# Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

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### Are morphisms of algebraic varieties determined by their underlying functions?

Let $K$ denote an algebraically closed field. Define that an algebraic variety over $K$ is a ringed space that can be covered by open sets, each of which is isomorphic to an affine algebraic variety, ...
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### Showing that $\mathbb{P}^1$ is two copies of $\mathbb{C}$ glued together; is there a high-power tool that can help here?

An algebraic geometry assignment asks the reader to show that $\mathbb{P}^1(\mathbb{C})$ is two copies of $\mathbb{C}$ "glued together." The meaning of "glued together" isn't defined but the emphasis ...
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### The 'Locally Ringed' condition in the definition of a scheme.

Is the 'Locally Ringed' condition in the definition of a Scheme redundant? My question is, if it admits a cover by Affine Schemes, does it follow that the Ringed space is Locally Ringed? More ...
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### Does the locally-ringed spaces viewpoint on topology actually do what we want?

There's a post here about how we know that the morphisms of smooth manifolds as locally-ringed spaces are the same as the morphisms of smooth manifolds as charts and atlases. I didn't understand the ...
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### Do we have $\operatorname{Ker}f^\sharp =\operatorname{Ker}g^\sharp$?

Let $f: Y\to X$ and $g:Z\to X$ be two closed immersions of locally ringed spaces. If $Y\simeq Z$, do we have $\operatorname{Ker}f^\sharp =\operatorname{Ker}g^\sharp$?
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### About gluing of sheaves on a cover

Suppose we are given a cover $\{U_i\}_{i \in I}$ of a space $X$ and a gluing data $( \mathcal{F}_i, \psi_{ij} )_{i,j \in I}$ for the sheaves of sets with respect to this covering. I want to show ...
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### Ideal sheaf which is locally generated by sections.

My question is about this Example in the stacks project. Let $(X,\mathcal{O}_X)$ be a locally ringed space with a sheaf of $\mathcal{O}_X$-ideal $\mathcal{I}$. Then the support of the $\mathcal{O}_X$...
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### Open vs Closed Immersions of Locally Ringed Spaces

I'm reading Qing Liu's book at the moment and I'm trying to figure out why open immersions of locally ringed spaces are required to be isomorphisms on stalks, but closed immersions are only required ...
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### What properties single out $\operatorname{Spec}(\mathbb{k})$-schemes that are quasi-projective varieties over $\mathbb{k}$?

I have a question in algebraic geometry that I would like to ask: Let $\mathbb{k}$ be an algebraically closed field. Is there a property $P$, phrased in the language of schemes, such that every ...
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### Correspondence between vector bundles and locally free sheaves

I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...
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### When are sheafification and the embedding of sheaves into presheaves exact functors?

Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
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### Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors  U_{\...
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### Composition of morphisms of locally ringed spaces

I have a specific question about defining the composition in (locally) ringed spaces. The definition I had formulated myself while reading Hartshorne, since he conveniently neglected to suggest any ...
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### $\Psi$ is a morphism of ringed spaces iff it is smooth in local coordinates

Here is what I have to show: Let $(M,\mathcal{F})$ and $(N,\mathcal{F}')$ be smooth manifolds of class $C^{\infty}$ and let $\Psi:M\to N$ be a continuous map. Show that the following conditions ...
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### About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ ...
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### Algebraic approach to Local Analytic Complex Geometry

I'm attending a second course in Complex Analysis from a geometrical point of view. In the final part of the course we have discussed about germs of complex analytic sets and their algebraic ...
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### locally ringed space $(X,\mathcal{O}_X)$ isomorphic as Ringed Spaces to $Spec(A)$ but not isomorphic as Locally Ringed Spaces.

I'm starting studying Hartshorne Chapter II and is the first time that I'm studying Schemes. I'm looking for some intuition viewing some examples. I'm looking for an example of a locally ringed ...
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### sections of tensor product of sheaves of modules

I am confused about notation concerning tensor products of sheaves of modules. I know that given a ringed space $X$ and $\mathcal{O}_X$-Modules $\mathcal{F}$ and $\mathcal{G}$ their tensor product is ...
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### When is a morphism of $k$-ringed spaces the morphism induced by pullbacks?

Let $(X, O_X)$ and $(Y,O_Y)$ be $k$-ringed spaces ($k$ a field), and let $(f,P)$ be a morphism between them. When is the case that $P: (O_Y(U)) \to O_X(f^{-1}(U))$ is given by precomposition with $f$? ...
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### A rigorous characterization for a ringed spaces to be isomorphic to an affine scheme.

On page 21-22 of the book The Geometry of Schemes by Eisenbud and Harris there is a characterization for when a ringed spaces $(X,\mathcal{O}_X)$ is isomorphic to an affine scheme \$(\text{Spec}(R),\...