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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.

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. Reference: Wikipedia.

The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometrical information.

421 questions
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Precise connection between Poincare Duality and Serre Duality

The statements of Poincare duality for manifolds and Serre Duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. I have heard tangential statements from some people that there is indeed some connection…
user1119
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What local system really is

I know a local system is a locally constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, $\tilde{X}$ is the universal covering space of $X$, and $V$…
abc
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Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that in this case they intersect. Suppose that they do not…
guest31
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Category of quasicoherent sheaves not abelian

Wikipedia mentions that the category of quasicoherent sheaves need not form an abelian category on general ringed spaces. Is there a `naturally occurring' example of this failing, even for locally ringed spaces?
Jonathan Gleason
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Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite satisfactorical, since this yields a better…
Patricio
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Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - as $\mathcal O_X$-modules - to be locally free; we…
Arrow
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Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. What follows is an excerpt from Dieudonné's History…
Arrow
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Hartshorne exercise II.5.12(b)

I've been working on the Hartshorne exercise in the title for quite a while, which goes like this: let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes, $\mathscr{L}$ a very ample invertible sheaf on $X$ relative to $Y$, and $\mathscr{M}$ a…
Justin Campbell
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Formal Schemes Mittag-Leffler

Here is a question that is similar to my last one. I've been trying to learn about Grothendieck's Existence Theorem, but it seems that there aren't very many places that talk about formal schemes and even less that come up with examples. Suppose…
Matt
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Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving myself as concrete an understanding as possible of…
Ben Blum-Smith
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Freeness of stalk Implies locally free

Let $ A $ be a Noetherian ring, and $ M $ a finitely generated $ A $ module. Suppose that $ \mathfrak { p } \in M $ such that $ M_{\mathfrak{p}} $ is free. Show that there is a $ f \in A \setminus \mathfrak{p} $ such that $ M_{f} $ is free over $…
Hodge-Tate
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Vector bundle associated to a locally free sheaf

I am starting to study vector bundles over schemes and I have encountered two different definitions of the vector bundle associated to a locally free sheaf. I tried to understand why this was the case, but the reason eludes me. Eisenbud and Harris…
Pedro
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is the pushforward of a flat sheaf flat?

Let $f:X \to Y$ be a morphism of schemes and let $F$ be an $\mathcal{O}_X$-module flat over $Y$. Is $f_*F$ flat over $Y$? What's wrong with this argument? [EDIT: as Parsa points out, the (underived) projection formula does not hold for arbitrary…
Jacob Bell
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Null-correlation and Tango bundles on $\mathbb{P}^3$

Let $V$ be a four-dimensional complex vector space and $\mathbb{P}^3=\mathbb{P}(V)$. There are two interesting bundles $N$ and $T$ on $\mathbb{P}^3$, both of rank 2, called respectively a null-correlation bundle and a Tango bundle. Let me briefly…
mahavishnu
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Coherent Sheaves on Smooth Curves

Let $X$ be a locally noetherian integral non-singular scheme of dimension 1 (I'm really interested in the case $X=\mathbb{P}^1_K$) and $\mathcal{E}$ a coherent sheaf over $X$, I've already shown that $\mathcal{E}$ is either locally free (of finite…
espacodual
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