# 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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### 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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### 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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### 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 ...