Articles of quasicoherent sheaves

Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $ \mathbb{C}^* $, and I’m confused. I don’t have much background in this, so I was hoping a very concrete example would clear up a lot of this confusion. So far: We have this cochain of $ \mathbb{C}[x,x^{-1}]$-modules: $0 \longrightarrow \mathbb{C}\longrightarrow \mathbb{C}[x,x^{-1}] \longrightarrow {\Omega}_{\mathbb{C}[x,x^{-1}]/\mathbb{C}}^1 \longrightarrow […]

Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules

The question is basically like this: Prove that if $S_{\cdot}$ is a finitely generated (in degree 1) graded ring over a field $k$ and $M_{\cdot}$ is finitely generated, then the saturation map $M_{n}\rightarrow\Gamma(\text{Proj}S_{\cdot},\widetilde{M(n)_{\cdot}})$ is an isomorphism for large $n$. The hint in [Hartshorne] says we can follow the hint in the proof of Theorem 5.19. […]

First Ext group of a sheaf

Let $F$ be a quasicoherent sheaf on a scheme $X$, which is supposed to be sufficiently nice. Does one then have a canonical isomorphism $Ext^1(F,F) \simeq H^1(X, \underline{End}(F))$, where with $\underline{End}(F)$ I denote the sheaf of endomorphisms of $F$. I know that this holds for $F$ locally free, but I read an article where this […]

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.

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 could define quasi-coherent sheaves to be those $\mathcal O_X$ […]

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 $\mathcal{F}$ a quasicoherent sheaf on X. Show that we have a […]

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 particular, we have the following result about affine schemes: If $X=(X,\mathcal O_X)$ is […]

Global sections on quasi coherent sheaves on affine scheme

This is a lemma from Hartshorne’s Algebraic Geometry. Let $X=\text{Spec}(A)$ be an affine scheme $f\in A, D(f)\subseteq X$. Let $\mathcal{F}$ be a quasi coherent sheaf on $X$. If $s\in \Gamma(X,\mathcal{F})$ is such that $s|_{D(f)}=0$ then for some $n>0$, $f^ns=0$. If $t\in \Gamma(D(f),\mathcal{F})$ then for some $n>0$ $f^nt$ extends to a global section of $\mathcal{F}$ over […]

Modules over a functor of points

I have a question on the ”functor of points”-approach to schemes and $\mathcal{O}_X$-modules. Please let me first write up a defintion. Let $Psh$ denote the category of presheaves on the opposite category of rings $Rng^{op}$. So $Psh$ is the category of functors from the category of rings $Rng$ to the category $Set$ of sets. Fix […]

Inverse image of the sheaf associated to a module

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 it in simple way. Could you help me?