Articles of sheaf cohomology

Skyscraper sheaf?

I was wondering whether someone could provide me with an easy definition of a skyscraper sheaf and, more importantly, with a visualization of it. Thanks in advance.

Question about Tate resolution and cohomology groups of nonzero coherent sheaves.

Let $\mathcal{F}$ be a nonzero coherent sheaf on the projective space $\mathbb{P}_{k}^m$. I am trying to show that for every integer $d$ there is $j$ for which $h^j\mathcal{F}(d-j) \neq 0$. My professor said we can use the dual of Tate resolution and self-injective concept to show that. Let $\mathcal{F}= \widetilde{M}$, and $E$ the exterior algebra […]

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 […]

cohomology of the pullback of a sheaf

Let $X$ a degree $d$ curve in $\mathbb{P}^n$ not contained in any hyperplane and $i:X\hookrightarrow \mathbb{P}^n$ the corresponding closed immersion. Then I would like to prove that $dim H^0(X, i^{*}O_{\mathbb{P}^n}(1))=n+1$. I think that the proof of this fact lies on the fact that it is not contained in any hyperplane so you can count linearly […]

Skew symmetry of indices in cocycles of Cech cohomology

In Cech cohomology, the coboundary operator $$\delta:C^p(\underline U, \mathcal F)\to C^{p+1}(\underline U, \mathcal F) $$ is defined by the formula $$ (\delta \sigma)_{i_0,\dots, i_{p+1}} = \sum_{j=0}^{p+1}(-1)^j \sigma_{i_0,\dots, \hat {i_j},\dots i_{p+1}}{\Huge_|}_{U_{i_0}\cap\dots\cap U_{i_{p+1}}}. $$ In the book by Griffiths and Harris, the authors claim that $\delta\sigma=0$ implies the skew-symmetry condition $$ \sigma_{i_0,\dots,i_p} = -\sigma_{i_0,\dots, i_{q-1},i_{q+1},i_q,i_{q+2},\dots,i_p}. $$ How […]

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don’t really have time to learn all basics of sheaves theory and the connection to vector bundles. I just need a definition […]

Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of integers on $X$. In proving this, I have seen several sources proceed more or less the same way: Let $\tilde{\mathcal{S}}^k$ be the sheafification of the presheaf $\mathcal{S}^k$ of singular cochains […]

Calculate the cohomology group of $U(n)$ by spectral sequence.

Here $U(n)$ is the unitary group, consisting of all matrix $A \in M_n (\mathbb{C})$ such that $AA^*=I$ Problem How to calculate the integer cohomology group $H^*(U(n))$ of $U(n)$? What if $O(n)$ replace $U(n)$? My primitive idea is that: as for $U(n)$, it is a Lie group and can acts transitively on the unit sphere with […]