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8
votes
2answers
193 views

Clifford algebras for quadratic modules over ringed spaces

What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ...
7
votes
0answers
230 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
294 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
338 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 ...
12
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
518 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 ...