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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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(...
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Is one sheafification enough for the module inverse image?

Let $\newcommand{\G}{\mathcal{G}}\newcommand{\O}{\mathcal{O}} f: (X, \O_X) \to (Y, \O_Y)$ be a morphism of locally ringed spaces and $\G$ a sheaf of $\O_Y$ modules. The module inverse image of $\G$ is ...
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Constant Presheaf: An Error on the Wiki Page for Sheaf?

I know there are known subtleties in correctly defining a constant sheaf, and I know there have already been issues discovered on Wikipedia relating to this, but currently in paragraph 4 of Overview ...
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Twisted section corresponding to rational function

I am following Vakil's notes on Foundations of algebraic geometry. We have just introduced twisting sheaves ${\cal O}(n)$ on $\mathbb P^1_k$ by gluing the structure sheaves on standard open subsets ...
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Do stalks determine a sheafification?

Today I was thinking about an explicit example of sheafification, namely the sheafification of the presheaf of bounded continuous functions on $\mathbb{R}$. I am working with sheafification as the ...
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Serre's definition of a sheaf in FAC

In Serre's FAC, on page 9 (http://achinger.impan.pl/fac/fac.pdf), he states that "One verifies immediately that the data (a) and (b) satisfy the axioms (I) and (II)." It's clear to me why the map $f\...
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So many different 'varieties', which one is this? Serre's algebraic variety

Anyone who has ever tried to study algebraic geometry has experienced the phenomenon of being burdened by countless types of varieties (variety, affine variety, projective variety, quasi-affine ...
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Inducing Sheaf of Local Rings to Locally Closed Subspace

In this English translation of Serres FAC [working on p.38], for $X = K^r$ in the Zariski topology, $K$ algebraically closed, we define a locally closed subspace $Y$ as usual: the intersection of ...
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Relative Spec (the structure map)

Given a scheme $S$ and a quasi coherent sheaf $\mathcal{F}$ of $\mathcal{O}_S$ algebras, we want to define a scheme $X = \mathrm{Spec}(\mathcal{F})$ over $S$. To do so, we define it in three stages: ...
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Understanding sheaves on a $2$-element set

I'm working through the Geometry of Schemes and wanted some clarification for an exercise. Exercise I-5 considers a two-element set $X=\{0,1\}$ with the discrete topology and asks the reader to find ...
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Are projective (pre)sheaves summands of free (pre)sheaves?

It is well known that for a ring $R$ any projective $R$-module is the summand of a free $R$-module. Let now $(X,\mathcal{O})$ be a site with a sheaf of rings. Are projective $\mathcal{O}$-premodules ...
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Exact sequence of structure sheaves implies exact sequence of twisted sheaves?

Assume I have an onto morphism $\pi: Y\longrightarrow X$, where $X$ and $Y$ are both projective curves over an algebraically closed field $k$. Also, assume that the following exact sequence of ...
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Restriction section of a sheaf to a closed set?

I am doing an exercise from Hartshorne (II Ex 6.2) on divisors and I have come across an abuse of notation that I am not entire sure how to interpret. Let $X \hookrightarrow \mathbb{P}_{k}^{n}$ be a ...
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The stalk $\mathscr{O}_x$ can be embedded in $\mathscr{F}(X)_x$

Let $X = K^r$ for $K$ an algebraically closed (thus infinite) field. Equip $X$ with the Zariski topology. Let $\mathscr{F}(X)$ be the sheaf of germs of functions on $X$ with values in $K$, so that $$\...
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Serre's Structure Sheaf on Affine Space - Clarification

I am working out of this English translation of Serre's FAC. In Section 31, p.38 we begin to put the structure sheaf on the affine space equipped with Zariski topology. I have a few clarification ...
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Motivation of the sheaf associated to $M$ on Proj $S$

I am wondering why the $\tilde{M}$ is defined like that. To be more precise, when $S=\mathbb{C}[x_0,...,x_n]$. $X:=$Proj $S=\mathbb{P}_\mathbb{C}^n$. $\tilde{M}$ is similar to the sheaf defined on ...
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Definition of the sheaf $GL_n(\mathcal{O}_X)$ of invertible $\mathcal{O}_X-$linear functions

Let $(X, \mathcal{O}_X)$ be a ringed space. Is there such a thing as the sheaf of invertible linear functions $GL_n(\mathcal{O}_X)$? The point is that I cannot see how to define the restriction ...
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The kernel of a morphism of quasi-coherent sheaves on a scheme $(X,\mathcal{O}_X)$ is quasi-coherent.

Let $(X,\mathcal{O}_X)$ be a scheme. I know that an $\mathcal{O}_X$-module $\mathcal{F}$ is quasi-coherent if for each $x \in X$ there exists an open neighborhood $U$ of $x$ and an exact sequence of $\...
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Definition of $\mathcal{O}_X(n)$

Let $S$ be a graded ring, and let $X = Proj S$, we define the sheaf $\mathcal{O}_X(n)$ to be $S(n)^\sim$. Here can you explain what $S(n)$ is?
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Why is this function regular?

In this English translation of Serres FAC, on page 62 he is giving the structure of $\mathbb{P}_{r}(K)$. He sets $t_i$ to be the $i$-th coordinate function on $K^{r+1}$, and defines $$V_i = \{ x \in ...
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Sheaf Axiom for Presheaves on Sites

My question concerns a statement about two equivalent (why ?) characterisations of sheaves on sites introduced in https://en.wikipedia.org/wiki/Grothendieck_topology#Sites_and_sheaves We start with a ...
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Importance of Vanishing Cohomology

As part of my masters project I have been working through Serre's FAC. Below are three closely related results I will be presenting as part of my defense. These results are from n$^{°}$ 52, page 63 ...
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Sheaves on a GIT quotient

As stated in the title, my question regards sheaves on a GIT quotient. Let me fix the notation: $G$ is the group scheme acting on the scheme $X$ and both $X$ and $G$ are $k$-schemes. Searching online ...
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Serre says open covers do not form a set, why? Directed sets and limits.

The following selection is from Serre's FAC (Chapter 1, §3, n°22, page 26). The relation `$\mathfrak{U}$ is finer than $\mathfrak{V}$' (which we denote hencforth by $\mathfrak{U} \prec \mathfrak{...
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Constructing an invertible sheaf from a Cartier divisor?

Let $X$ be a scheme. By definition, a Cartier divisor $D$ is a global section of the sheaf $\mathcal{M}_X^{\times}/\mathcal{O}_X^{\times}$, where $\mathcal{M}_X^{\times}$ is the sheaf of ...
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Characterization of irreducible spaces in terms of sheaves

The following exercise appears in the algebraic geometry textbook by Görtz and Wedhorn (Exercise 2.13 (b)): Let $X$ be a connected topological space and assume that there exists a sheaf $\mathcal{...
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Clarification on a proof that the rank of a locally free sheaf is the same everywhere if $X$ is connected.

I have seen the answer in this previous post. My question is as follows. Given a locally free sheaf $F$ over a connected scheme $X$. Why is it true that if $U$ and $V$ are two open sets in $X$ such ...
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Sheafification of constant real sheaf on smooth manifold and sheaf of smooth functions

I have a question that has been motivated by this one. Since $H^k_{dR}(M)\cong \hat{H}^k(M;\mathbb{R}_M)$ I was wondering if the constant sheaf $\mathbb{R}_M$ was isomorphic to the sheaf of $C^{\...
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The ampleness of canonical sheaves and the proof of “$X \simeq \mathrm{Proj}\left(\bigoplus_k H^0(X, \omega_X^k)\right)$”.

In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the ...
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Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?

Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $\mathbb{G}_m$ and its subgroup $\mu_p$, the $p$-th roots of unity. It is well known that the quotient ...
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Dimension of an Open Covering - Serre's FAC

I am reading Serre's Faisceaux Algébriques Cohérents (Henceforth FAC) and he uses some terminology I have not seen. I have searched around a bit but can't get a clean and clear definition. Question:...
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Where is the finiteness of product used in this proposition from Hartshorne?

See this question: Link I have exactly the same question, but I feel none of the questions explain why the proof fails in the infinite case. I am not looking for a counterexample. I have two related ...
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Pushforward of the structure sheaf on $\mathbb{P_\mathbb{C}^1}$

Today I had the final exam of the lesson Algebraic Geometry. There was a question was asked: Let $X=Y=\mathbb{P_\mathbb{C}^1}$ the homogeneous coordinate $(x_0,x_1)$ and $(y_0,y_1)$, respectively. ...
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Injectivity of locally free sheaf

Let $F$, $G$ be two locally free sheaf on $X$, and let $\phi:F\rightarrow G$ is injective. Then why is that $\phi$ may not be injective on all fibers? Are we still regarding this map as morphism of ...
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Restriction of a sheaf of modules

Let $X$ be a scheme and $Y$ be a closed subscheme. For $\mathcal{F}$ a sheaf of modules on $X$ to be the pushforward of a sheaf of modules on $Y$ via the inclusion $i: Y \rightarrow X$ is necessary ...
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Example of sheaf on $\mathrm{Ring}$ that does not come from $\mathrm{Sch}$.

At the end of Remarque 2.3.6 (p. 221-222) of EGA I, the author says that there are functors in $\mathbf{Fais}|_{\mathbf{Ann}}$ (sheaf on the category of Rings) that are not isomorphic to sheaves that ...