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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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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59 views

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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53 views

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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28 views

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 ...
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Proof in EGA I Chapter 0, point 3.2.2

I am reading the proof of a necessary and sufficient condition for a presheaf over a base of a topology to be a sheaf, from Élements de Géometrie Algébrique I, chapter 0, point 3.2.2. There is a line ...
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On torsion sheaf of a coherent sheaf of $\dim X$

$\underline {Background}$:Let,$E$ be a coherent sheaf on a Noetherian,integral scheme $X$ and $\dim E$=$\dim X$. Then we have the unique torsion filtration of that coherent sheaf as $0\subset T_0(E)...
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How to prove $\text{Sh}_G(X)\simeq \text{Sh}(G\backslash X)$ when $X$ is a free $G$-space?

Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which act on $X$ continuously from the left....
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67 views

Are sections of an injective sheaf of abelian groups themselves injective abelian groups?

Let $X$ be a topological space, $\mathcal I$ some injective sheaf of abelian groups on $X$ and let $U \subset X$ be open. Is $\mathcal I(U)$ an injective abelian group? If not what requirements would ...
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On action of sheaf of symmetric algebra

$\underline {Background}$:Let,$X$ be a projective scheme over a field $K$. Let, $\phi:\mathcal F\to\mathcal F\otimes\mathcal L$ be a morphism of $\mathcal O_X$ modules for some vector bundle $\...
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315 views

Trace of a morphism of sheaves

$\underline {Background}$:Suppose $X$ be a scheme and $\mathcal F$ and $\mathcal G$ are two sheaves of $\mathcal O_X$ modules. Also assume that $\exists ${U}$ $ a cover of $X$ by open sets such that ...
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Why is the exterior differential independent of the chosen basis?

Let M be as smooth manifold, W a k-dimensional vector space with basis $(e_1,...,e_k)$ and $\Omega^r(M,W)$ be the differential r-form on M with values in W. Let $\omega \in \Omega^r$ and $v_1,..., v_r ...
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63 views

On isomorphism of sheaves

$\underline {Background}$: Suppose $X$ is a scheme and $\mathcal F$ is a sheaf of $\mathcal O_X$ module. Suppose we have, $\mathcal {F}\oplus \mathcal {O}_X \cong \oplus_r \mathcal {O}_X$ as sheaves ...
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Group Action on a Scheme

Let $X$ be scheme and $G \subset Aut(X)$ be a subgroup of automorphism group of $X$. By definition $G$ acts espectially on local sections $\mathcal{O}_X(U)$ for open $U$ and one can therefore define ...
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Galois Action on Coherent Sheaves Exact Functor

Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined ...
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42 views

A non-zero sheaf that vanishes on an open covering

Let $X$ be a topological space, $\mathcal F$ a sheaf of abelian groups on $X$ and $X=\bigcup_iU_i$ an open covering such that $\mathcal F(U_i)=0$ for all $i$. Does this imply $\mathcal F=0$? I think ...
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Sheaf of sets on a two point discrete space?

Let $X=\{0,1\}$ be equipped with the discrete topology. Let us consider $\mathcal{F}$ a sheaf of sets on $X$. Then $\mathcal{F}(\emptyset)=*$, any singleton set. $\mathcal{F}(\{0\})=A$ and $\mathcal{...
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Stalks of the pushforward sheaf.

I know that in general, the stalks of the pushforward of a sheaf need not be the same as the original stalk. That is, $(f_{*}\mathcal{F})_{f(p)}=\mathcal{F}_p$ is not true in general. But when $X \...
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Transfer modules for closed embeddings

Given a map $f:X\to Y$ of smooth complex manifolds, the transfer module $\mathcal D_{X\to Y}$ is the left $\mathcal D_X$-module $\mathcal O_X\otimes_{f^{-1} \mathcal O_Y} f^{-1} \mathcal D_Y$. I'm ...
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60 views

Support of a quasi-coherent sheaf on an affine scheme

Let $A$ be a commutative ring, $X = \text{Spec }A$, and let $M$ be a $A$-module. The $\mathcal{F} = \tilde M$ is a sheaf on $X$. Exercise II.5.6 in Hartshorne's Algebraic Geometry states that if $A$ ...
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What is a smooth function on the closure of an open set?

Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{\infty}(\bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $\bar{U}$ which are the restrictions ...
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102 views

Morphisms between torsion sheaves

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $Y,Z\subset X$ be closed subvarieties of $X$. Denote the embeddings of $Y$ and $Z$ into $X$ by $i_Y$ and $i_Z$ respectively. What is the ...
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45 views

Relation between pullback and fiber product.

Consider the following cartesian diagram of schemes: $$\begin{array} AX^{'} & \stackrel{v}{\longrightarrow} & X \\ \downarrow{u} & & \downarrow{f} \\ \mathrm{Spec}A & \stackrel{g}{\...
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Johnstone, Topos theory: families of arrows inducing the same sheaf condition

Johnstone, Topos theory, 0.3, page 13, asserts that, given a Grothendieck pretopology $P$, if the equalizer condition on a presheaf $F$ is satisfied for a family of arrows $R=\{U_i\to U\}$, then it is ...
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Graded global sections of Proj(S) for S a polynomial ring and more general

Throughout, suppose $S$ is a graded ring which is finitely generated by $S_{1}$ and an $S_{0}$-algebra. Let $X = \text{Proj} S$. There is the usual associated graded module given by $$ \Gamma_{\bullet}...
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1answer
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When is it the case that all closed immersions of all irreducible components are flat?

Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,\ldots,X_r$ be the irreducible components of $X$ and let $f_i: ...
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Exact sequences of $\mathcal{O}_X$-modules, sections over X minus a point, and splitting

Let $X$ be a (let's say irreducible) scheme, let $x$ be a closed point, put $U = X - \{x\}$. Let $0\to\mathcal{F}\to\mathcal{G}\to\mathcal{H}\to 0$ be an exact sequence of $\mathcal{O}_X$-modules. If ...
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56 views

Adjoint property of $f^{-1}$ and $f_*$.

Let $f: X \rightarrow Y$ be a continuous map. Then a standard exercise is to show that the functors $f_*: \text{Sh}(X) \rightarrow \text{Sh}(Y)$ and $f^{-1}: \text{Sh}(Y) \rightarrow \text{Sh}(X)$ are ...
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45 views

Explicit sections after sheafification

Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $\mathscr{F}$ over a topological space $X$, the sheafification of $\mathscr{F}$, $\mathscr{F}^+$, as $$ \mathscr{F}^+(U) = \{ ...