The concept of scheme mimics the concept of manifold obtained by glueing pieces isomorphic to open balls, but with different "basic" glueing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, spectrum of a ...

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60 views

Can Hom(X,Y) be given a natural scheme structure?

Let $X$ and $Y$ be $S$-schemes. Is there a "natural" scheme structure on $\operatorname{Hom}_S(X,Y)$, maybe subject to some conditions on the schemes in question? Interpret "natural" however you'd ...
5
votes
2answers
256 views

Closed immersion definition

Hartshorne defines a closed immersion as a morphism $f:Y\longrightarrow X$ of schemes such that a) $f$ induces a homeomorphism of $sp(Y)$ onto a closed subset of $sp(X)$, and furthermore b) the ...
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vote
1answer
116 views

Example of Gluing Schemes

Let $k$ be a field. $U_0 = \mathbb{A}^1_k = \operatorname{Spec}(k[T])$ and $U_1=\mathbb{A}^1_k = \operatorname{Spec}(k[S])$. $U_{01} = D(T) = \mathbb{A}^1_k\backslash \{0\} = ...
2
votes
0answers
44 views

Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
2
votes
0answers
128 views

Closed points of a scheme (locally) of finite type over an algebraically closed field

If $X$ is an arbitrary scheme, I can prove that the set $X(k)$ of $k$-valued points is in bijective correspondence with the set $$\{(x,\iota) \, | \, x \in X, \, \iota:\kappa(x) \hookrightarrow k\},$$ ...
3
votes
0answers
66 views

Noetherian schemes and varieties

What types of varieties (e.g. projective, affine,...) over a field $k$ (char = $0$) are Noetherian schemes?
6
votes
1answer
104 views

Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
5
votes
1answer
75 views

Existence of a morphism of schemes from $X$ to $\mathbb P_A^n$?

In exercise §II.III.VIII. of Algebraic geometry and arithmetic curves by Qing.Liu. one is asked the following: Let $X$ be a scheme over a ring $A.$ Let $f_0, \cdots,f_n\in\mathcal O_X(X)$ be ...
3
votes
0answers
142 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
2
votes
1answer
60 views

If $B$ is a graded $A$-algebra, then $\operatorname{Proj}B$ is an $A$-scheme

If $B$ is a graded ring, then for me is clear that $\operatorname{Proj}B$ with affine covering given by $D_+{(f)}\cong\operatorname{Spec} B_{(f)}$ is a scheme. The problem arises when $B$ is a graded ...
6
votes
1answer
256 views

When is the global section functor exact?

Given sheaves $\mathcal{F}_1,\mathcal{F}_2, \mathcal{F}_3$ on some scheme $X$, and an exact sequence $$ 0\rightarrow \mathcal{F}_1\rightarrow \mathcal{F}_2\rightarrow \mathcal{F}_3\rightarrow 0 ...
5
votes
1answer
96 views

Why is the preimage of a Spec $A$ the Spec of its integral closure in a morphism of curves?

Let $\phi:X\to Y$ be a morphism of nonsigular complete curves. Let Spec $A:=U\subset Y$ be an open set. Why is $\phi^{-1} (U) =$ Spec $B$, the spectrum of the integral closure of $A$ in $K(X)$? Can we ...
3
votes
1answer
64 views

About two isomorphic schemes

My question is related to an answer I read on MO: http://mathoverflow.net/questions/157973/classical-algebraic-varieties-vs-k-schemes-vs-schemes In the accepted answer, the user Julian Rosen claims ...
6
votes
1answer
151 views

From the residue field at a point to a scheme

Consider a scheme $X$; for every $x\in X$ with residue field $k(x)$, we have the canonical surjection $\mathcal O_{X,x}\longrightarrow k(x)$ that induces the morphism of affine schemes ...
6
votes
1answer
100 views

Example of nonempty scheme with no closed points

I know that when a scheme $X$ is quasicompact, every point has a closed point in it's closure. This of course means that every nonempty quasicompact scheme has a closed point. If we drop the ...
5
votes
2answers
248 views

What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
3
votes
2answers
344 views

Groups acting on schemes: the quotient scheme doesn't always exist.

Preliminary notion: Consider the action of a group $G$ on an object $X$ of some category $\mathcal C$. We have a group homomorphism $\rho:G\longrightarrow\operatorname{Aut}(X)$ which sends $g$ in ...
3
votes
1answer
49 views

$K$-rational points and algebraic sets

Notation: If $X$ is a $K$-scheme, then a point $x\in X$ is said $K$-rational if its residue field $k(x)=\frac{\mathcal O_{X,x}}{\mathfrak m_{X,x}}$ is isomorphic to $K$. The set of all $K$-rational ...
1
vote
1answer
58 views

Closed subsets and closed subschemes

Consider a scheme $(X,\mathcal O_X)$; a closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$ such that: $Z$ is a closed subset of $X$ There is a morphism of schemes ...
2
votes
1answer
142 views

Exercise 2.3 from Hartshorne's algebraic Geometry.

2.3) A scheme $(X,\mathcal{O}_X)$ is reduced if for every open set $U\subset X$, the ring $\mathcal{O}_(U)$ has no nilpotent element. b) Let $(X,\mathcal{O}_X)$ be a scheme. Let ...
3
votes
2answers
45 views

The two projection maps are different?

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..." Essentially my problem is the following ...
9
votes
2answers
483 views

How to think of the Zariski tangent space

The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. While I do appreciate this definition, I find it hard to work with, because we are not given ...
4
votes
2answers
68 views

The projective line is defined over $\mathbb Q$

Notations: A variety $X$ over a field $C$ is an integral $C$-scheme such that the structure morphism $p: X\longrightarrow \textrm{Spec } C$ is separated and of finite type. We say that a variety $X$ ...
8
votes
1answer
163 views

Global generation of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $\mathcal{E}$

I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: ...
1
vote
1answer
53 views

Dimension of the closure of points on a scheme

I would like to prove the following fact. Let $X$ be a scheme, and $x\in X$. Show that $\text{dim}(\mathcal{O}_{X,x})=\text{codim}(\bar{\{x\}},X) $, with $\bar{\{x \}}$ the closure of the subset ...
3
votes
1answer
60 views

Reducibility of a Hilbert scheme in projective space

My question concerns the computation of the Hilbert scheme $\mathsf{Hilb}_{3}^{2x+1}$, which parametrizes all curves of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ algebraically closed. ...
1
vote
1answer
81 views

Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
2
votes
1answer
52 views

affine morphism of schemes

Let $X$ be a smooth projective variety over a field $k$ and $\mathcal{E}$ a locally free sheaf of finite rank. Consider the total space ...
1
vote
2answers
78 views

In a quasicompact scheme every nonreduced point has a closed nonreduced point in its closure

Let $x\in X$ be a non reduced point. Then I can find a nilpotent $f\in \Gamma(U, \mathcal O_X)$, where $U$ is some open neighborhood of $x$. I also know that when $X$ is assumed to be quasicompact, ...
3
votes
0answers
52 views

Valuative criteria with varieties

Let $X$ be an algebraic prevariety over an algebraically closed field $k$ (I.e. an integral scheme of finite type over $k$). In the valuative criteria for separatedness and properness of $X$ over ...
2
votes
0answers
77 views

Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
3
votes
2answers
109 views

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 ...
2
votes
2answers
202 views

Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
3
votes
2answers
182 views

Example of an affine scheme where closed points aren't dense.

I'm looking for an example of an affine scheme where closed points aren't dense. It's easy to show (using Hilbert's Nullstellensatz) that if $A$ is a finitely generated algebra over a field, then the ...
2
votes
1answer
88 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
4
votes
1answer
159 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
3
votes
1answer
323 views

pull-back and push-forward of quasi-coherent sheaves on affine schemes

Let $f:Y\to X$ be a map of affine schemes, where $X=\text{Spec}A$ and $Y=\text{Spec}B$. Let $M,N$ be modules over $A$ and $B$, respectively. I know the following three facts: The functors $f^{*}$ ...
1
vote
1answer
58 views

Associated sheaf to a $\mathbb{C}$-module

In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A. Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. ...
4
votes
0answers
26 views

Is a nonzero divisor locally nonzero divisor?

Suppose $X$ is a scheme (or locally ringed space), $a\in \Gamma (X,O_X)$, not a zero divisor. Can we show $a_x$is also a nonzero divisor in $O_{X,x}$? (It is true if $X$ is an affine scheme)
2
votes
1answer
88 views

Finite fiber- unramified morphisms

I'm in trouble understandig the proof of Proposition 3.2 Chapter 1 of Milne's Book "Étale Cohomology". Let $f:Y\rightarrow X$ be locally of finite-type. The following are equivalent. $(a)$ f is ...
4
votes
1answer
99 views

Two definitions of Scheme theoretic image

Suppose $f:X \to Y $ is a morphism. I saw two definitions of scheme theoretic image. The first one requires $f$ to be quasi-compact and quasi-separated, or quasi-compact, which ensures the kernel ...
6
votes
1answer
153 views

Morphisms from a proper scheme to the affine line over a field must be constant.

Let $k$ be a field. Let $X$ be a (non empty) connected proper $k$-scheme. I would like to prove the maximum principle, that is for any $k$-morphism $\varphi \colon X \to \mathbb A_k^1$, the image of ...
2
votes
1answer
51 views

Universally closed morphism and closed morphism

If $f:X\longrightarrow Y$ is a universally closed morphism of schemes (that is the map obtained by base extension is closed), then does it imply $f$ is closed? Or, is the assumption of $f$ being ...
0
votes
1answer
66 views

Representability of transformations by morphisms

This is a very naive question. Suppose I'm given two functors $F,G\colon(Schemes/S)^{op}\to (Sets)$, over some base scheme $S$, represented by $S$-Schemes $X_F$ and $X_G$ respectively. There are ...
3
votes
2answers
107 views

Field automorphisms and varieties

Let $C$ be an algebraically closed field and consider a variety $X$ over $C$. In the language of schemes $X$ is a separated, integral scheme over $C$ with a morphism of finite type $f:X\longrightarrow ...
1
vote
1answer
38 views

An action of an automorphism of a field $C$ on a variety $X$ over $C$

My question comes from the reading of the first two pages of the article "Bernhard Köck - Belyi's Theorem Revisited". Consider a field $C$ and a variety $X$ over $C$. In particular $X$ is a ...
3
votes
2answers
83 views

Separated prevarieties and schemes

As it is shown in "Goertz,Whedorn - Algebraic geometry I" there is an equivalence of categories between the category of prevarieties over $k$ (field algebraically closed) and the category of integral ...
2
votes
1answer
184 views

Sum of Homogeneous Ideals vs Homogeneous Ideal of Intersection

Sorry for the recent question spam, but I've been striving furiously over the last few days to solve a problem in Hartshorne (Algebraic Geometry) discussed in this question. The problem (II.8.4a) ...
1
vote
1answer
46 views

Image of the diagonal map in a scheme

My question is similar to the question in The image of the diagonal map in scheme. I saw a hint to my question in the comments, but I was not able to prove it. Let $f:X\longrightarrow Y$ is a ...
6
votes
0answers
73 views

For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...