Mathematics

Questions tagged [quasicoherent-sheaves]

238 questions
36
votes
1 answer

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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34
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4 answers

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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26
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1 answer

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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21
votes
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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
  • 14,560
20
votes
3 answers

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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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
votes
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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15
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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
votes
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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2 answers

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
votes
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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12
votes
0 answers

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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12
votes
1 answer

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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11
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1 answer

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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