# Questions tagged [quasicoherent-sheaves]

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### Hartshorne proposition II.5.4

I am really confused on the proof of Proposition II.5.4 on Hartshorne's algebraic geometry book. Especially, I am not sure how does the previous Lemma II.5.3 work on this proposition. Let me state the ...
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### Why quasi coherent sheaves?

So i was wondering why one considers quasi-coherent sheaves in algebraic geometry. I have read a lot that they are closely linked to the geometric properties of the underlying space. This means that ...
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### Quasi-coherent $\mathcal{F} \cong \tilde{M}$ on $\operatorname{Proj}B$ does not determine $M$

I would like to understand in detail the fact that given $\tilde{M}\cong \tilde{N}$ on $\operatorname{Proj}B$, then it is not always true in general that $M \cong N$. I know this has to do with ...
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### Reconciliation of definition of presentable module/quasi-coherent sheaf with the fact that all modules have free presentation

Presentable module is defined in nLab (https://ncatlab.org/nlab/show/presentable+module) as “the cokernel of a homomorphism of free modules”. What I’m confused about is that, according to https://en....
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### Hartshorne Chapter 2 Exercise 5.6. d)

In this exercise 5.6. d) in chapter 2 from "Algebraic Geometry" by Hartshorne one has to show that for any ideal $\mathfrak a \subset A$ of a noetherian ring $A$ and $A$-module $M$ the sheaf ...
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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 ...
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### Internal hom of equivariant sheaves

Let $G$ is affine algebraic group over $\mathbb{C}$ acting on a smooth scheme $X$ over $\mathbb{C}$, let $\mathcal{F},\mathcal{G}$ be two quasi coherent equivariant sheaves on $X$. Is there a natural ...
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### 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 ...
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### sheaves on a scheme

I work within the framework of Demazure & Gabriel`s book. (My schemes are all functors). Let $X$ be a scheme. I can define an underlying topological space $|X$| of $X$. Its points are equivalence ...
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### Quasicoherent sheaves over manfiolds

For any locally ringed space $\left(X,O_X\right)$, a quasicoherent sheaf is a sheaf of $O_X$-modules which are locally the quotient of free modules. Considering a manifold (Haussdorf, second-countable)...
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### Equivalent definitions of flat morphism

Suppose $\pi : X \to Y$ satisfies that pullback on quasi-coherent sheaves is exact, how do I prove that $\pi$ is flat via the local definition; i.e stalkwise $O_{X,p}$ is a flat $O_{Y,q}$ module ...
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I have a question about a part from the proof of Theorem 7.4.16 on page 290 from Liu's "Algebraic Geometry and Arithmetic Curves". The claim is Theorem 7.4.16 Let $f : X \to Y$ be a finite ...