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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Canonical homomorphism $(f_* \mathcal{O}_P(1))^{\otimes n} \to f_*\mathcal{O}_P(n)$ surjective

let $X$ be a Notherian scheme and $P:=\mathbb{P}^m_X$ projective scheme endowed with structure morphism $f: P \to X$. let $\mathcal{O}_P(n)$ the $n$-the Serre twist. in Qing Liu's Algebraic Geometry ...
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The spectrum of a not necessarily quasi-coherent sheaf of Algebras and a related vague question.

See this answer on Mathoverflow and this wikipedia section. These links claim that one can construct $Spec \mathcal{A}$ for any sheaf of algebras over a scheme(and even for any locally ringed space). ...
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Prime ideals and localisation of the global section ring of the structure sheaf

Let $(X, \mathcal O_X)$ be a Noetherian scheme. Let $A=\mathcal O_X(X)$ be the global section ring. For every $x\in X$, let $\mathfrak m_x$ be the unique maximal ideal of the local ring $\mathcal O_{X,...
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Why are these homomorphisms equal?

I am asking this question because it is included in an exercise solution, but it is possible I don't have enough knowledge yet to know why this should be obvious... Let $\varphi:A\rightarrow B$ be a ...
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50 views

Hom sheaf of a torsion sheaf

Let $\mathcal{F}$ be a torsion sheaf in a smooth projective scheme $X$. Is it true that $\mathcal{Hom}(\mathcal{F}, \mathcal{O}_{X}) = 0$? If so, where can I find the demonstration of this result? ...
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Confusion about the definition of a sheaf of rings on Spec A for some ring A in Hartshorne's Algebraic Geometry

On page 70 of Hartshrone's Algebraic Geometry he defined a sheaf $O$ of rings on $\operatorname{Spec} A$ for some ring $A$. In particular, $O(U)$ is the set of functions $s: U \rightarrow \coprod_{p \...
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Chern class of tangent sheaf of quintic threefold in $\mathbb{P}^4$

$\newcommand{\ch}{\mathrm{ch}}$Let $X \subset \mathbb{P}^4$ be a quintic threefold, and the ground field $\mathbb{C}$. I'm following Wikipedia's method of computing the (total) Chern class of $\...
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Is pullback an exact funtor for locally free sheaves?

Let $f: X \longrightarrow Y$ be a morphism of smooth projective schemes. Consider $$0 \longrightarrow \mathcal{F}_{n} \longrightarrow \cdots \longrightarrow \mathcal{F}_{0} \longrightarrow \mathcal{...
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On affine morphism and Picard group

Let $X,Y$ be Noetherian schemes and $f: X \to Y$ is an affine morphism ( $f^{-1}(U)$ is affine for every affine open $U \subseteq Y$ ). Is it true that $H^1 ( X, \mathcal O_X^{\times}) \cong H^1 ( Y,...
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Realizing sheafification via compatible families

Let $P$ be a presheaf on a topological space $X$. The following quotient set seems to be a good candidate to realize the sheafification of $P$. Below $(U_i)\twoheadrightarrow U$ means $(U_i)$ is a ...
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How to see the action of $GL(p)$ on the universal coherent sheaf $\mathcal{U}$ over $X\times Q(\mathcal{O}_X^p/P)$?

In Newstead's book 'Introduction to Moduli Problems and Orbit Spaces' (Chapter 5, Page 110) he says The group $GL(p)$ may be identified with the group of automorphisms of $\mathcal{O}_X^p$; thus $...
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What is the local ring of the sheaf of regular functions with respect a component T

I'm studying the theory of abelian coverings of varieties. Let $\mathcal{O}_X$ be the sheaf of regular functions of a varieties $X$ and let $T$ be a closed subset of $X$. What means the symbol $\...
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More confusion about the definition of smooth morphisms of schemes

Let $f: X \rightarrow Y$ be a finite type morphism of noetherian schemes with $x \in X$ and $f(x) = y$. Then $f$ is smooth at $x$ if, 1) $f$ is flat at $x$; 2) $\Omega_{X/Y}$ is locally free of rank ...
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The definition of sheaf $K^*/\mathcal{O}^*$

The sheaf $K^*/\mathcal{O}^*$ is defined as the sheafification of the cokernel from the inclusion $\mathcal{O}^*\to K^*$. For the pre sheaf $(K^*/\mathcal{O}^*)^{pre}$, $(K^*/\mathcal{O}^*)^{pre}(U):=...
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Constructing projective Calabi-Yau varieties

A nice way of constructing a non-projective Calabi-Yau threefold is to take the total space $$Y:= \mathrm{Tot}(\omega_S) = \mathbf{Spec}(\mathrm{Sym}^\bullet(\omega_S^\vee))$$ of the canonical ...
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Subfunctor of finitely definable elements

Suppose that $\mathcal{C}$ is a category of sets and functions and $F:\mathcal{C}\to Set$ is a presheaf. We can define a subfunctor of $F$, $F^*$, as follows: $F^*(A) := \{a \in F(A)\mid$ for some ...
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Zero section of quasi-coherent bundle

Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define ...
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Definition of $\mathcal{O}(D)(U)$

Let $L(D)$ denote the space of meromorphic functions $f$ on $M$ (complex manifold), s.t. $D+(f)\ge0$. Then we can find a global meromorphic section of $\mathcal{O}(D)$ $s_0$ with $(s_0)=D$. Then we ...
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Help in understanding a sheaf of modules over $\mathcal{R}$

I read the following in "Sheaves on Manifolds by Kashiwara" on page 87 The relevant definition is Let $\mathcal{R}$ be a sheaf of rings on $X$. An $\mathcal{R}$-module $M$ (or a sheaf of modules ...
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Lemma of devissage from Mumford's AGII

In Mumford's & Oda's Algebraic Geometry II, on page 81 the authors give a proof for 6.12: Lemma of devissage. after a carefully reading of the proof, I failed to understand an argument: Theorem ...
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How is the degree defined in this case?

By Daniel Huybrechts, we have: Definition: Let $E$ be a coherent sheaf of dimension $d = \text{dim}X$. The degree of $E$ is defined by: $$\text{deg}(E) = \alpha_{d-1}(E) - \text{rk}(E).\alpha_{d-1}(...
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Scheme $\text{Spec}(S^*\text{Hom}(V,W)^{\vee})$ from “The Geometry of Moduli Spaces of Sheaves”

I have a couple of questions about the notations & their meaning used in "The Geometry of Moduli Spaces of Sheaves" by Huybrechts & Lehn, in Example 2.2.2 (page 38): $V$ is assumed to be a be ...
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Tensor product of quasi-coherent $\mathcal{O}_X$-modules on a ringed space

Let $(X,\mathcal{O}_X)$ be a ringed space. Show that the tensor product $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G}$ of two quasi-coherent $\mathcal{O}_X$-modules $\mathcal{F}, \mathcal{G}$ is ...
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Well-definedness of conormal sheaf

In the definition of the conormal sheaf, we are given a locally closed immersion $X \to Y$, which factors through some closed subscheme $Z$ so that we have $X \to Z \to Y$, where the first map is a ...
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Euler sequence, exact sequence and Blow Up

In $ X = \mathbb{P}^{n}$ we have the Euler sequence: $$0 \longrightarrow \mathcal{O}_{X} \longrightarrow \mathcal{O}_{X}(1)^{\oplus (n+1)} \longrightarrow T_{X} \longrightarrow 0 $$ Let $Y \subset X$ ...
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Does sheaf of smooth functions recover the smooth structure?

Given a smooth manifold $M$, one can easily define its sheaf of smooth functions. I hope to arrive the following statement: Sheaves are just good devices that keep track of extra structures other ...
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Surjective maps of Sheaves

My question is mainly along the lines of another post: Surjectivity on stalks implies surjectivity on sheaves I’m trying to prove that given a map of sheaves $\psi : \mathcal{F} \rightarrow \mathcal{...
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Equality of subsheaves of a sheaf

Given two subsheaves $\mathcal{F}$ and $\mathcal{G}$ of a sheaf $\mathcal{O}$ on a topological space X, prove that $\mathcal{F}_p = \mathcal{G}_p$ for every p $\in$ X $\Rightarrow$ $\mathcal{F}(U) = \...
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Examples of presheaves which are not sheaves

1) Consideremos $\mathbb{C}$ con la topología usual. Definimos el prehaz de las $\textbf{funciones acotadas}$ $\mathcal{F}:\textbf{Top}(X)\to \textbf{Ab}$ de la siguiente manera: $$\mathcal{F}(U)=\{f:...
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Support of a sheaf need not be closed

To prove that the support of a sheaf is not necessarily closed I consider this sheaf: $\mathcal{F}:=\oplus_{p_i \in [0,1)}\mathrm{Sky}_{p_i}\mathbb{Z}$. Then we have that $\mathrm{Supp}(\mathcal{F})=...
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Equivalent definitions of $\mathcal{O}_X$ module of finite type

I have two different definition of $\mathcal{O}_X$-module of finite type. One definition is that given $\mathcal{F}$ an $\mathcal{O}_X$-module we have that $\forall x \in X$ there exists $U$ open ...
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$\mathcal{O}_X$-submodule generated by a sheaf of sets.

Let $\newcommand{\m}{\mathcal}(X, \m{O}_X)$ be a ringed space, $\m{F}$ an $\m{O}_X$-Module and $\m{A}$ a sheaf of sets on $X$ with an injection $\m{A} \hookrightarrow \m{F}$ (so $\m{A}$ is a subsheaf ...
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Blow up and relationship between tangents sheaves

Let $X = \mathbb{P}^{n}$ and $Y \subset X$ a smooth subvariety of $X$. Let us consider the blowup morphism of $X$ along of $Y$, denoted by $\pi : \widetilde{X} \longrightarrow X$ with exceptional ...
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Noetherian $R$-algebra corresponds to a coherent sheaf of rings on $\operatorname{Spec}(R)$

Let $R$ be a ring and $A$ a Noetherian $R$-Algebra. Let $\newcommand{\m}{\mathcal} \m{A} = \tilde{A}$ be the corresponding $\m{O}_X$-Module, where $(X, \m{O}_X) = \operatorname{Spec}(R)$. I would like ...
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Does $\mathsf{Top}$ have interesting Grothendieck topologies, and do they have applications?

In algebraic geometry, the importance of non-trivial Grothendieck topologies is very well-known. One starts out with the Zariski topology on $\mathsf{Sch}$, but concludes that it is 'too coarse' for ...
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Equality of subsheaves on stalks implies equality of subsheaves

My question is in context of Sheaves with the same stalks are not necessarily isomorphic another question posted earlier. Why is it true that given two subsheaves F and F’ of a sheaf G on a ...
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$\mathcal{O}_Y= f_* \mathcal{O}_X$ preserved by base change

Let $f:X→Y$ be a proper + dominant map with connected fibres (i.e. $X_y$ connected topological space for all $y \in Y$) between integral proper schemes over an algebraically closed field $K$ of ...
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Surjectivity on stalks implies surjectivity on sheaves

Let $\phi : F \rightarrow G$ be a map of sheaves. Let $\phi _p : F_p \rightarrow G_p$ be the induced maps of stalks at a point p $\in$ X (where F and G are sheaves over some topological space X) ...
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Conormal and Tangent Sheaves of a Distribuition in $\mathbb{P}^{n}$ when $n = 2$ and $n = 3$

Definition 1): A codimension one distribution of degree $d \geq 0$ in $\mathbb{P}^{n}$ is given by an exact sequence: $$\mathscr{F}: 0 \longrightarrow T_{\mathscr{F}} \longrightarrow T\mathbb{P}^{n}\...
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Vanishing of Sheaf cohomology over (affine ) schemes

Let $QCoh(X)$ be the category of Quasi coherent sheaves on a scheme $ X$ . If $X$ is affine and $\mathcal F \in QCoh(X)$ then it is known that the higher sheaf cohomologies are all zero i.e. $H^j(X,\...
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Degree of a coherent sheaf after a blow up

Let $Y$ be a smooth projective scheme and $X$ a projective subscheme of $Y$. Let us consider the blowup morphism of $Y$ along $X$, denotated by $\pi_{X} : \widetilde{Y} \longrightarrow Y$. Let $\...
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Reduced schemes from schemes

My homework problem is to prove that the embedding of the category of reduced schemes into the category of all schemes has a right adjoint. The most natural thing to do is to set $O'_x(U) : = O_x(U)/\...
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How to show, that the ring of regular functions is a a sheaf. (Hartshorne Chapter one, page 42)

I don't really see, why the ring of regular functions, defined as $\mathcal{O}(U)=\bigcap_{P\in U}R_P$ on Page 42 in the Hartshorne book, is a sheaf. I know, that i need to show: (F1) if $U=\...
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What is a constant morphism of schemes

I am dealing with some basic notions on schemes and I was asking myself what does it mean for a morphism between two schemes $$f:(X,\mathcal{O}_X)\rightarrow(Y,\mathcal{O}_Y)$$ to be constant. The ...
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Cohomology of weakly constructible sheaf on affine line

Let $X = \mathbb{A}^1(\mathbb{C}) \cong \mathbb{C}$ and $F$ be a sheaf on $X$. We call $F$ to be a weakly constructible sheaf (in this particular case) if there exists finite set of points say $S$, ...
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Sheafification of moduli functor of vector bundles.

I´m studying the construction of the moduli space of vector bundles and something came up to my attention. Suppose that we want to study the representability of the functor $$Vec_{X}:\{\text{k-schemes}...
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On Isomorphism of stalk of a line bundle

Let, $X$ be a scheme and $L$ be a line bundle on $X$,then we know that at any point $x \in X$ we have $L_x \cong \mathcal O_x$. At this point my question is the following : Under this isomorphism is ...
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Section's local behaviour of locally constant sheaves

Let $F$ be a locally constant sheaf on $X$ and $U$ is an open subset and $F|U$ is a constant sheaf. Let $x\in U$, now let $s,s'$ be two sections from $F(U)$ s.t. $s(x)=s'(x)$, can we say $s=s'$ in a ...
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Why is the empty set assigned the trivial group in the definition of a pre-sheaf?

I'm currently taking a course on Complex Geometry, where we were introduced to the concept of a pre-sheaf. A pre-sheaf $\mathscr{F}$ of abelian groups on a topological space $X$ is an assignment to ...
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Tensor product of two sheaves on surface

Let, $X$ be a smooth projective surface and $C \subset X$ be a an irreducible curve over an algebraically closed field $\mathbb K$ of characteristic $0$.Let $j:C \to X$ be the inclusion , $A \in Pic(...