# Questions tagged [quasicoherent-sheaves]

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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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### Quasi-coherent sheaves $\supset$ locally free sheaves?

This question has been completely reformulated following the guidelines in the comments below, to make it clearer where I was able to get and where I can't get out of. Such comments helped me a lot. ...
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### Homomorphism of quasi coherent sheaves [closed]

I was trying to solve the problems of Liu's book and wanted to show that if both $F$ ,$G$ are coherent then $Hom(F,G)$ is also coherent... but I realised that I really need to understand the meaning ...
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### product of quasi-coherent sheaves is not a quasi-coherent sheaf [duplicate]

Reading the basics of $\mathcal{O}_{X}$ modules, where $\left(X,\mathcal{O}_{X}\right)$ is a fixed ringed space, i understood that arbitrary direct products of quasi-coherent $\mathcal{O}_{X}$ modules ...
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We have the definition: Definition. Let $X$ be a scheme. Let $Z,Y⊂X$ be closed subschemes corresponding to quasi-coherent ideal sheaves $\mathcal{I},\mathcal{J}⊂\mathcal{O}_X$. The scheme theoretic ...
### Explicit description of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a line bundle
I understand the construction of $\mathcal{O}_{\Bbb{P}^1}(-1)$ as a sheaf on $\Bbb{P}_\Bbb{C}^1$, but I'm trying to understand how exactly does this define a line bundle and why people call this the &...