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7
votes
0answers
201 views

Colimits of quasi-coherent sheaves on a ringed space

Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...
3
votes
2answers
287 views

Smooth submanifolds defined by Subrings

To be honest, I don't really know, whether or not the following is a research level question: Let $M$ be a smooth manifold, $C^\infty(M)$ the smooth function ring on $M$ and suppose $R\subset ...
5
votes
2answers
333 views

Coherence for pull-backs and push-forwards

Let $p:X \to S$ and $q:Y\to S$ be two objects in the category of ringed spaces over the ringed space $S$, and let $f:X \to Y$ be a morphism over $S$. Given a sheaf $\mathcal{F}$ of ...
11
votes
5answers
1k views

Cohomology of Structure Sheaves: Algebraic, Constructible and more

I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...
3
votes
0answers
482 views

quasi-coherent modules outside algebraic geometry?

Let $X$ be a ringed space. A quasi-coherent module on $X$ is a module which has locally a presentation, i.e. locally on $X$, it is the cokernel of a map between free modules. If $X$ is a scheme, then ...