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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Is the sheafified Cech complex for an étale cover a resolution?

Suppose I have an étale cover of a scheme $X$ by etale $X$-schemes $U_{i}$, let $U:=\coprod_{i=1}^{n} U_{i}$ and write $f:U\rightarrow X$ for the induced map. Then for every quasicoherent sheaf $\...
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Vanishing of cohomology of affine scheme

In EGA I 5.1, more specifically the proof of 5.1.9, which states that $X$ is affine iff the closed subscheme defined by a quasi-coherent sheaf of ideals $\mathscr{I}$ such that $\mathscr{I}^n = 0$ for ...
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Isomorphism of cohomology related to Kunneth formula

Let $S$ be a smooth complex (rational) algebraic surface, and $\mathcal{F}$ be a quasi-coherent sheaf such that $H^2(S, \mathcal{F}) = 0$. Then, by Kunneth formula, we have $H^2(S \times S, \mathcal{...
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First sheaf cohomology of closed 1-forms on a homogeneous space

Given a smooth quasi-projective variety $X$ over $\mathbb C$, is there a good way to compute the first sheaf cohomologiy $H^1(X,\Omega_{X,cl}^1)$ of closed 1-forms $\Omega_{X,cl}^1$? What I'm mostly ...
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Ideal sheaf of diagonal embedding of projective rational surface $S$ in $S \times S$.

Let $S$ be a smooth (projective) complex rational surface, and $\Delta$ is the diagonal embedding $S \rightarrow S \times S$. Consider the ideal sheaf $I_{\Delta}$ of $\Delta$ on $S \times S$. My ...
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Cohomology of a monad.

Definition. A monad over a projective variety $X$ is a complex $$M : 0 \longrightarrow \mathcal{A} \stackrel{f} {\longrightarrow} \mathcal{B} \stackrel{g} {\longrightarrow} \mathcal{C} \longrightarrow ...
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The quotient of the constant sheaf by the structure sheaf is flasque.

Let $X$ be an integral scheme of finite type over $k$ with $\dim(X)=1$. Let $\mathcal{K}_{X}$ be the constant sheaf on $X$ with value $K(X)$, where $K(X)$ is the function field of $X$. I want to ...
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How does the leading symbol of an elliptic differential operator define an element of $K(T^{*}X)$?

I am trying to understand the analytic index of an elliptic operator as a map $K(T^{*}X) \to \mathbf{Z}$. My primary source is Differential Topology and Quantum Field Theory by Charles Nash and the ...
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Question about cohomology of differential sheaf.

In $\mathbb{P}^1_{\mathbb{C}}$, given affine covers $U=(v\neq0),V=(u\neq0)$, we write element $(a,b)$ of $C^0(\{U,V\},\Omega^1)=\{(a,b): a\in \Omega^1(U), b\in \Omega^1(V)\}$ as $a=\sum^{\infty}_{n=...
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Connecting map in de Rham cohomology

Let $A$ be an smooth manifold,$\quad\partial A =B\times C, \quad dimB=\nu$. $B$ and $C$ are smooth manifold $$f: \Omega^{*}(A)\longrightarrow\Omega^{*}(\partial A)\longrightarrow\Omega^{*-\nu}(C)$$ ...
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Euler-Poincare characteristic of $L \otimes Sym^k(\Omega^1_S)$

Let $S$ be a smooth complex rational surface, and $L$ be an ample line bundle on $S$. I want to compute the Euler-Poincare characteristic of $L \otimes Sym^k(\Omega^1_S)$, i.e., $$\chi(S, L \otimes ...
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Čech cocycles and monodromy

It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
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de Rham cohomology—the connecting map [closed]

How does the connecting map act in a long sequence exact of de Rham cohomology of the pair $(M,\partial M)$?
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If $X$ is a K3 surface and $X\to C$ is an elliptic fibration, then $C\simeq \Bbb{P}^1$

Sometime ago I heard someone say in a talk that if an algebraic surface $X$ is K3 and has an elliptic fibration $\pi:X\to C$ (where $C$ is a curve), then $C$ must be rational. The speaker just said ...
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Understanding etale cohomology versus ordinary sheaves

I am a physicist trying to understand etale cohomology from Shafaverich, and I would like to check a misunderstanding, undoubtedly. When defining etale cohomology, it seems it is sheaf cohomology in ...
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Flat cohomology as a subspace of local Galois cohomology

Consider an elliptic curve $E$ over a local field $k$ with good reduction and with Neron model $\mathcal{E}/\mathcal{O}_k.$ The Galois group $G_k$ acts on $E[m]$, so we have a cohomology group $H = H^...
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Long exact sequence of sheaf cohomology from the normal bundle of $\mathbb{P}^1$ in $\mathbb{P}^2$

Let $i: \mathbb{P}^1 \to \mathbb{P}^2$ denote the inclusion $[x_0,x_1] \mapsto [x_0,x_1,0]$. In other words, we identify $\mathbb{P}^1$ as the subvariety defined by ${x_2 = 0}$ in $\mathbb{P}^2$. The ...
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Why does the vanishing of this cohomology say that?

Consider the following proposition found in the article: Stable Vector Bundles of Rank 2 on $\mathbb{P}^{3}$ - Hartshorne. Proposition 3.1. Let $\mathscr{E}$ be a rank 2 bundle on $\mathbb{P}^{...
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Soft question: are presheaves worth to study?

I'm taking a second course in algebraic geometry and getting acquainted with the notion of sheaf and presheaf. In the meanwhile, there is this topological space that I'm interested in (a personal ...
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Examples for failure of cohomology presheaf to be a sheaf

Let $f:X\to Y$ be a continuous map and $F$ a sheaf on $X$. Consider the cohomology presheaf on $Y$ defined by $V\mapsto \mathrm H(f^{-1}V;F)$. I am looking for some instructive examples to understand ...
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Map comparing Alexander-Spanier and Čech cohomology

I'm trying to follow the argument in Bredon's Sheaf Theory (page 29) supposedly constructing an isomorphism between the Alexander-Spanier cohomology $H_{AS}^n (X; G)$ and Čech cohomology $\check H^n(X;...
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Explicitly Understanding a Case of Dolbeault's Theorem.

If $\Omega^p$ is the sheaf of holomorphic $p$-forms on a complex manifold $M$, then Dolbeault's theorem states $$ H^{p,q}(M) \cong H^q(M,\Omega^p) $$ Setting $p=0,q=1$, we get $$ H^{0,1}(M) \cong H^1(...
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A help in exact sequence and Castelnuovo-Mumford regularity

Let $F$, $F'$ and $F''$ be coherent sheaves on $\mathbb{P}^{3}$. Consider the following exact sequence $$0 \longrightarrow F' \longrightarrow F \longrightarrow F'' \longrightarrow 0 $$ Suppose ...
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Mayer-Vietoris Sequence for Cohomology with Supports

I am working on problem III.2.4 in Hartshorne, and I have been quite stuck in showing the existence of a Mayer-Vietoris sequence for cohomology with supports. To be more precise, I have $Y_1,Y_2\...
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Sheafification in the derived category.

Let ${\cal F}$ be a presheaf on a scheme $X$ which associates each abelian group ${\cal F}(U) \in {\mathrm{Ab}}$ for each open $U \hookrightarrow X$. Then we can always associate its associated sheaf $...
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Algebraic geometry reference

I believe similar reference requests have been asked previously but I think mine is somewhat specific. I am interested in learning algebraic geometry. My experience so far has been with complex ...
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Does Direct limit/union of subspaces commute with sheaf cohomology

Let $X$ be a topological space and $\mathcal{F}$ an abelian sheaf on $X$. Furthermore let $0=X_0 \subset X_1 \subset X_2 \subset \ldots$ be an increasing sequence of subspaces of $X$ such that $X=\...
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Is the sheaf cohomology only a sheaf for flabby sheaves?

Let $X$ be a topological space and let $\mathcal{F}$ be an abelian sheaf on $X$. Choose an arbitary open cover $\{U_i\}_i$ of $X$. Then the Mayer-Vietoris long exact sequence is $$0\rightarrow \...
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Simplicial approach to Cech cohomology?

Is there an entirely simplicial approach to constructing Cech cohomology? Maybe we can define what a "sheaf on a simplicial set" would mean and show that for any open cover $\{U_i\}$ and sheaf on a ...
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Given principal bundle $P_O(E)\to E$ with fibration $O_n\to P_O(E)\to X$, there is an exact sequence…

Consider $E$ a rank $n$ real vector bundle over $X$. Let $P_O(E)$ be the orthonormal frames of $E$. This is a principal bundle $P_O(E)\to X$ with $O(n)$ action. Thus there is $i:O(n)\to P_O(E),\pi:P_O(...
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Resolution of the constant sheaf $\mathbb C$.

The differential-graded algebra of complex-valued smooth differential operators $\mathcal{A}_{X, \mathbb C}^{\bullet}$ on an $n$-dimensional complex manifold $X$ (real dimension $=2n$) is acyclic ...
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different definitions of fine sheaves / partitions of unity

I have the feeling that in the literature there are two different definitions of what a partition of unity for a sheaf is supposed to be. A partition of unity for a sheaf of abelian groups $\...
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Theorem on normal singularities from Badescu's Algebraic Surfaces

Reading Lucian Badescu's Algebraic Surfaces I have encountered a proof I can't understand. That's Theorem 3.28 (M. Artin) at pages 41/42: Theorem 3.28 (M. Artin). Let $(Y, y)$ be a two-dimensional ...
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Cohomology of the sheaf $\mathcal{M}^*$ of invertible meromorphic functions on a Riemann surface

Let $X$ be a Riemann surface and denote by $\mathcal{M}^*$ the sheaf of invertible (i.e. not constantly zero on any connected component) meromorphic functions. I have seen claims that $H^1(X, \...
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A complex to compute Čech-Cohomology

Let $\newcommand{\F}{\mathcal{F}} \F$ be a sheaf of abelian groups on a paracompact (Hausdorff) space $X$. The Čech-cohomology of $\F$ is defined as $$ \check H(X, \F) := \varinjlim_{\mathcal{U}} \...
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Blow up and Castelnuovo-Mumford Regularity

Let $\pi : Z = \widetilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowing up of $\mathbb{P}^{3}$ along an irreducible non-degenerate smooth curve $\mathcal{C}$ of degree $d$. According ...
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Doubt over a proof about higher direct image functors in Hartshorne

For reference, this is Chapter III Proposition 8.5 in Hartshorne. The claim is this Let $X$ be a noetherian scheme and let $f: X \rightarrow Y$ be a morphism of $X$ to an affine scheme $Y = \text{...
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IC complex on $\Bbb C = \Bbb C^* \sqcup \{0\}$

Let $X = \Bbb C$ stratified as $\Bbb C^* \sqcup \{0\}$. Let $\mathscr L$ be a local system on $\Bbb C^*$. How to describe $IC(U, \mathscr L)$ ? By describe I mean : compute its stalk at $0$, ...
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Cohomology beyond Hartshorne's Algebraic Geometry

I have worked through most of Hartshorne's Algebraic Geometry text, and I'd like a recommendation for a book or set of lecture notes, which go beyond in particular chapter 3 on cohomology. To be ...
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Čech cohomology and sheaf cohomology

Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X.$ It is clear to me that $\check{H^{0}}(X,\mathcal{F})=H^{0}(X,\mathcal{F}).$ It is also true that $\check{H^{1}}...
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Infinitesimal neighborhoods of an affine bundle

Let $S$ be an $\mathbb{A}^1$-bundle over $\mathbb{P}^1_{\mathbb{C}}$. We denote by $\Delta$ the diagonal of $S \times S$ (so, $\Delta \simeq S$). Let us consider the $k$-th infinitesimal neighborhoods ...
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Neron Severi group: two descriptions [duplicate]

I have a question on a comment from Daniel Hyubrechts' Complex Geometry Complex Geometry on pages 133/134. Let $X$ be a compact Kähler manifold. Consider the exponential sequence on cohomology $$ .....
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Sheaf cohomology and Cech cohomology on paracompact spaces

I read here that on paracompact (Hausdorff) spaces Cech and sheaf cohomology agree, with a reference to Godement's book Topologie algébrique et théorie des faisceaux, thm. 5.10.1. I have been trying ...
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Confused of the direct limit and the quotient of a disjoint union

I am a beginner in sheaf cohomology (especially on Riemann surface). Before clarifying my confusion, let us start at the direct limit, it can be obtained as a quotient of a disjoint union:$$\...
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Ext groups of sheaves and sheaf cohomology in the context of duality theorems

I am currently trying to understand Serre duality from a more formal point of view, with the goal of eventually looking at Grothendieck duality. I'm following Hartshorne III.6-7. For our sake, let's ...
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Why does $(L, L') = 0$ imply $H^0(L) = 0$, if $L'$ is ample, and the variety is a K3 surface?

I tried reading the proof that the intersection pairing on K3 surfaces is non-degenerate, in Huybrecht's Lectures on K3 Surfaces. Let $(-,-)$ denote the intersection pairing of invertible sheaves on ...
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For what classes of schemes can we compute cohomology via standard methods?

I am trying to really get comfortable with sheaf cohomology by actually sitting down and computing a whole heap of examples. But I am wondering what schemes can we actually reasonably compute the ...
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Compute the cohomology of projective schemes (clarifying an old answer)

I want to clarify an answer in this question. I realise there is a comment function there but I thought since the answer is five years old it might be hopeless asking there. Let me know if there are ...
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Hilbert's 90 Theorem proof clarification (Milne's Étale Cohomology)

So I'm reading Milne's Étale Cohomology and stuck in trouble to understand the proof of Proposition 4.9 Hilbert's Theorem 90), from §4; page 124: The canonical maps $$ H^1(X_{Zar}, \mathcal{O}_X^*) \...
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Trying to understand why Cech cohomology computes derived functor cohomology

I am trying to get a better understanding of what Cech cohomology actually is. In particular, I want to see why it computes derived functor cohomology. I am familiar with the orthodox proof of this ...

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