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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 …

Let $X,Y$ be analytic spaces with $X$ compact reduced and $Y$ arbitrary (i.e. maybe with nilpotents in its structure sheaf). Then Douady showed in his thesis that the set of holomorphic maps $Hol(X,Y) …

I) The most elementary, but yet quite useful, use of a vanishing theorem is via Riemann-Roch for a smooth complete curve $X$ over the algebraically closed field $k$. Riemann-Roch for a divisor $D$ on …

Let $A$ be a noetherian ring and $X=\operatorname {Spec}A$ the corresponding affine scheme. There are three rings which might reasonably be called the ring of rational functions on $X$. a) The total r …

The ring $\mathcal K(X)$ is called the ring of meromorphic functions on $X$ in EGA, the Stacks project or by Kleiman and many others. The terminology is disastrous if one considers schemes over $\math …

Since your question might interest other readers, allow me to expand it. Given a scheme $T$, you can associate to it the contravariant functor $h_T: \mathcal{ Schemes}^\text{opp} \to \mathcal{Sets}$. …

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 …

If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals. The converse is false for a silly reason : in the real or comp …

About your new question: Let $Y$ be a projective variety and let $X\subset Y$ be an open subset with complement the closed subset $S:=Y\setminus X$. Call $f:X\hookrightarrow Y$ the inclusion. Let $\ma …

In 1963 Grothendieck introduced the algebraic de Rham cohomolog in a letter to Atiyah, later published in the Publications Mathématiques de l'IHES, N°29. If $X$ is an algebraic scheme over $\mathbb C …

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 …

The answer is Yes for complex manifolds of dimension one. Indeed for any open subset $U\subset X$ the long exact cohomology sequence associated to the exponential sequence you mention yields the fr …

Grothendieck and Dieudonné prove in $EGA_{II}$ (Proposition 5.5.5.(ii), page 105) that if $f:X\to Y, g:Y\to Z$ are projective morphisms of schemes and if $Z$ is separated and quasi-compact, or if t …

Since in 2007-2008 you evoked [ Class 24, §1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially co …

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 $\newcommand{\F}{\mathcal F}H^i(X,\F)$ …