Mathematics Stack Exchange
Mathematics Stack Exchange

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.

150 questions
22
votes
1 answer

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
  • 689
16
votes
2 answers

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
  • 7,996
16
votes
1 answer

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
  • 4,761
16
votes
2 answers

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
  • 1,258
14
votes
1 answer

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
  • 2,227
13
votes
1 answer

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
  • 275
12
votes
1 answer

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
  • 16,589
10
votes
3 answers

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
  • 629
10
votes
2 answers

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
  • 35,843
8
votes
2 answers

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
  • 5,679
  • 1
  • 26
  • 57
8
votes
1 answer

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
  • 2,536
7
votes
5 answers

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
  • 7,155
7
votes
0 answers

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 are somehow natural or interesting? Edit: perhaps one…
J. T. B.
  • 121
7
votes
1 answer

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 (this post on mathoverflow gives a counterexample on…
Federico
  • 1,815
  • 12
  • 20
7
votes
0 answers

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
  • 44,216
1
2 3
…
9 10