Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

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4
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0answers
49 views

Natural example of a ringed space that is not a locally ringed space

It wouldn't be too difficult to cook up some contrived examples of ringed spaces that aren't locally ringed spaces; however, are there any such examples that appear "in the wild," i.e. that ...
6
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3answers
518 views

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 ...
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0answers
33 views

Non zero sections of sheaves which vanish on all fibers

I was stuck on the following question and I was wondering if someone more familiar with sheaf theory might be able to help me with it. $\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$ ...
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0answers
62 views

Existence of Quotients in Locally Ringed Spaces

This is part of an exercise (2.14) in Qing Liu's book on Algebraic Geometry. I'm going to be verbose to see if there's something fundamental I'm missing. Actual questions will be numbered. Let $G$ act ...
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0answers
31 views

Morphism of ringed space and its induced morphism of stalk

It is known that: Let $\varphi:\mathfrak{F}\to\mathfrak{G}$ be sheaves of abelian groups over a topological space $X$. Then $\varphi$ is an isomorphism if and only if the induced map on the stalk $\...
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0answers
55 views

Why is the ring of local-ring valued points of a ring scheme a local ring.

I'm confused on a supposedly easy claim: Let $T$ be a base scheme, and let $\mathbf{R}$ be a ring scheme over $T$, i.e. a scheme $\mathbf{R} \to T$ such that for all $E \in \operatorname{Sch}_{/T}$ $\...
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0answers
23 views

Map of global sections is inclusion map

I have a question related to the answer to this question: Morphism of ringed spaces not induced by homomorphism of rings. I understand the entire argument but I'm having trouble how I can see that ...
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0answers
47 views

Locally ringed point space is not a scheme?

Let $X$ be the point space $\{*\}$, and let the structure sheaf $\mathcal{O}_X$ be given by: $$\mathcal{O}_X (X) = \mathcal{O}_X (\{*\}) := \mathbb{Z}_p$$ $$\mathcal{O}_X (\varnothing) := 0$$ Why is ...
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0answers
58 views

Computing $\mathscr O_X (X − \{x\})$ for $X$ an affine scheme

Let $R=\Bbb Z[T], X = Spec\ R, \mathfrak m := (T − 2, 3)$ where $\mathfrak m$ is a maximal ideal of $R$. Compute $\mathscr O_X (X − \{x\})$ where $x\in X$ corresponds to $\mathfrak m$. We have $X − \...
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2answers
35 views

Characterisation of isomorphisms of ringed spaces

An isomorphism of ringed spaces is a morphism of ringed spaces $(f,\theta) :(X, \mathcal{O}_{X})\rightarrow (Y,\mathcal{O}_{Y})$ $(f:X\rightarrow Y$ continuous, $\theta: \mathcal{O}_{Y}\rightarrow ...
3
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0answers
44 views

Is $\textbf{MAN}$ a full subcategory of $\textbf{LRS}$? [duplicate]

Let MAN be the category of differentiable manifolds with smooth maps as morphisms. Let LRS be the category of locally ringed spaces. Now I know that there exists a functor $\textbf{MAN}\to\textbf{LRS}$...
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1answer
85 views

A closed subset of a prevariety is a prevariety

My question comes from Gathmann's notes https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/alggeom-2014.pdf on page 42 Exercise 5.13. Let $Y$ be a closed subset of a prevariety $X$, ...
2
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1answer
59 views

Condition of closed embedding

One condition, (2) of Definition 25.4.1, for a morphism of ringed spaces $i:Z\rightarrow X$ to be a closed immersion is that $$O_X \rightarrow i_*O_Z$$ is surjective. I have two confusions (a) ...
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0answers
46 views

The spectrum of a not necessarily quasi-coherent sheaf of Algebras and a related vague question.

See this answer on Mathoverflow and this wikipedia section. These links claim that one can construct $Spec \mathcal{A}$ for any sheaf of algebras over a scheme(and even for any locally ringed space). ...
2
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1answer
66 views

When the contravariant Hom functor of two finite type integral $k$-schemes are isomorphic on points

Let $k$ be an Algebraically closed field. Let $\mathcal C_k$ be the category of integral $k$-schemes of finite type over $k$ (the morphisms between two objects being the morphism of schemes that also ...
2
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0answers
66 views

Is there a “natural” categorical description of (pre-)sheaves of modules?

I've been wondering about the following: Is there a $\textit{neat}$ description of the category $[\mathcal P]\mathcal{Mod}(\mathcal O)$ of [pre]sheaves of modules on a sheaf of rings $\mathcal O$? ...
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0answers
51 views

Is flatness condition for proper base change actually needed?

In his 1988 Paper on Resolutions of Unbounded Complexes, Spaltenstein proves that $Lg^*Rf_! \cong RF_! LG^*$ where $\require{AMScd}$ \begin{CD} A @>F>> B\\ @V G V V @VV g V\\ C @&...
3
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1answer
97 views

Proper base change for ringed spaces

$\newcommand{\Oo}{\mathcal{O}}$ Let $f:X\to Y$ be a proper map of topological spaces. Then we can consider the basechange along any continuous map $g:Y'\to Y$. Denote by $X'=X\times_Y Y'$ the base ...
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1answer
82 views

What is the point of the topological space of a (locally) ringed space?

My main motivation is in trying to intuitively understand schemes. As I currently understand them, schemes are tools which allow us to recover points (in the traditional sense) as morphisms or encode ...
0
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1answer
25 views

Bijective $\mathcal{O}_X$-Module Homomorphisms

Let $(X,\mathcal{O}_X)$ be a ringed space, $\mathcal{F}$ and $\mathcal{G}$$\,\,\,$$\mathcal{O}_X$-modules, and $\varphi:\mathcal{F}\to\mathcal{G}$$\,$ an $\mathcal{O}_X$-module homomorphism. If $\...
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0answers
45 views

Determinant of a nonfree module

Is there a definition of a determinant which can be applied to a module with no basis? We can produce a module with noncommutative rings, without knowing a basis for these rings, i.e. without units. ...
6
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1answer
72 views

$X_f$ 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 ...
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0answers
18 views

Why adjunction appear to preserve stalk?

Question: Let $(f,f^{\flat}) \colon (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ be a morphism of locally ringed space. $f \colon X \to Y$ is a continuous map and $f^{\flat} \colon \mathcal{O}_Y \to f_* ...
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1answer
57 views

Definition of the sheaf $GL_n(\mathcal{O}_X)$ of invertible $\mathcal{O}_X-$linear functions

Let $(X, \mathcal{O}_X)$ be a ringed space. Is there such a thing as the sheaf of invertible linear functions $GL_n(\mathcal{O}_X)$? The point is that I cannot see how to define the restriction ...
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1answer
80 views

If restricted morphism of ringed spaces are equal, then they are actually equal

Given any two ringed spaces $(X, \mathcal O_X)$ and $(Y, \mathcal O_Y)$, let $\{ U_\lambda\}_{\lambda \in \Lambda}$ be an open covering of the topological space $X$. If $$f,g: (X, \mathcal O_X) \to (Y,...
4
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1answer
162 views

Locally free $\mathcal{O}_{X}$ modules are not projective

Let $(X , \mathcal{O}_{X})$ be a locally ringed space. I know that in general it is not true that locally free $\mathcal{O}_{X}$ modules are projective in the category of $\mathcal{O}_{X}$ modules (...
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1answer
50 views

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, ...
2
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1answer
80 views

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 ...
3
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0answers
67 views

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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0answers
45 views

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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0answers
141 views

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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1answer
120 views

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 $\mathcal{O}_X$...
0
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1answer
129 views

Description of tangent space via locally ringed space arrows from dual numbers

In the book Manifolds, Sheaves, and Cohomology by Wedhorn appears the following equality. $$\mathrm T_pX= \left\{ \substack{t:\mathrm{pt}[\varepsilon]\to M\text{ morphism of locally} \\\text{ringed ...
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0answers
542 views

What is the ideal sheaf of a closed subset of a scheme?

Let $(X, \mathcal{O}_{X})$ be a scheme and let $Y \subseteq X$ be a closed subset. What is meant by the "ideal sheaf of the closed subset $Y$"? Normally to define the ideal sheaf of a closed subscheme ...
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1answer
428 views

Universal properties of immersion of schemes

In this answer of Martin Brandenburg he refers to the universal properties of the open and closed immersion. As this looks kind of useful I tried to find a reference but I couldn't find any. So my ...
0
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1answer
43 views

Ways of approaching the study of a Hausdorff space $X$ which has the structure of a complex manifold.

In Holomorphic Functions of Several Variables by L. Kaup and B. Kaup, the authors have the following discussion in the opening of section 31, which concerns ringed spaces. If a Hausdorff space $X$ ...
2
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1answer
134 views

Relation between $\mathcal{O}_Y\rightarrow f_*\mathcal{O}_{X}$ and $f^{-1}\mathcal{O}_Y\rightarrow \mathcal{O}_X$ being epimorphism/monomorphism.

Let $f:(X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ be a morphism of ringed spaces. This is the data of a map $f:X\rightarrow Y$ between the topological spaces and either a morphism of sheaves $$f^{...
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0answers
121 views

Trying to understand vector bundles on manifolds via locally free sheaves

My background is primarily in algebra and topology/geometry with my primary interest lying in algebraic geometry. I am learning about locally free sheaves in the context of schemes, and they always ...
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0answers
51 views

Product of two analytic spaces

Let $X,Y$ be two analytic spaces. Does the product $X\times Y$ exist? (in the category of analytic spaces and the category of locally ringed spaces) I try to mimic the proof of schemes. We can do it ...
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1answer
182 views

introducing locally ringed space definition necessary at all?

Hartshorne defined locally ringed space $(X,O_X)$ as for every $p\in X$, $O_{X,p}$ stalk is local ring where $O_X$ is the structural sheaf over $X$. For $X=\operatorname{Spec}(A)$ for unital rings, $...
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0answers
37 views

Why is the group of sections of a sheaf of modules given by the group of morphisms of sheaves?

Let $(X, \mathcal{O}_{X})$ be a ringed space and let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X}$-modules. Why is it true that $$ \text{Hom}_{\mathcal{O}_{X}|_{U}} \left( \mathcal{O}_{X}|_{U} , \...
12
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1answer
250 views

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, ...
7
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1answer
457 views

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 ...
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0answers
74 views

What is the restriction of a locally ringed space to an open subset U?

Let $(X, \mathcal{O}_X)$ be a locally ringed space and $U\subset X$ an open subset. For the definition of a scheme one considers $(U, \mathcal{O}_X|_U)$. What is $ \mathcal{O}_X|_U$ formally? Is it ...
1
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1answer
237 views

Stalks and closed immersion

I have a very simple question regarding calculating the stalk of a certain sheaf: I refer to Proposition 2.24 (and also Lemma 2.23) of Liu's Algebraic Geometry. Suppose we have $$(f,f^\sharp):(Y,\...
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1answer
74 views

Is the adjunct of the comorphism for a closed immersion of ringed spaces an isomorphism?

Consider a closed immersion of ringed spaces $(i, i^\sharp): (Z, \mathcal{O}_Z) \to (M, \mathcal{O}_M)$. That is, $i: Z \hookrightarrow M$ is an embedding with closed image and $i^\sharp: \mathcal{O}...
17
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1answer
1k views

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 $...
2
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1answer
123 views

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 ...
2
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0answers
87 views

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 ...
3
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1answer
701 views

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 ...