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59 votes

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 ...
Georges Elencwajg's user avatar
28 votes

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? ...
Mariano Suárez-Álvarez's user avatar
18 votes
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Detail in the proof that sheaf cohomology = singular cohomology

To elaborate on Dmitri Pavlov's comment, your questions have been answered in a recent paper by Yehonatan Sella. Sella give an example (Example 0.3) of a locally contractible space $X$ for which $\...
Eric Wofsey's user avatar
16 votes

Why only consider Dolbeault cohomology?

$\newcommand{\dd}{\partial}$To supplement the existing (excellent) answers: If $E \to M$ is a holomorphic vector bundle, the transition functions of $E$ are "constant with respect to $\bar{\dd}$", so ...
Andrew D. Hwang's user avatar
13 votes
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Why only consider Dolbeault cohomology?

Note that $H^{0,0}_{\bar{\partial}}(X) = \ker\bar{\partial} : A^{0, 0}(X) \to A^{0, 1}(X)$ is precisely the collection of holomorphic functions on $X$ which, historically, were of more interest than ...
Michael Albanese's user avatar
11 votes
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Sheaf cohomology intuition

It is hard to know what will give you an intuition without knowing what you want and what background you already have. If you are asking roughly what kind of geometric or topological information Cech ...
Colin McLarty's user avatar
11 votes

Why was Sheaf cohomology invented?

Hartshorne says that cohomology was first introduced to abstract algebraic geometry by Serre in his Faisceaux Algebriques Coherents paper (translated to English by a friend of mine). The FAC says We ...
xyzzyz's user avatar
  • 7,654
11 votes
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Understanding etale cohomology versus ordinary sheaves

As mentioned in the comments, acyclic sheaves for the Zariski site are not the same as those which are acyclic sheaves for the étale site. (see the edit for the reason.) Consider $\mathbf{P}^1_{\...
shubhankar's user avatar
10 votes

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 ...
David E Speyer's user avatar
10 votes
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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 ...
Roland's user avatar
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9 votes
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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 ...
hunter's user avatar
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8 votes

Skyscraper sheaves cohomology

A skyscraper sheaf is flasque, hence has no cohomology: Hartshorne Chap. III, Prop.2.5, page 208.
Georges Elencwajg's user avatar
8 votes
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Relative vs. local sheaf cohomology?

These notions are "dual". More precisely, there are two distinguished triangle in the derived category of sheaves on $X$ : $$ j_!j^{-1}\rightarrow 1\rightarrow i_*i^{-1}\overset{+1}\rightarrow $$ and $...
Roland's user avatar
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8 votes
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Proof of Serre Duality in Hartshorne

So it seems I got it backwards: If you start with applying Thm 7.6 to $H^i(X, \mathcal{F})$ you get: $$H^i(X, \mathcal{F}) = Ext^{n-i}(\mathcal{F}, \omega_X^\circ)' = Ext^{n-i}(\mathcal{O}_X, \...
red_trumpet's user avatar
  • 8,951
8 votes
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Vector Bundle Transition Functions as Cech Cocycles

The condition you suspected, namely $g_{\alpha\beta}(g'_{\alpha\beta})^{-1} = \lambda_{\beta}\lambda_{\alpha}^{-1}$, is the correct condition if the coefficient group is abelian. The latter condition, ...
Michael Albanese's user avatar
8 votes

Why are line bundles called principal $\mathbb G_m$-bundle?

There is an equivalence of categories between the groupoid of line bundles on $X$ and the groupoid of principal $\mathbb{G}_{\mathrm{m}}$-bundles which works as follows. Given a principal $\mathbb{G}_{...
Sasha's user avatar
  • 17.5k
8 votes
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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 ...
Mark Saving's user avatar
7 votes
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A question on Hartshorne Chapter III Proposition 2.6

The functor $\Gamma(X,-)$ is initially defined on page 207 as a functor $Sh(Ab, X) \to Ab$, where $Sh(Ab, X)$ is the category of sheaves of abelian groups on $X$. You have the forgetful functor $U : ...
Watson's user avatar
  • 23.8k
7 votes
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Doubt over a proof about higher direct image functors in Hartshorne

Your assertion that These higher direct image functors, and indeed cohomology functors are defined out of the category of $\mathcal{O}_{X}$-modules. is correct but not the whole story: indeed, we ...
KReiser's user avatar
  • 66.1k
7 votes
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Explicitly Understanding a Case of Dolbeault's Theorem.

Consider a class $[\alpha]\in H^{0,1}(M)$, we can find a fine enough open cover $\{U_i\}_{i\in I}$ so that $\alpha$ is locally $\bar\partial$-exact, so $\alpha|_{U_i}=\bar\partial \beta_i$ for some ...
lEm's user avatar
  • 5,487
7 votes

How are (pre)sheaves even well-defined on sites?

This is an abuse of notation. $F(U)$ is not just a function of $U$ as an object, it's a function of the morphism $f : U \to X$, which is being dropped from the notation. That is, it's shorthand for $F(...
Qiaochu Yuan's user avatar
7 votes
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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 ...
KReiser's user avatar
  • 66.1k
6 votes

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 ...
Shubhodip Mondal's user avatar
6 votes

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 ...
Jonathan's user avatar
  • 268
6 votes
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Where we use assumption that resolution is locally free (Hartshorne III.6.5)

If $\mathscr{L}_.$ is not a complex of locally free sheaves, then $h^i(\mathscr{H}om(\mathscr{L}_.,\mathscr{G}))$ is not a $\delta$-functor in $\mathscr{G}$. Indeed, given a short exact sequence $0\...
Roland's user avatar
  • 12.5k
6 votes
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How to calculate $Ri^!$?

The main tools at your disposal are duality and the localization triangles. I come from algebraic geometry where everything is orientable and there is a factor 2 between dimension and cohomological ...
Roland's user avatar
  • 12.5k
6 votes
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Canonical morphism $H^{n}(Y,f_* \mathscr{F}) \to H^{n}(X, \mathscr{F})$ in sheaf cohomology.

As MooS said, in this particular situation, you can use the category of $\mathcal{O}$-module to construct the morphism you seek. In general this is a good method because it leads to a morphism ...
Roland's user avatar
  • 12.5k

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