Questions tagged [ringed-spaces]
For questions on ringed spaces or locally ringed spaces
146 questions
0
votes
1
answer
88
views
What is Grothendieck's subsheaf $\mathcal{F}$ of $\underline{\mathbb{Z}}$ with $\mathcal{F}_0 = \{0\}$ and $\mathcal{F}_x = \mathbb{Z}$ else?
Let $\newcommand{\cO}{\mathcal{O}}(X, \cO_X)$ be a ringed space. In EGA 1, Chapter 0, item 5.1.1 Grothendieck defines what it means for an $\cO_X$-module $\newcommand{\cF}{\mathcal{F}}\cF$ to be ...
- 6,937
0
votes
0
answers
52
views
Clarification: Stalks of the structure sheaf of an affine scheme (and more generally, stalks of a sheafification)
Given a presheaf $\mathcal{F}$ of abelian groups on a space $X$, let us take the definition of the sheafification as in Hartshorne (compatible germs). Now, one way to show that the stalks are ...
- 337
0
votes
0
answers
31
views
Example of a direct sum of injective sheaves which is not flabby
Let $X$ be a scheme, $(I_k)_{k \in K}$ is a family of injective $O_X$-modules, here $K$ is an infinite set. Is there an example of $X$ and such family that $\oplus_{k \in K} I_k$ is not a flabby sheaf?...
- 6,589
1
vote
0
answers
33
views
Checking the definition of morphism of locally ringed spaces using adjunction between pushforward and inverse image sheaves
A morphism of locally ringed spaces $(X,\mathcal{O}_X)\xrightarrow{}(Y,\mathcal{O}_Y)$ consists of a continuous map $f:X\xrightarrow{}Y$ between the underlying topological spaces and a sheaf ...
- 104
3
votes
0
answers
50
views
Zariski tangent space of an infinite product of Lie groups
I am trying to understand the proof of the following lemma in Maret's Character varieties:
Lemma 2.3.2. The Zariski tangent space to $G^\Gamma$ at any point identifies with $\mathfrak g^\Gamma$.
...
- 1,573
0
votes
1
answer
127
views
Qing Liu's Algebraic Geometry Exercise 3.1.6
The exercise states:
Let $\pi : T\rightarrow S$ be a morphism of schemes. Supposing that $\pi : T\rightarrow S$ is an open or closed immersion, or that
$S=$Spec$A$, and $\pi $ is induced by a ...
- 134
1
vote
1
answer
70
views
Manifolds as ringed spaces and values of functions
I am currently reading a book on supergeometry which also introduces sheaves and ringed spaces, and in particular it proposes a definition of a differential manifold as a locally ringed space $(M,\...
- 33
1
vote
1
answer
68
views
Group actions on ringed topological spaces
Letting $(X,\mathcal{O}_{X})$ be a ringed topological space, and $G$ a group of automorphisms on $X$, I'm confused on how an element $g\in G$ is supposed to act on an abelian group $\mathcal{O}_{X}(U)$...
- 134
1
vote
1
answer
40
views
Locality of closed immersions of schemes
Let $(\varphi,\varphi^\#):(X,O_X)\to (Y,O_Y)$ be a morphism of schemes. Let $V\subset Y$ be an open set, and $U:=\varphi^{-1}(V)$. Choosen $x\in U$, the diagram below commutes: $\require{AMScd}$
$$\...
- 439
1
vote
1
answer
134
views
Correspondence between sheaves of ideals and closed immersion
Let $(X,\mathcal O_X)$ be a locally ringed space.
Let $\mathcal J$ be a sheaf of ideals over $\mathcal O_X$. Then we can construct a closed immersion of locally ringed spaces: $$(Z(\mathcal J),i^{-1}(\...
- 439
1
vote
1
answer
51
views
Direct image of locally ringed space
Let $(X,\mathcal{O}_X)$ be a ringed space and $Y$ be topological space and $f: X \to Y$ a continuous map. Then $(Y,f_*\mathcal{O}_X)$ is obviously a ringed space, where $f_*\mathcal{O}_X$ is the ...
- 4,584
0
votes
1
answer
60
views
Definition of a Prevariety - Specifically a covering by ringed spaces
I'm reading Gathmanns's lecture notes on Algebraic Geometry where he defines a Prevariety as ringed space $V$ which has a finite open cover of affine varieties $U_i$. I assume this not just a set-...
2
votes
1
answer
66
views
Closed immersion of locally ringed spaces vs closed immersion determined by ideal sheaf
Let $f: X \to Y$ be a closed immersion of locally ringed spaces, that is,
$f$ is a homeomorphism onto a closed subset of $Y$,
$f^{\#}:\mathscr{O}_Y \to f_*\mathscr{O}_X$ is surjective,
$\mathscr{I} = ...
- 55
0
votes
1
answer
101
views
Do morphisms of locally ringed spaces which agree on stalks agree on the whole of the locally ringed space?
Let $f,g:X\rightarrow Y$ be morphisms of locally ringed spaces, such that the topological maps $f$ and $g$ are identical, and the maps of sheaves $f^\sharp,g^\sharp:\mathcal{O}_Y\rightarrow f_*\...
- 4,163
0
votes
0
answers
64
views
Let $(X,\mathcal{O}_X)$ be a locally ringed space, does there exists a monomorphism $\iota:U\rightarrow X$ for all open $U$?
Let $(X,\mathcal{O}_X$ be a locally ringed space, and suppose that $U\subset X$ is an open set equipped with the sheaf of rings $\mathcal{O}_X|_U$ defined by $\mathcal{O}_X|_U(V)=\mathcal{O}_X(V)$ for ...
- 4,163
0
votes
0
answers
76
views
Defining algebraic varieties in general
I have encountered two general notions of algebraic variety when reading different texts in algebraic geometry, and wanted to ask whether they were equivalent or whether one is stronger than another.
...
- 404
1
vote
1
answer
67
views
If $\mathfrak{p}_y$ the preimage of $\mathfrak{m}_x$ by $\mathcal{O}_Y(Y)\rightarrow\mathcal{O}_X(X)\rightarrow \mathcal{O}_{X,x}$, show that $f(x)=y$
I'm trying to solve the following problem:
Let $(X,\mathcal{O}_X)$ be a locally ringed space and $Y$ an affine
scheme. Let $f:X \rightarrow Y$ be a morphism of locally ringed
spaces. Then, for any $x\...
- 355
0
votes
1
answer
136
views
Pullback of structure sheaf along point inclusion into locally ringed space is residue field?
I'm coming from a differential geometry background (though I'm pretty familiar with category theory) and trying to learn a bit about ringed spaces.
Let $(X,\mathscr{O}_X)$ be a locally ringed space, ...
- 11.1k
2
votes
0
answers
139
views
Closed subvariety is a subvariety
I am attempting to solve Exercise 4.4.7 from this note:
Given a variety $X$ and a closed subset $Y \subseteq X$ equipped with the induced topology. For $V \subset Y$ open define
$$\mathcal{O}_Y(V)=\{f:...
- 608
0
votes
1
answer
50
views
Closed subspace cut off by the ideal sheaf coming from a locally closed immersion
$\def\sO{\mathcal{O}}
\def\sI{\mathcal{I}}$Given a locally ringed space $Y$ and an ideal sheaf $\sI\subset\sO_Y$, we can consider the closed subspace of $Y$ cut off by $\sI$, i.e., the closed ...
3
votes
0
answers
78
views
When can $\textbf{SpecMax}(R)$ be a scheme?
Let $R$ be a commutative ring, and let $(\text{Spec(R)},\mathcal{O}_R)$ be the affine scheme associated to $R$.
Let $\text{SpecMax}(R)$ the subspace the $\text{Spec(R)}$ of the maximal ideals.
My ...
0
votes
0
answers
73
views
In what sense the "pullback at $ x $" map if functorial?
Let $ X $ and $ Y $ be differentiable manifolds, and let $ f\colon X\to Y $ be a smooth map.
Given $ x\in X $ one can define the canonical pullback at $ x $ map
$$
f_x^*\colon \mathscr C_{Y,f(x)}^\...
0
votes
1
answer
81
views
A surjecive homomorphism of $ \mathbb R $-algebras
The main question
Let $ A $ and $ B $ be two algebras over the real numbers, and let $ J $ be an ideal of $ B $. Let $ f\colon A\to B $ be a homomorphism of $ \mathbb R $-algebras, and suppose $ f^{-1}...
3
votes
0
answers
66
views
More examples of morphisms of ringed spaces that aren't local?
$\def\Spec{\operatorname{Spec}}$All questions and answers that I've found in MSE regarding a morphism of ringed spaces between affine schemes that isn't a morphism of locally ringed spaces are the ...
0
votes
1
answer
141
views
Characterization of isomorphisms of ringed spaces
Let $ (X,\mathscr O_X) $ and $ (Y,\mathscr O_Y) $ be ringed spaces over the same unspecified commutative ring.
My book defines a morphism between $ (X,\mathscr O_X) $ and $ (Y,\mathscr O_Y) $ as a ...
1
vote
1
answer
168
views
Subvarieties of an abstract affine variety
I'm really confused on how to transfer constructions from "concrete" affine varieties (i.e. zeroes of polynomial equations in an affine space) and (abstract) affine varieties (i.e. ringed ...
- 3,324
2
votes
1
answer
202
views
Is there a morphism of ringed spaces between smooth (resp., complex) manifolds that is not local?
$\def\bbC{\mathbb{C}}
\def\sO{\mathcal{O}}
\def\hom{\operatorname{Hom}}
\def\rs{\mathsf{RS}}
\def\lrs{\mathsf{LRS}}
\def\swf{\mathsf{SWF}}
\def\k{\operatorname{K}}
\def\ent{\mathrm{ent}}
\def\spec{\...
0
votes
1
answer
69
views
Do restrictions preserve ring structure of ringed space?
Let $(X, \mathcal{O})$ be a ringed space, i.e. $X$ is a topological space and $\mathcal{O}$ is a sheaf of rings on the open subsets of $X$.
I would like to show that for two global sections $a, b\in \...
- 1,239
0
votes
1
answer
36
views
"Punctured stalk" of a locally ringed space (at a closed point) is the fraction field of the stalk?
Let $(X,\mathcal{O})$ be a locally ringed space. For $x \in X$ a closed point (i.e. $\{x\}$ is closed in $X$), let $\mathcal{O}_x$ denote the stalk of $\mathcal{O}$ at $x$; and define the "...
- 2,753
1
vote
1
answer
169
views
On a ringed space: If a section has zero germ at $x$, must it be zero on some neighborhood of $x$?
Let $F$ be a sheaf of commutative rings or Abelian groups on a topological space $X$, let $x \in X$ be a point, let $U$ be an open neighborhood of $x$ in $X$.
Let $f \in F(U)$, and suppose the germ $...
- 2,753
0
votes
0
answers
53
views
Locally, every coherent sheaf is isomorphic to the cokernel of a homomorphism $\phi: \mathcal{A}^q \to \mathcal{A}^p$
Say, we have a topological space $X$ and a sheaf $\mathcal{F}$ over a sheaf of rings $\mathcal{A}$ on $X$.
The sheaf $\mathcal{F}$ is said to be coherent, if the following two conditions are satisfied:...
- 569
0
votes
1
answer
64
views
The radical presheaf is not a sheaf
$\def\sO{\mathcal{O}}
\def\sI{\mathcal{I}}
$Let $(X,\sO_X)$ be a ringed space. Let $\sI\subset\sO_X$ be an ideal sheaf. We define the radical presheaf of $\sI$, denoted $\sqrt[p]{\sI}$, as the ...
1
vote
1
answer
218
views
Is a morphism from a quasi-affine variety to a quasi-projective variety given by globally defined regular maps?
$\def\bbA{\mathbb{A}}
\def\bbP{\mathbb{P}}
\def\sO_{\mathcal{O}}$The following discussion is strictly classical. Throughout this question, I will use the notions of (i) sheaf of $k$-algebras, (ii) $k$-...
0
votes
2
answers
149
views
How can I think about a morphism of locally ringed spaces?
A locally ringed space is a pair $(X,\mathcal{O}_X)$ of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$. Then we say that $(f,f^b):(X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is a ...
- 2,028
1
vote
1
answer
131
views
Is the module sum presheaf a sheaf?
$\def\O{\mathcal{O}}
\def\M{\mathcal{M}}
\def\N{\mathcal{N}}
\def\P{\mathcal{P}}
$Given a ringed space $(X,\O{_X})$, an $\O_X$-module $\P$ and $\mathcal{O}_X$-submodules $\M,\N\subset\P$ we define the ...
0
votes
0
answers
100
views
function on structure sheaf and its values at points
I have some trouble understanding the concept of a function in Vakil's lecture notes.
First, let $X = Spec(A)$ for some ring $A$. A function $f$ is a section in $\mathcal{O}_X(X)$, thus an element of ...
- 569
1
vote
1
answer
68
views
Existence of morphism from a locally ringed space $X \to Spec(\mathbb{F}_p)$
Let $X$ be a topological space, such that $(X, \mathcal{O}_X)$ is locally ringed. Let $A$ be a ring. We showed in the lecture that there is a natural bijection of $Hom((X, \mathcal{O}_X),(Spec(A), \...
- 569
0
votes
1
answer
114
views
Surjectivity of $\mathscr G\to f_*f^{-1}\mathscr G$
Let $f:X\to Y$ be a continuous map, and let $\mathscr G$ be a sheaf on $Y$. There is a canonical morphism $\varphi:\mathscr G\to f_*f^{-1}\mathscr G$, hence a map $\varphi_y:\mathscr G_y\to f_*f^{-1}\...
- 47
0
votes
1
answer
58
views
Function vanishing on ringed spaces
In Vakil's notes on locally ringed spaces, he claimed that "we can't even make sense of the phrase of 'function vanishing' on ringed spaces in general. "
Could someone explain what this ...
- 385
0
votes
1
answer
208
views
Morphisms of algebraic varieties are regular?
I want to understand a proof that establishes the fact that every map between abstract algebraic varieties (ie, a ringed space on k-algebras which is locally isomorphic to a Zariski closed on the ...
- 59
1
vote
1
answer
46
views
For a locally ringed space $(X,\mathcal{O})$, is there a sheaf of ideals whose stalks are the max ideals of the stalks of $\mathcal{O}$? [closed]
For $X$ a locally ringed space with structure sheaf $\mathcal{O}$, for each $x \in X$
let $\mathcal{M}_x$ denote the max ideal of the stalk $\mathcal{O}_x$ of $\mathcal{O}$ at $x$.
In general, does ...
0
votes
1
answer
64
views
Is the forgetful functor from locally ringed spaces to topological spaces a full functor? Faithful? What about when restricted to schemes?
In detail, given locally ringed spaces $X,Y$ with underlying topological spaces $X_0,Y_0$,
can every continuous map $f_0 : X_0 \rightarrow Y_0$ lift to a morphism of ringed spaces $f : X\rightarrow Y$...
- 2,753
2
votes
1
answer
291
views
Is the "sheaf of derivations" locally free?
Let $k$ be a field. We require all algebras to be associative commutative, and when unital we require morphisms between them to respect the identity element.
Let $X$ be a topological space, equipped ...
- 2,753
0
votes
1
answer
137
views
Constructing a "sheaf of vector fields" for a flasque sheaf of $k$-algebras
Let $k$ be a field. We require all algebras to be associative and commutative. Unital algebra morphisms are required to preserve the multiplicative identity.
Let $\mathcal{O}$ be a sheaf of unital $k$...
- 2,753
3
votes
0
answers
40
views
Does the "sheaf of diffeologically-smooth real-valued functions" functor reflect isomorphisms?
(This is a follow-up question to this earlier one.)
Setup: let $u : \mathrm{Diff} \rightarrow\mathrm{Set}$ denote the forgetful functor on the category of smooth manifolds.
Let $\tilde{X} \subset
\...
- 2,753
0
votes
0
answers
183
views
Definition of restriction of morphism of ringed space?
I have a question. Is there a definition of restriction of morphism of (locally) ringed space?
Let $(f,f^{\flat}): X \to Y$ bea morphism of ringed spaces ; i.e., $f:X\to Y$ is a continuous map and $f^{...
- 3,072
0
votes
1
answer
104
views
Well-definedness of ring operations on stalks
Let $(X, \mathcal{O}_X)$ be a locally ringed space. As a sanity check for myself, I'd like to show that the addition and multiplication of the germs at a point $p \in X$ are well-defined. I was able ...
- 625
3
votes
0
answers
139
views
Is the ideal product presheaf a sheaf?
Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\mathcal{I},\mathcal{J}\subset\mathcal{O}_X$, we define the ideal product presheaf $\mathcal{I}\cdot_p\mathcal{J}$ as the ideal presheaf
$$
...
1
vote
1
answer
366
views
A detail in the proof that tensor product of sheaves of $\mathcal{O}_X$-modules commutes with pullback
Given a morphism of ringed spaces $f:(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$ and $\mathcal{O}_Y$-modules $\mathcal{M}$ and $\mathcal{N}$, here it is proven that
$$
f^*(\mathcal{M} \otimes_{\mathcal{O}...
1
vote
1
answer
162
views
Stacks Project proof that gluing locally ringed spaces which happen to be schemes gives a scheme
I'm currently reading the Stacks Project section on gluing schemes. I can understand the proof of Lemma 01JB, but it is hard for me to understand the proof of Lemma 01JC. By constructions of the ...
- 334