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Here is the great answer given by another of my brilliant friends: Let $X$ be $\mathbb C, U_0$ be the open complement in $X$ of the closed disk $\bar D=\{z\in \mathbb C\vert \vert z\vert \leq1 \}$ and …

Here is the solution obtained by one of my brilliant geometer friends evoked in the question: Take $X=S^2$, the unit $2$-sphere with equation $x_1^2+x_2 ^2+x_3^2=1$, and cover it by the three open str …

Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{D …

Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$. My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal …

a) Using Morse theory Hamm proved a theorem here implying that every Stein complex space $X$ of complex dimension $n$ is homotopy equivalent to a CW-complex of (real) dimension $\leq n$. Notice tha …

0) I would guess that the compact spaces you are looking for are extremely rare. 1) For example the extremely simple contractible space $I=[0,1]$ is not suitable: Consider the inclusion $j\colon …

the field of rational functions $\mathcal K_X$ on an integral scheme $X$ ( for example an algebraic variety) is flasque and so is the sheaf of its invertible elements $\mathcal K^\ast_X$. This has a …

Dear Justin, let me address your first question. First of all, the local quotient of two analytic functions is called meromorphic. In dimension one you can see meromorphic functions as regular functio …

Dear Daniel, the reason you couldn't find a proof of your statement nor locate one in the literature is that it is false ; so you were quite right to "have doubts now" ! Here are two (essentially equ …

Dear J, here is a little technical warning which might be relevant to your question. If you open Hartshorne and read the definition of "coherent" (Chapter II, §5, page 111) you might get the impress …

Dear roger123, let $R$ be a commutative ring and $M$ an $R$-module ( which I do not suppose finitely generated). In order to minimize the risk of misunderstandings, allow me to introduce the followin …

Although we clearly all have more or less the same answers, here is how I like to organize things. I) Let $\mathcal F$ be a sheaf of abelian groups on the topological space $X$. It is said to be sof …

Here is an elegant application of sheaves seen as étalé spaces. Consider a complex manifold $M$. It automatically comes with a holomorphic local isomorphism $\pi: \mathcal O_M \to M $ described as fol …

Dear Victoria: here is a summary of the main comparison results I know of between Grothendieck cohomology (which is usually just called cohomology and written $\newcommand{\F}{\mathcal F}H^i(X,\F)$ …

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero …