# Tag Info

## Hot answers tagged sheaf-cohomology

### Why was Sheaf cohomology invented?

Sheaves and sheaf cohomology were invented not by Serre, but by Jean Leray while he was a World War II prisoner in Oflag XVII (Offizierlager=Officer Camp) in Austria. After the war he published his ...

### Why was Sheaf cohomology invented?

Sheaf cohomology is just the elaboration of the following problem: you have a space and a covering, and you can do something you want on each set of the covering: can you do it on the whole space? ...
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### Comparing the proofs of Riemann Roch & Serre Duality in Forster's and Miranda's book

$\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}\def\cK{\mathcal{K}}$There are three definitions of coherent sheaf cohomology I'll want to compare here: (1) Miranda's definition (which is pretty close to the ...
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### Exactness and obstruction in sheaf cohomology

Well of course a sheaf is by definition something that can lift local datas to global ones in the following sense : if $\{U_i\}$ forms a cover of $X$ and if $f_i\in \mathcal{F}(U_i)$ are sections such ...
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### why 0-th sheaf cohomology is the global section

I remember getting tripped up on the same thing! In the definition of derived functors, you have to "chop off" the left-most object in the injective resolution (i.e., the object you started ...
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### Skyscraper sheaves cohomology

A skyscraper sheaf is flasque, hence has no cohomology: Hartshorne Chap. III, Prop.2.5, page 208.
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### Two definition of sheaves: thoughts?

The first definition is what I will call a sheaf (of Abelian groups). The second definition I will call an etale space (of Abelian groups). The two categories are equivalent. Let’s begin with an etale ...
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### Is Hartshorne's Remark III.2.9.1 actually a valid argument?

The key is to get more inventive with your diagram. Instead of taking the colimit all at once, build it up in pieces: if $I$ is an infinite set, then we can consider the diagram made up of finite ...
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### Sheaf cohomology with support

Construct a counterexample as follows. Let $X = \text{Spec} A$. (Assume $A$ to be Noetherian.) Choose an ideal $\mathfrak a$ such that $U= X \setminus V(\mathfrak a)$ is NOT an affine scheme.(Later ...
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### Skyscraper sheaves cohomology

You can also see this using Čech cohomology: Consider an open cover $\mathfrak{U}=(U_i)_{i\in I}$ of your space $X$. You can always refine this cover so that only one of the sets $U_i$ contains the ...
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