X\times F})$ also a constant sheaf on $X$ with ...
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Why is $H^0(X_n, \mathcal{O}_{X_n})$ a local artin ring.

Let $V$ be regular, proper and of dimension 2 over $S = $ spec $R$, for $R$ a complete discrete valuation ring with uniformizer $t$, maximal ideal $\mathfrak{m}$, and algebraically closed residue ...
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How do one show that the quotient space is a projective manifold?

I want to prove the following statement. Let $\Omega$ be a bounded domain, and $\Gamma \subset \text{Aut}(\Omega)$ be the subgroup acting totally discontinuously on $\Omega$ without fixed points such ...
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How to caculate $H^2(X,\mathcal{O^*})$

Let $X$ be a complex torus of dimension 2. I wonder that if there is a torsion free normal subgroup of $H^2(X,\mathcal{O^*})$. I intended to use Selberg's Lemma to deduce, but it requires that $H^2(X,\...
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Grothendieck trace formula over $\mathbb{F}_{p^{n}}[[t]]$?

Let $n\in\mathbb{N}$, $k=\mathbb{F}_{p^{n}}$, $S=\operatorname{Spec}(k[t])$, $F$ the Frobenius automorphism on $S$ and $\mathcal{E}$ a sheaf of modules over $X$ such that $\mathcal{E}_{0}$ is of ...
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Global functions on arithmetic varieties

Let $f:X\to\operatorname{Spec} O_K$ be an arithmetic variety where $K$ is a number field and $O_K$ is its rings of integers. We assume that $X$ is integral, projective, regular and $f$ is flat. If ...
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On a particular exact sequence in cohomology

The setup for my question is as follows (from page 9 of Deschamps' expository notes on the Artin-Winters proof of semi-stable reduction here). We want to prove that for $X$ the special fiber of a ...
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cohomology of projective space over non-noetherian ring

The cohomology of projective spaces over a noetherian ring $A$ is computed in e.g. Hartshorne Chapter III.5. In particular, we know that $H^i(\mathbb{P}_A^n,\mathcal{O}_{\mathbb{P}})$ are finitely ...
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Pushforwards from a projective bundle corresponding to a coherent sheaf

Consider a normal Noetherian scheme $X$. In the case I am most interested in, it's even Cohen-Macaulay. For a locally free coherent sheaf $E$ its projective bundle $\mathbb{P}(E)$ is usually defined ...
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Is the Lefschetz Hyperplane Theorem true for analytic spaces?

The LHPT is usually stated as something like the following. Given an $n$-dimensional quasiprojective variety $X\subset \mathbb{C}\mathbb{P}^n$, and a generic hyperplane section Y, the natural map on ...
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Higher direct image of a locally constant sheaf

I am trying (struggling) with understanding the local system structure of the higher direct image of a locally constant sheaf. Say I have a locally trivial fibration $f: X\longrightarrow B$, with ...
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Representing sheaf/Cech cohomology as de rham cohomology for a non-constant sheaf

I'm having trouble understanding the relation between these two cohomology theories. I know that for a constant sheaf, the sheaf cohomology is essentially singular cohomology and by the de Rham ...
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If $\mathscr{C}$ is the sheaf continuous real-valued functions on $X$, then when is $H^1(X,\mathscr{C})=0$?

This is related to Hartshorne's Exercise III.2.7 here one shows that $H^1(S^1,\mathscr{R})=0$. Here, $\mathscr{R}$ is the sheaf of germs of continuous real-valued functions. The method of proof is to ...
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Characteristic class of principal bundle

I've been told that principal $G$-bundles $E \to M$ are classified by specifying a characteristic class $c(E) \in H^2(M,\pi_1(G)) ≈ \pi_1(G)$ I have a few questions Given a bundle $E$ how do we get a ...
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$H^n(X, S)$ and $\check {H^n}(X,S)$ when $X\setminus S$ is an $n$-manifold (Hawaiian earring related)

I am dealing with a compact topological space $X$, such that $X\setminus S $ is an $n$-manifold, here $S\subset X$ is a compact subset of $X$ ($S$ stands for singular locus). An example could be the ...
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Why is this de Rham 1-cochain a cocycle on an elliptic curve?

I am trying to understand a proof about the de Rham cohomology of an elliptic curve in Kedlaya's notes, p7 Example 1.5. The motivation for this question has overlap with my previous question Let $X = \...
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3 votes
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If $M$ is a compact Riemann surface, then $H^1(M,\mathbb R)\cong H^1(M,\mathcal O)$

I am wondering why we have the isomorphism stated in the title. Concretely, we have the following exact sequences of sheaves: $$0\rightarrow\mathbb R\rightarrow\mathcal O\rightarrow\mathcal O/\mathbb ...
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Proper push-forward of sheaves does not necessarily have right adjoint

Let $f:X\to Y$ be a continuous map of topological spaces. Denote $\operatorname{Sh}(X)$ to be the category of sheaves of abelian groups (or $k$-modules where $k$ is a field) on $X$. We can define the ...
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Conormal exact sequence [closed]

Let $X$ be a smooth variety and $Y\subset X$ a smooth subvariety with ideal sheaf $\mathcal{I}_{Y/X}$. For $n\geq 2$ is there an analogue of the exact sequence $$0\rightarrow \mathcal{I}_{Y/X}/\...
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Cohomology of sheaf restriction

I want to know if the follow is true and if this is true, prove it. Let $X$ be a variety and let $Z\subset X$ be a subvariety of $X$. If $\mathcal F$ is a coherent sheaf then $H^i(Z,\mathcal F|_Z)=H^...
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Sheaf homology defined in terms of Tor

By the general philosophy of cohomology, cohomology is essentially derived $\operatorname{Hom}$ (i.e. $\operatorname{Ext}$), and homology should be derived tensor product (i.e. $\operatorname{Tor}$). ...
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Direct product totalization in the definition of hypercohomology

Let $X$ be a topological space, and $\mathcal{F}^\bullet$ be a cochain complex of sheaves on $X$. The hypercohomology $\mathbb{H}^i(X,\,\mathcal{F}^\bullet)$ is defined as $$\mathbb{H}^i(X,\,\mathcal{...
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A mistake in Grothendieck's Tôhoku paper? Theorem 5.2.1

I was reading the ``Sur quelques points d'alegbre homologique'' English translation when I came across the spectral sequences for equivariant cohomology shown in Theorem 5.2.1 (in the original French ...
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Canonical mapping from locally finite cozero cover inducing projection map on the level of Cech Cohomology

My question comes from Nagami's "Dimension Theory". There's a bit of windup. Let $X$ be a normal Hausdorff space, and $A\subseteq X$ a closed subset. For an open cover $\mathcal{U}$ of $X$ ...
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Computation using projection formula for $\ell$-adic cohomology

I'm not familiar with derived tensor product, hence it may be a stupid question. Let $f: X\to Y$ be a separated morphism of finite type between schemes. Let $A$ be a torsion ring, and $D^-(X,A)$ (resp....
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Cohomology of pushforward of a line bundle

Let $X$ be a smooth projective surface and $C$ be a smooth curve on $X$ ( let $j$ denote the embedding of $C$ inside $X$). Let $L$ be a line bundle on $C$. Assume that it's known that $h^0( j_*(L) \...
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For an oriented sphere bundle, is the transgression of the fibre wise fundamental class the obvious yoneda extension?

Let $E\xrightarrow{\pi} B$ be an oriented sphere bundle of dimension $n$. Then transgression of a chosen fibre generator yields a class in $H^{n+1}(B,\mathbb{Z})$. Such a class also leads to an ...
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Simple proof that $H^1(X,\mathcal M_X^\ast)=0$ for $X$ a compact Riemann surface.

Let $X$ be a compact Riemann surface, $\mathcal M_X$ the sheaf of meromorphic functions, $\mathcal M_X^\ast$ the sheaf of invertible meromorphic function (a sheaf in abelian groups for the ...
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A question regarding the homotopy method in "Sheaves on Manifolds" by Kashiwara and Schapira

I am reading the book "Sheaves on Manifolds" by Kashiwara and Schapira (1990 version). (I can only find a one-page errata in https://webusers.imj-prg.fr/~pierre.schapira/BooksMono/Errata.pdf....
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Finiteness of cohomology of coherent sheaf for proper morphisms

It has been stated and proved that the Cohomology of a coherent sheaf $F$ on a closed projective subscheme X of $\mathbb P^n_A$ where A is Noetherian is finite dimensional A-module. If $f: X \to Spec ...
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Vanishing first local cohomology group

I have been reading about local cohomology from Hartshorne's notes on the same and I have the following question. Let $X$ be a topological space, and let $Z\subset X$ be closed. Then we have a long ...
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Creating a sheaf out of cohomology

I have been learning about the cohomology of sheaves, and I have the following questions. Suppose we have a sheaf $\mathscr{F}\in Sh_X$ where $X$ is some topological space. Now, for any open $U\subset ...
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Beauville Fact III.22

I had the following doubt regarding the proof of Fact III.22 in Beauville's book on algebraic surfaces: What I don't get is the second application of the projection formula. How are we applying the ...
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why 0-th sheaf cohomology is the global section

Let $\mathcal{G}^*$ be some injective resolution for sheaf $\mathcal{F}$,hence we have the exact sequence $0\to \mathcal{F} \to \mathcal{G}^*$. Then we have the chain complex after taking global ...
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Comparison morphism between Čech and sheaf cohomology

Let $X$ a topological space (ie a scheme) and $E$ be a sheaf of abelian groups on $X$. Now we can associate to $X and E$ sheaf cohomology groups $H^i(X,E)$. For an explicit construction choose an ...
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Why $\operatorname{Pic}^0(\mathbb{P}^n)=0$?

I try to prove that $\operatorname{Pic}^0(\mathbb{P}^n)=0$. We have the exponential exact sequence $$0 \to \mathbb{Z} \to \mathbb{C} \to \mathbb{C}^* \to 0$$ and the exponential exact sequence of ...
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Surjectivity of induced map on cohomology for projective scheme

Note: Below, cohomology means Čech cohomology. (Liu, exercise 5.3.13) Let $X,S$ be locally Noetherian schemes, $f:X\to S$ a projective morphism and $\mathcal{F,G}$ quasicoherent sheaves on $X$ with a ...
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global section of inverse image and etale cohomology

Let $f: X\to Y$ be a morphism of schemes. If it is needed, we can assume that $X,Y$ are quasi-compact and quasi-separated, and $f$ is affine. Consider the two small etale sites $X_{et}$ and $Y_{et}$. ...
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Show Cech cohomology and alternating Cech cohomology have the same cohomology groups without axiom of choice

Let $\mathcal{F}$ be a sheaf of abelian groups over a topological space $X$ with an open covering $\mathcal{U}=\{U_i\}_{i\in I}$. The Čech complex of $\mathcal{F}$ over $\mathcal{U}$ is denoted as $\...
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Top Dolbeault cohomology group vs top De Rham cohomology group of a compact complex manifold

I cannot reconcile two simple facts for a compact complex manifold $X$ with $dim_{\mathbb{C}}=n$. Let $A^{p,q}(X)$ be differential forms of degree $(p,q)$ on $X$. One the one hand, $$H^{2n}_{DR} (X)= ...
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How does the Frobenius really act on Weil sheaves in $\ell$-adic cohomology?

Let $X_0$ be a connected scheme defined over $\mathbb F_{p}$ and let $X$ be the product $X_0 \times_{\mathbb {F}_p} \overline{\mathbb F_p}$, as usual, with the natural map $\pi:X \to X_0$. Given an ...
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Equivalent forms of MV axiom in definition of algebraic motives

Let $F$ be a sheaf on a smooth scheme $X$ with values in $C(\mathcal{A})$, $\mathcal{A}$ an abelian category. In a course on algebraic motives I encountered a statement which after restriction from ...
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Can one recover the set of connected components through the global sections of a locally constant sheaf?

Let $X$ be a noetherian scheme over a field, or any other topological space such that the following makes sense. Let $\mathcal F$ be a locally constant sheaf on $X$ (for the Zariski topology, or on ...
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Cohomological dimension of truncated algebraic De Rham complex.

Is true that the cohomological dimension of the (fine) truncated algebraic De Rham complex $\tau^{\leq q}\Omega^{\bullet}$ is at most $2q$? i.e. its sheaf cohomology vanishes above $2q$. If necessary ...
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Hypercohomology of a direct image of a complex of sheaves

$\newcommand{\hy}{\textrm{-}}$ Let $\mathcal{R}_X$ be a sheaf of rings on a smooth manifold $X$, let $\mathcal{R}_X\hy \mathsf{Mod}$ be the abelian category of $\mathcal{R}_X$-modules and let $D(X)=D(\...
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Doubts about a short exact sequence of sheaves

I am attending a course on Complex Manifolds and we are dealing with sheaves theory. Our professor made a statement which is not really clear to me, so I am hoping for some clarification here. Given ...
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Hartshorne problem III.4.4

Again I'm stuck on a problem in Hartshorne! In this problem we are supposed to show that if we take the limit of all coverings in on a topological space then the Cech cohomology agrees with derived ...
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7 votes
1 answer
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An example that Čech cohomology is not equal to derived cohomology with on an affine scheme with Zariski topology.

There is an example in 3.1.10 $\mathbb{A}^1$-homotopy theory of schemes demonstrating that the Čech cohomology on an affine scheme with Zariski topology can be different from the derived cohomology. ...
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Are stalks of q-singular cochain equal to 0?($q>0$)

Let $\mathcal{S}^{q}$ be the presheaf of singular q-cochains on a topological space $X$, that is the functor $\mathcal{S}^{q} \colon \mathcal{O}(X)^{op} \to \mathbb{Z}$-$\mathsf{Mod}$ which is given ...
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Proof of Birger Iversen "cohomology of sheaves"Scolium 5.2

In Birger's cohomology of sheaves. In page 101, Scolium 5.2,I don't know how to fill in the dotted arrow to make the square homotopy commutative. Is there anyone who have read this? the link is here. ...
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