Let $X$ be a topological space and $$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$ be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if for …

Let $X$ be a topological space, $\mathcal{C}$ be a locally small category with "good properties" (such as having small inverse limit, small filtrant inductive limit...etc.) and $\m …

Given a scheme $X$ with structure sheaf $\mathcal{O}_X$, we can associate to each $\mathcal{O}_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the str …

I am slightly confused about sheafification at the moment. I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of …

After digesting the Presheaf definition by the very first time, one feels (at least I felt) a strange sensation noticing the existence and uniqueness conditions to graduate that Pr …

Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X),\text{Sh} …

A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to …

So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf specifically lemma 3.4. The setup is as follows: Let $X/\mathbb{C}$ be …

Recall from my previous question the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids i …

Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the c …

It's known that the moduli stack ${\cal M}$ of semistable sheaves on a given polarized projective variety, with a fixed Hilbert polynomial, is compact (meaning that ${\cal M}$ has …

Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem: Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acycl …

A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-b …

Alright, this is a follow-up to my previous question (http://mathoverflow.net/questions/94151/spectral-sequences-in-hypercohomology-of-sheaves), sorry I took so long to reply. Let …

I have a technical question coming from reading Toen's master course on stacks. If we view schemes as locally ringed spaces then there we could define a morphism to be surjective …