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Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

A ringed space is a topological space together with a sheaf of commutative rings. Of particular interest are locally ringed spaces, which are ringed spaces in which each stalk of the sheaf of rings is a local ring. Morphisms of locally ringed spaces are also required to satisfy a locality condition.

Locally ringed spaces provide a general setting for "geometry". Important examples include schemes, complex analytic spaces, and smooth manifolds.

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Definition of smooth manifold using sheaves.

While defining differential manifolds using the concept of sheaves wikipedia gives the following definition. A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is…
HarshCurious
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What is the "dimension" of a locally ringed space?

Let $(X,\mathscr{O}_X)$ be a locally ringed space. If it is a scheme, the natural notion of dimension is the dimension of the subjacent topological space (the size of the biggest chain of irreducible closed subsets). But if $X$ is a manifold, I…
Gabriel
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On a *ringed space*, show that the non vanishing set of $f$ is open, and that it is invertible there

This is an exercise of Ravi Vakil that I solved by a very trivial argument without using the hint. For this reason I'm worried that I might have missed something. If $f$ is a function on a locally ringed space $X$, show that the subset of $X$ where…
Rodrigo
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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 $$ U_{\mathbf{LRS}}: \mathbf{LRS} \to \mathbf{RS} $$ (the…
user8463524
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Non-isomorphic locally ringed spaces which represent isomorphic functors $\mathsf{CommRing} \to \mathsf{Set}$.

It's well known that the restricted Yoneda functor $よ : \mathsf{Schemes} \to \operatorname{Fun}(\mathsf{CommRing},\mathsf{Set})$ is an embedding, so that (in particular) if $X$ and $Y$ are schemes such that $よ(X) \cong よ(Y)$, then $X \cong Y$. This…
diracdeltafunk
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Is there a notion of "schemeification" analogous to that of sheafification of a presheaf?

So this may seem like an odd question, but hear me out. In the Stacks Project, tag 01I4, we find that not only does the category of affine schemes live inside the category of locally ringed spaces, but that limits of affine schemes can be computed…
Joe
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Does the category of locally ringed spaces have products?

The category of schemes has all fibered products, but the proof uses affine schemes in a crucial way. I want to understand whether this is true for the category of locally ringed spaces. The standard sources for categorical properties (nLab and…
Dmitry
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When considering a finite-type scheme as a ringed space, is it enough to look at its $k$-points?

I am reading a set of notes by Michel Brion about automorphism groups of projective varieties. The following claim appears in the proof of a theorem stating that if G is a connected group scheme, $X$ a $G$-scheme, and $\pi:X \to Y$ a proper morphism…
user7090
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Can we define MaxSpec as a locally-ringed space?

I will start with a motivation. Let $X$ be a compat hausdorff space and let $A$ be the ring ($\mathbb{R}$-algebra) of continuous functions $X\to \mathbb{R}$. We define $Y=MaxSpec(A)$ to be the set of maximal ideals of $A$ and we can endow it with…
KotelKanim
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Definition of a morphism of locally ringed spaces

Let $(X, \mathcal O_X), (Y, \mathcal O_Y)$ be locally ringed spaces. A morphism of ringed spaces is defined to be a pair $(f,f^{\#}):(X, \mathcal O_X) \rightarrow (Y, \mathcal O_Y)$, where $f:X \rightarrow Y$ is continuous, and $f^{\#}: \mathcal…
D_S
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Hartshorne's Exercise II.5.1 - Projection formula

I'm trying to solve Exercise 5.1 of Chapter II of Hartshorne - Algebraic Geometry. I'm fine with the first $3$ parts, but I'm having troubles with the very last part, which asks to prove the projection formula: Let $f:X\to Y$ be a morphism of…
Abramo
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Relation between a generalization of Weil's abstract varieties and algebraic schemes

We would like to generalize this question when the base field $k$ is not necessarily algbraically closed. We fix an algebraically closed field $\Omega$ which has infinite trancendence dimension over the prime subfield. Let $k$ be a subfield of…
Makoto Kato
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Details of gluing sheaves on a cover

I am sure this is a simple question, but I am really not able to think straight at the moment and this is bugging me. I am doing Exercise 1.22 from Hartshorne. It is the classic gluing of of sheaves on a cover given the cocycle condition question.…
Joe
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Ring theory conventions - Zero ring, local homomorphisms

Just wondering about conventions dealing with the zero ring and the zero scheme. Does the category of schemes have an inital object? Is the zero ring considered local? For the purposes of scheme theory, is a map of sheaves that induces on stalks a…
Patrick Nicodemus
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$X_f$ of locally ringed space $(X, O_X)$.

Let $(X, O_X)$ be a locally ringed space. $f \in \Gamma(X,O_X)$ be a global section. $$X_f:= \{ x \in X \, ; \, f_x \text{ is invertible in } O_{X,x} \} $$ It is claimed that $X_f$ is an open subset The image of $f$ in $\Gamma(X_f, O_X)$ is…
Bryan Shih
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