Questions tagged [ringed-spaces]

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References on topological ringed spaces

This is a follow up to this question of mine. First of all, let me fix some terminologies, which may or may not be standard: Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)...
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Localification of a ringed space

Call a ringed space local it if it lies in the image of the obvious faithful, non-full functor from locally ringed spaces to ringed spaces. Given a ringed space, is there a map $f$ from it to some ...
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1 answer
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About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...
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9 votes
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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 ...
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9 votes
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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 ...
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3 votes
2 answers
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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 C^\...
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2 answers
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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 $\mathcal{O}_Y$-...
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Sheaves of $\mathbb Z$-modules = sheaves of abelian groups

In his "Algebraic Geometry", Hartshorne proves that for any ringed spaces $(X,\mathcal O_X)$, category $Mod(X)$ of sheaves of $\mathcal O_X$-modules has enough injectives. If we take $\...
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19 votes
5 answers
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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 ...
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4 votes
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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 ...
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