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Questions tagged [quasicoherent-sheaves]

228 questions
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Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is equivalent to the category of $S$-modules. In…
M Turgeon
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When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In the following we have $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ a quasi-coherent sheaf on $X$ and I am referring to proposition 5.8 page 115 in Hartshorne. To prove that the pushforward of a quasi-coherent sheaf is quasi-coherent,…
Stefano
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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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20
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Inverse image of the sheaf associated to a module (Hartshorne, Proposition II.5.2)

In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows from the definition. But I don't know how to prove…
user46336
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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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19
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2 answers

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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17
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2 answers

Reference request: Bundles in Algebraic Geometry

I heard many times that quasi-coherent sheaves of $\mathcal O_X$-modules are morally the same thing as the sheaves of sections of a bundle $V\to X$ over $X$. We think of a ring $A$ as of the ring of functions of the imagined space…
Nico
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Why is the sheaf $\mathcal{O}_X(n)$ called the "twisting sheaf" (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
sti9111
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14
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1 answer

Classifying Quasi-coherent Sheaves on Projective Schemes

I know some references where I can find this, but they seem tedious. Both Hartshorne and Ueno cover this. I am wondering if there is an elegant way to describe these. If this task is too difficult in general, how about just $\mathbb{P}^n$? Thanks!
BBischof
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13
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Stalk of a pushforward sheaf in algebraic geometry

Excuse me if this is a naive question. Let $f : X \to Y$ be a morphism of varieties over a field $k$ and $\mathcal{F}$ a quasi-coherent sheaf on $X$. I know that for general sheaves on spaces not much can be said about the stalk $(f_*\mathcal{F})_y$…
Justin Campbell
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12
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2 answers

Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to the product as $\mathcal{O}_{X}$-modules, but I…
user8249
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Morphisms with connected fibers

Let $f\colon X\to Y$ be a morphism of schemes. I am interested in the following property (too long for the title): $$ f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y} \quad \text{(P)}$$ Under very reasonable assumptions ($f$ projective between noetherian…
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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The projection formula for quasicoherent sheaves.

I am looking for a certain way of proving the following : Let $f. X \rightarrow S$ be a morphism of schemes. Suppose that f is quasiseparated and quasicompact, or that X is noetherian. Let $\mathcal{G}$ be a locally free sheaf on S and $\mathcal{F}$…
Dedalus
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10
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2 answers

Gaga and quasicoherent sheaf

Let $X$ be a complete algebraic variety over $\mathbb{C}$. Ad Serre GAGA stases, its analytification $X^{an}$ is compact and the analytification functor induces an equivalence of categories between $Mod_c(\mathcal{O}_X)$ and…
Tommaso Scognamiglio
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