# Questions tagged [coherent-sheaves]

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space.

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• 2,438
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### coherent sheaves annihilated by ideal sheaves and morphisms between them

Let $X$ be a Noetherian scheme and $\mathcal I\subseteq \mathcal O_X$ be a coherent ideal sheaf defining a closed subscheme $Z$ of $X$. Let $i:Z\to X$ be the closed immersion. I have the following ...
• 423
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• 642
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### Lazarsfeld's proof of Mumford's regularity Theorem 1.8.3

I want to apologize first of all if my English is bad, I hope my message will still be understandable. I’m currently reading Lazarsfeld’s book "Positivity in Algebraic Geometry I" and I’m ...
• 555
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### Sheaves of modules isomorphic after pullback - when isomorphic in general?

Let $f: X \to Y$ be a morphism of schemes and $\mathcal{M}, \mathcal{N}$ be $\mathcal{O}_Y$-modules. Suppose $f^*\mathcal{M} \cong f^* \mathcal{N}$. Under which assumptions can I conclude that already ...
• 797
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### Is a Noetherian sheaf of rings stalkwise Noetherian?

Let $X$ be a scheme such that $O_X$ is a coherent $O_X$-module. Assume that for every open subset $U\subset X$ and every family of coherent ideal sheaves $\{I_i\}_i$ of $O_U$, $\sum_iI_i$ is a ...
• 1,308
1 vote
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• 423
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### Autoequivalences of $\operatorname{Coh}(X)$

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero. Is there a description of $\operatorname{Aut}(\operatorname{Coh}(X))$, i.e. the autoequivalences ...
• 1,872
282 views

### Hartshorne Exercise II 6.11 (c)

Exercise II 6.11: Let $X$ be a nonsingular curve over an algebraically closed field $k$. (c) If ${\mathscr{F}}$ is any coherent sheaf of rank $r$ (means that its stalk at the generic point has ...
• 521
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### Locally free resolution of coherent sheaves on nonsingular curves

This question is from Exercise II 6.11 of Hartshorne. Let $X$ be a nonsingular curve over an algebraically closed field $k$. For any coherent sheaf $\mathcal{F}$ on $X$, show that there exist ...
• 521
1 vote
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### Do analytic coherent sheaves remain coherent in the analytic Zariski topology?

Let $X$ be a compact Kähler manifold. Let $F$ be a coherent $O_X$-module. Is there always an open cover $\{U_i\}_i$ for $X$ such that for every $i$: 1.$X\setminus U_i$ is an analytic subset of $X$; ...
• 1,308
1 vote
158 views

### Support of the direct image sheaf equals the image?

$\def\sO{\mathcal{O}} \def\supp{\operatorname{Supp}} \def\sI{\mathcal{I}} \def\sC{\mathcal{C}} \def\colim{\operatorname{colim}}$I am studying complex spaces using Grauert, Remmert, Coherent Analytic ...
1 vote
153 views

### Support of a section of an $\mathcal{O}_X$ module

Given a coherent sheaf $F$ over a space $X$, I understand from https://en.wikipedia.org/wiki/Support_of_a_module that the support of the sheaf of modules is all those points of $X$ such that the stalk ...
• 724
1 vote
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• 1,111
1 vote
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### How to show saturation map (w.r.t quasicoherent sheaves) isnt always injective / surjective

This question is motivated by problem 15.4.D(a) in Vakil, but to give some setup since the notation / terminology may differ: let $S_\bullet$ be a nice graded algebra (finitely generated, generated in ...
• 417
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### When is an extension a vector bundle?

Let $X$ be a smooth threefold and $C \subset X$ be a smooth (but not necessarily irreducible) curve with ideal sheaf $\mathcal{I_C}$. I am looking for an answer to the question of when an ...
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### Hilbert scheme of $\mathbb{P}^2$ is not a product?

Fixing a connected component in the Hilbert scheme of $\mathbb{P}^2$ is the same as fixing the Hilbert polynomial $ax+b$. Taking a primary decomposition of any subscheme in this connected component ...
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