Questions tagged [ringed-spaces]
For questions on ringed spaces or locally ringed spaces
0
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1answer
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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}$$\,$ an $\mathcal{O}_X$-module homomorphism.
If $\...
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0answers
32 views
A trivialization of line bundle is same as nonvanishing section
I am reading Lemma 17.22.10. My fundamental confusion is how is Nakayama Lemma applied.
Claim: Let $X$ be a ringed space. Assume that each $O_{X,x}$ is a local ring with maximal ideal $m_x$. Let $ ...
1
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0answers
25 views
Determinant of a nonfree module
Is there a definition of a determinant which can be applied to a module with no basis?
We can produce a module with noncommutative rings, without knowing a basis for these rings, i.e. without units. ...
6
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1answer
44 views
$X_f$ of locally ringed space $(X, O_X)$.
Let $(X, O_X)$ be a locally ringed space. $f \in \Gamma(X,O_X)$ be a global section.
$$X_f:= \{ x \in X \, ; \, f_x \text{ is invertible in } O_{X,x} \} $$
It is claimed that
$X_f$ is an open ...
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0answers
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Why adjunction appear to preserve stalk?
Question:
Let $(f,f^{\flat}) \colon (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ be a morphism of locally ringed space. $f \colon X \to Y$ is a continuous map and $f^{\flat} \colon \mathcal{O}_Y \to f_* ...
0
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1answer
37 views
Definition of the sheaf $GL_n(\mathcal{O}_X)$ of invertible $\mathcal{O}_X-$linear functions
Let $(X, \mathcal{O}_X)$ be a ringed space. Is there such a thing as the sheaf of invertible linear functions $GL_n(\mathcal{O}_X)$? The point is that I cannot see how to define the restriction ...
1
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1answer
70 views
If restricted morphism of ringed spaces are equal, then they are actually equal
Given any two ringed spaces $(X, \mathcal O_X)$ and $(Y, \mathcal O_Y)$, let $\{ U_\lambda\}_{\lambda \in \Lambda}$ be an open covering of the topological space $X$. If $$f,g: (X, \mathcal O_X) \to (Y,...
3
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1answer
65 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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0answers
40 views
What's the definition of the restriction of a sheaf of rings to a closed set?
For the ringed space $(\operatorname{supp}(\mathcal O_X/\mathcal I),(\mathcal O_X/\mathcal I)|_{\operatorname{supp}(\mathcal O_X/\mathcal I)})$, what's the definition of the restriction of $\mathcal ...
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1answer
40 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
77 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 ...
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0answers
55 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
33 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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38 views
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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61 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
41 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 $\mathcal{O}_X$...
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0answers
61 views
Is the unit of the adjunction $f^*\dashv f_*$ compatible with restriction?
Let $f\colon X\to Y$ be a morphism of ringed spaces (or schemes), and let $f^*\dashv f_*$ be the induced adjunction between the respective categories of sheaves of modules. Consider open subsets $U\...
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1answer
76 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.
$$\mathrm T_pX= \left\{ \substack{t:\mathrm{pt}[\varepsilon]\to M\text{ morphism of locally} \\\text{ringed ...
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0answers
337 views
What is the ideal sheaf of a closed subset of a scheme?
Let $(X, \mathcal{O}_{X})$ be a scheme and let $Y \subseteq X$ be a closed subset. What is meant by the "ideal sheaf of the closed subset $Y$"? Normally to define the ideal sheaf of a closed subscheme ...
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1answer
254 views
Universal properties of immersion of schemes
In this answer of Martin Brandenburg he refers to the universal properties of the open and closed immersion. As this looks kind of useful I tried to find a reference but I couldn't find any. So my ...
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1answer
38 views
Ways of approaching the study of a Hausdorff space $X$ which has the structure of a complex manifold.
In Holomorphic Functions of Several Variables by L. Kaup and B. Kaup, the authors have the following discussion in the opening of section 31, which concerns ringed spaces.
If a Hausdorff space $X$ ...
2
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1answer
107 views
Relation between $\mathcal{O}_Y\rightarrow f_*\mathcal{O}_{X}$ and $f^{-1}\mathcal{O}_Y\rightarrow \mathcal{O}_X$ being epimorphism/monomorphism.
Let $f:(X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ be a morphism of ringed spaces. This is the data of a map $f:X\rightarrow Y$ between the topological spaces and either a morphism of sheaves $$f^{...
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0answers
77 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
38 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 ...
1
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1answer
121 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
34 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
$$
\text{Hom}_{\mathcal{O}_{X}|_{U}} \left( \mathcal{O}_{X}|_{U} , \...
11
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1answer
159 views
Is there a notion of “schemeification” analogous to that of sheafification of a presheaf?
So this may seem like an odd question, but hear me out. In the Stacks Project, tag 01I4, we find that not only does the category of affine schemes live inside the category of locally ringed spaces, ...
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1answer
238 views
Details of gluing sheaves on a cover
I am sure this is a simple question, but I am really not able to think straight at the moment and this is bugging me. I am doing Exercise 1.22 from Hartshorne. It is the classic gluing of of sheaves ...
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0answers
63 views
What is the restriction of a locally ringed space to an open subset U?
Let $(X, \mathcal{O}_X)$ be a locally ringed space and $U\subset X$ an open subset.
For the definition of a scheme one considers $(U, \mathcal{O}_X|_U)$. What is $ \mathcal{O}_X|_U$ formally? Is it ...
1
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1answer
140 views
Stalks and closed immersion
I have a very simple question regarding calculating the stalk of a certain sheaf: I refer to Proposition 2.24 (and also Lemma 2.23) of Liu's Algebraic Geometry. Suppose we have
$$(f,f^\sharp):(Y,\...
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1answer
53 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}...
13
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1answer
781 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 $...
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1answer
92 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 ...
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82 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
494 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 ...
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0answers
109 views
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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1answer
92 views
Stalks of ringed space
Let $X$ be a locall ringed space (more narrowly a scheme, if you like) and $A=\Gamma(X,\mathcal{O}_X)$ its ring of global sections. Given a point $x\in X$, is there a prime ideal $p$ of $A$ such that $...
2
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0answers
54 views
Flatness of a counit for the inverse/direct image adjunction for a finite map of (locally Noetherian?) schemes
Let $f:X\to Y$ be a finite map of locally Noetherian schemes. In fact it's not clear to me whether the hypotheses of finiteness or local Noetherianity are ultimately relevant for my question (I would ...
2
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1answer
472 views
Minimal prime ideals of $\mathcal O_{X,x}$ correspond to irreducible components of $X$ containing $x$
Let $X$ be an algebraic variety over an algebraically closed field $K$. By definition, $X$ is a separated prevariety, and $x \in X$. I'm trying to show
(i): The minimal primes of $\mathcal O_{X,...
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1answer
270 views
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_{\...
2
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1answer
219 views
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 ...
3
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1answer
93 views
$\Psi$ 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 ...
3
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0answers
141 views
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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0answers
135 views
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 ...
6
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1answer
191 views
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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1answer
324 views
morphism betweem sheaves that is an isomorphism in local sections of a basis
Let $\{U_{\alpha}:\alpha \in A\}$ be a basis of open sets for the topological space $X$.
Let $\mathscr{F},\mathscr{G}$ be sheaves over $X$. Suppose that there exist a morphism $\phi: \mathscr{F} \to \...
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0answers
93 views
Ringed spaces isomorphism
Which of the following ringed spaces are isomorphic over $\mathbb{C}$?
(a) $\mathbb{A}^1\backslash\{1\}$
(b) $V(x_1^2+x_2^2)\subset \mathbb{A}^2$
(c) $V(x_2-x_1^2, x_3-x_1^3)\backslash \{0\} \...
2
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0answers
306 views
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 ...
0
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1answer
126 views
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 $f$? ...
4
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1answer
191 views
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),\...
