Questions tagged [ringed-spaces]

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Is an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules quasi-coherent on a complex manifold?

On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there ...
Zhaoting Wei's user avatar
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Who introduced the notion of ringed spaces?

My question is very concise, please forgive it. Who introduced the concept of ringed space? My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
user234212323's user avatar
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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)...
Dat Minh Ha's user avatar
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2 votes
3 answers

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 ...
user avatar
10 votes
1 answer

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question on math.stackexchange , however I don't think I will ...
user40276's user avatar
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9 votes
2 answers

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 ...
Matthias Wendt's user avatar
9 votes
0 answers

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 ...
Martin Brandenburg's user avatar
3 votes
2 answers

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^\...
Nevermind's user avatar
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5 votes
2 answers

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 $\mathcal{O}_Y$-...
Daniel Bergh's user avatar
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8 votes
1 answer

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 $\...
Rafael Mrden's user avatar
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22 votes
5 answers

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 ...
Justin Curry's user avatar
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6 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 ...
Martin Brandenburg's user avatar