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Questions tagged [sheaf-theory]

A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

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Torsion Free Quasi Coherent Module

Let $C$ be regular curve. Consider a finite and locally free morphism $f: C \to \mathbb{P}^1$. (the latter mean that $f_* \mathcal{O}_{C}$ is a free $\mathcal{O}_{P^1}$ module) Let $\mathcal{F}$ be ...
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Support of Torsion Sheaf

Let $\mathbb{P}^1$ the projective line (considered as scheme) and $\mathcal{F}$ a quasi corent sheaf of finite type on on it. Denote by $\mathcal{F}_T \subset \mathcal{F}$ it's torsion subsheaf. My ...
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Hartshorne, Exercise III 4.2 (a): A morphism $\mathcal{O}^r \to f_* \mathscr{M}$ that is an iso over the generic point.

I'm having some trouble with Exercise III 4.2 a) in Hartshorne's Algebraic geometry. It is Let $f: X \to Y$ be a finite surjective morphism of integral noetherian schemes. Show that there is a ...
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Sheaf of a Closed Subset

I’ve been given the following definition: Let $(X,\mathcal{O}_X)$ be a ringed space which is locally isomorphic to an affine algebraic variety, and $Y\subseteq X$ be closed. Then for an open $V\...
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Concerning Hartshorne's proof of the Vanishing Theorem of Grothendieck (Hartshorne III 2.7)

At certain point of the proof of the theorem in Hartshorne's book, he mentioned the following in which I don't understand: In Step 3, he said we can reduce our consideration to a sheaf $\mathscr{F}$ ...
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Soft sheaves on $T_1$-space

Suppose we have a topological space $X$ and an open subset $U \subseteq X$ with inclusion $j \colon U \hookrightarrow X$. If we have a sheaf $\mathcal F \in \text{Sh}(X)$, then we know that in ...
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Direct limit of $\mathscr{F}(U)$ is the same as direct limit of $\mathscr{F}(X_f)$, where $P\in U$ and $f\notin P$

This question is from Mumford's The Red Book of Varieties and Schemes (Section I.4). Let $X\subseteq k^n$ be an irreducible algebraic set, $R$ its affine coordinate ring. Since $X$ is irreducible, $I(...
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Universal mapping property for projective schemes and globalising it to projective bundles

Let $Y = \text{Spec}A$ be a noetherian affine scheme. Let $S$ be a graded $A$-algebra which is finitely generated by $S_{1}$ as an $S_{0}$ algebra. In other words, $S$ looks like $$ S = A[x_{0}, x_{1},...
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methods for computing cohomology from data of an exact sequence

Let X be a topological space and $A$ be a ring. Suppose we have an exact sequence of sheaves of $A$-modules on X, $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, suppose we ...
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Is the sheaf of rings $\mathscr{O}$ the sheafification of a presheaf?

This a paragraph from Hartshorne's Algebraic Geometry: Next we will define a sheaf of rings $\mathscr{O}$ on $\text{Spec }A$. For each prime ideal $\mathfrak{p}\subseteq A$, let $A_{\mathfrak{...
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Sheaf Cohomology vs Singular Cohomology in Locally contractable Space

Let $X$ be a space and we denote by $\mathbb{Z}_X$ the sheaf of local constant sections on $X$. We are going to compare the sheaf cohomology $H^i(X,\mathbb{Z}_X)$ with singular cohomology $H^i(X,\...
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Morphism between Invertible Sheaves injective?

Let $X$ be a $k$-scheme and $\mathcal{L},\mathcal{N}$ two invertible sheaves on $X$. Assume that there exist a morphism $\mathcal{L} \to \mathcal{N}$ of $\mathcal{O}_X$-modules. Assume that $X$ has ...
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How to show that $\mathcal{F} \otimes \mathcal{F}^\vee \cong \mathcal{O}_X$

Let $\mathcal{F}$ be a rank $1$ locally free sheaf. If we define $\mathcal{F}^\vee = Hom_{\mathcal{O}_X}(F, \mathcal{O}_X)$, then how would one go about showing that $\mathcal{F} \otimes \mathcal{F}^\...
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If $\mathcal{F}$ and $\mathcal{G}$ are locally free sheaves, then $\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is a locally free sheaf.

If $\mathcal{F}$ and $\mathcal{G}$ are locally free sheaves on $X$ of rank $m$ and $n$ respectively, then $\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is a locally free sheaf of rank $mn$....
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Twisting sheaf of projective space

Let $A$ be a ring, $S=A[x_{0},...,x_{n}]$, and $X=$Proj $(S)=\mathbb{P}_{A}^{n}.$ Hartshorne defines the twisting sheaf $\mathcal{O}_{X}(n)=S(n)^{\thicksim}$. Since $\mathcal{O}_{X}(n)|_{D+(x_{i})}$ ...
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Sections of a quasi-coherent sheaf along the non-vanishing set of a section of a line bundle.

Let $(X,\mathcal O)$ be a quasicompact, quasiseperated scheme, $\mathcal L$ a line bundle on $X$ and $\mathcal F$ any quasi-coherent sheaf on $X$. Let $s \in \Gamma(X,\mathcal L)$ be any global ...
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Closure of category of sheaves under inverse limits.

How do I show that the category of sheaves on a space $X$ taking values in , a category $K$ admitting inverse limits, admits inverse limits? The limit is a presheaf, I tried to show that it is also ...
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Why are (Pre)sheaves more important than Co(pre)sheaves?

I'm learning Sheaf Theory, and this is an issue that's been bothering me. Fix a small category $\mathcal{C}$. A $\mathcal{V}$-valued presheaf on the small category $\mathcal{C}$ is a functor $F:\...
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Transition functions of the dual sheaf

Let $(X, \mathcal{O})$ be a ringed space and $\mathcal{F}$ an $\mathcal O$-module on $X$, which furthermore is assumed to be locally free of some finite rank $n \in \mathbb N$. Then the dual sheaf $\...
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Sections of a locally free sheaf

Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on a scheme $X$. We know that we can associate to it a vector bundle $F$ on $X$ such that $F_x \simeq \mathcal{F}(x)$, where with $\mathcal{F}(x)$ ...
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Inverse Image Sheaf and pullback of a Vector Bundle.

Let $M,N$ be smooth manifolds and $f:N\to M$ a smooth map. Denote by $\mathcal{O}_M,\mathcal{O}_N$ the corresponding sheaves of smooth functions. If we regard $\mathcal{O}_M$ as the sheaf of smooth ...
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Sheaf of Regular Functions and Localisation

I’m trying to prove the following statement: Let $V$ be an affine algebraic set, $\Gamma(V)$ its coordinate ring, and $\Gamma(D(f),\mathcal{O}_V)$ the sheaf of regular functions of $D(f)=\{x\in V\...
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Prove that $V(f_1,\ldots,f_k)=Supp(\mathcal{O}_U/\mathcal{J})$

Let $U\subset \mathbb{C}^n$ be open and $f_1,\ldots,f_k\in \mathcal{O}(U)$, where $\mathcal{O}$ is the sheaf of holomorphic functions in $\mathbb{C}^n$. Show that $V(f_1,\ldots,f_k)=Supp(\mathcal{O}...
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Is $H^0(C,\Omega_X)\cong H^0(C,\Omega_X\otimes\mathcal{O}(-p))$?

If $C$ is a complex curve, then any point is a hypersurface. To a point $p$ in $C$, suppose we have $w(p)=0$ for all $w\in\Omega_X$, then do we have $H^0(C,\Omega_X)\cong H^0(C,\Omega_X\otimes\...
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Equivariant Sheaf

This question concerns the meaning of the data defining a so called equivariant sheaf $F$ on a scheme X. Let denote by $\sigma: G \times_S X \to X$ an action of a group scheme $G$ on $X$ . Then a $O_X$...
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A function being “finite” over a point on non-normal schemes?

I recently came across a remark about Cartier divisors in a textbook on algebraic geometry. I'm not sure how to interpret the remark. I've attached the previous paragraph as well for context. The ...
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Why adjunction appear to preserve stalk?

Question: Let $(f,f^{\flat}) \colon (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ be a morphism of locally ringed space. $f \colon X \to Y$ is a continuous map and $f^{\flat} \colon \mathcal{O}_Y \to f_* ...
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Sheafification of a given presheaf

Let $\mathcal{F}$ be a presheaf on $\mathbb{R}$ such that $\mathcal{F}(U)$ is the abelian group of continuous functions with bounded support on $U$. Then what is the sheafification of $\mathcal{F}$? ...
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Computing a support of a sheaf

Let $Z=\{1/n:n\in\mathbb{Z}-\{0\}\}$. Now define the sheaf $\mathcal{J}_Z$ as $J_Z(U)=\{f:f\text{ is a holomorphic function on U with vanishing on }Z\}. $ Find $\text{Supp}(\mathcal{J}_Z).$ My ...
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Characterization of torsion free sheaves

In "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn a torsion free sheaf is defined as coherent sheaf $E$ on an integral Noetherian scheme $X$ s.t. for every $x\in X$ and every non-...
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Is there a fiber bundle/espace-etale interpretation of sheaves on a Grothendieck site

Whenever I'm doing sheaf things and I have a construction that involves sheafifying, I find it convenient to think "the thing that has the same stalks but sections must be locally trivial sections to ...
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Cohomology theories and sheaf cohomology

Let $X$ be a paracompact Hausdorff topological space, $\mathcal U$ an open covering of $X$ and $\mathcal N(\mathcal U)$ the nerve of the covering (https://en.wikipedia.org/wiki/Nerve_of_a_covering). ...
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A decomposition property of constructible sheaves

I am trying to understant this proof on a decomposition property of constructible sheaves on a Noetherian scheme. But I start to get lost from line 6:"Since F is constructible, there is a finite ...
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A coherent sheaf is a vector bundle over subvariety?

Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset? Thanks in advance.
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Easiest way to show these two functions are regular?

Set $X = \mathbb{A}_K^2 = K^2$, $K$ an algebraically closed field. I am considering $A \, \colon= X - \{(0,0)\}$ viewed as a locally closed subspace of $X$. By definition a locally closed subset is ...
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ext sheaf, homological dimension and locally free sheaves.

Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf in $X$. We define the homological dimension of $\mathcal{F}$, denoted $hd(\mathcal{F})$, to be the least lenght of a locally free ...
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On rank$1$ torsion free sheaves

$\underline {Background}$: Let,$\mathcal F$ be a torsion free sheaf of rank $1$. This means (according to the book by Huybrechts,Lehn (page $11$ definition $1.1.2$)) $$\frac {\alpha _d( \mathcal F)}{...
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Confusion about Exercise II .5.15 in Hartshorne

I'm a bit confused about Exercise II 5.15 in Hartshorne's Algebraic Geometry, especially part (b) and (c) which are (b) Let $X$ be an affine noetherian scheme, $U$ an open subset, and $\mathscr{F}...
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Closed embedding = very ample line bundle

Let $\pi \colon X \to \mathbb{P}^n$ be a closed embedding given via an invertible sheaf $\cal L$ with global sections $s_0, \dots, s_n$. Thus ${\cal L} \cong \pi^* {\cal O}_X$. Why is ${\cal L} \...
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About $O_X$-modules?

Consider $F$ an $O_X$-module is true in general that $Hom_{O_X}(O_X,F) \equiv F(X)$ ? Morover I need to prove that if $F$ is a flasque sheaf then $H^n(X,F)=0$ for $n>0$. I think is beacuse the ...
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$\mathcal{O}_Y \to f_*\mathcal{O}_X $ is an Isomorphism

Let $f: X \to Y$ be a morphism of ringed spaces. I'm looking for criteria for $f, X,Y$ such that the morphism $$\mathcal{O}_Y \to f_*\mathcal{O}_X $$ is an isomorphism. Obviously, a necessary ...
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Unit $ \mathcal{G} \to f_*f^*\mathcal{G}$ or Counit $ f^* f_*\mathcal{F} \to \mathcal{F}$ Isomorphism

Let $f: X \to Y$ be a morphism of ringed spaces. Fix a $\mathcal{O}_Y$-module $\mathcal{G}$ and a $\mathcal{O}_X$-module $\mathcal{F}$. It is well known that the fuctors $f^*, f_*$ are adjunct via ...
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What is the module and sheaf of differentials (actually)?

Throughtout, assume all rings are commutative with identity, and all schemes are separated. If $ A \rightarrow B$ is an $A$-algebra, I am having a lot of trouble understanding precisely what is meant ...
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Morphism of Sheaves of Rings

Let $f: X \to Y$ be a morphism of ringed spaces, $\mathcal{G}$ a $\mathcal{O}_Y$-module, $\mathcal{F}$ a $\mathcal{O}_X$-module. It is well known that the fuctors $f^*, f_*$ are adjunct via the ...
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The open affine subsets of an algebraic variety $X$ form an open base for the topology of $X$

Let $K$ be an algebraically closed field. Define an algebraic variety to be a pair $(X,\mathscr{O}_X$) for a topological space $X$ together with a sheaf $\mathscr{O}_X$ that is a subsheaf of the ...
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Describe the points and the sheaf of functions of some schemes.

I am reading Eisenbud and Harris's The Geometry of Schemes. Exercise I-20 in it is to calculate the points and sheaf of functions for some schemes. $1)$ $X=$Spec $\mathbb C[x]/(x^{2}-x)$. We know ...
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$\mathcal{O}_X\cong \mathcal{O}_{X_{h}}$?

If $X$ is $\mathbb{C}P^n$ as a projective variety, and $X_h$ is the corresponding analytic structure. Then do we have an isomorphism $\mathcal{O}_X\cong \mathcal{O}_{X_{h}}$ for the structure sheaves? ...
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(Pre)sheaf epimorphism admits a section?

Let $f: F\to G$ be an epimorphism of (pre)sheaves of sets on a Grothendieck site. Does it admit a section? By this I mean a morphism $g: G\to F$ with $f\circ g={\rm id}$. I'm particularly interested ...
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difference between graded ring and its twisted global sections

Let $S_{\bullet}$ be a graded ring, generated in degree $1$ with $S_0 = k$ (a field). One can associate to $S_{\bullet}$ the twisted graded ring $$ \Gamma_{\bullet} = \left( \ \Gamma(\mathrm{Proj} S_{\...
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Sheafs with $F(X)=\emptyset$

In the book "A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology" by Jean Gallier, the author states on page 214 that if F is a sheaf on a topological space X and for an open set U, $F(...