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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0
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1answer
43 views

The structure morphism of a projective variety induces a morphism of $k$-algeras

Suppose that $k$ is an algebraically closed field and that $X=\textrm{Proj}{\frac{k[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}}$ is a projective variety with a structural morphism $p:X\rightarrow\textrm{Spec} ...
15
votes
2answers
699 views

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
3
votes
1answer
46 views

Surjective étale morphisms on points. [closed]

Let $X$ and $Y$ be schemes over a field $K$. We assume, moreover, $X$ and $Y$ to be of finite type, separated and geometrically integral. Let $f:X \rightarrow Y$ be a surjective étale morphism. Is it ...
2
votes
1answer
89 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
4
votes
1answer
181 views

Hartshorne ex III.10.2 on smooth morphisms

I need some help with the following exercise: Let $f:X\rightarrow Y$ be a flat proper morphism between varieties over $k$, where variety means separated, finite type, integral, and $k$ not ...
3
votes
0answers
83 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
0
votes
1answer
74 views

Non-Noetherian ring $R$ with Spec($R$) a Noetherian Scheme

In looking at the examples of Non-Noetherian rings I knew/found I wasn't able to find one where I could conclude that Spec($R$) was a Noetherian scheme (not just merely a Noetherian topological ...
1
vote
0answers
17 views

Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...
4
votes
1answer
109 views

Is the global section ring of a Noetherian Scheme Noetherian as well?

As the title suggests, I am asked to prove that, given a Noetherian scheme $(X,\ \mathcal{O}_{X})$ and any open subset $U\subseteq X$, $\Gamma(U,\ \mathcal{O}_{X}):=\mathcal{O}_{X}(U)$ is a Noetherian ...
1
vote
0answers
54 views

What is the opposite of category of schemes

The category of schemes sits between $Aff=CRing^\text{op}$ and its free cocompletion $\hom(Aff^\text{op},Sets)$, as the locally representable functors. Dualizing, we have $CRing\subseteq ...
1
vote
0answers
103 views

Compare étale morphisms in Top and Schemes

Perhaps because I find the algebraic definition of étale morphisms (of schemes) hard to approach, I'm trying to bootstrap from my understanding of étale morphisms of topological spaces. In topological ...
5
votes
1answer
65 views

quasicompact scheme are finite union of affine scheme

This is a problem from Liu`s book. Show that a scheme $X$ is quasi-compact if and only if it is a finite union of affine schemes. If the scheme is quasicompact then it is obviously a finite union of ...
1
vote
1answer
58 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
0
votes
0answers
50 views

Sheaf of ideals [duplicate]

Let $(X, \mathcal{O}_X)$ be a scheme and and $f \in \mathcal{O}_X(X)$. We define the sheaf of ideals by the assignment $$ U \longmapsto f|_U \mathcal{O}_X(U) $$ and denote this sheaf by ...
0
votes
0answers
22 views

Equivariant Sheaves, Local system

Let $(G,m)$ be a group scheme unipotent, and $L$ a local system of rank 1 on $G$ such that: $m^*(L) \simeq L \boxtimes L $. Then why is $L$ an equivariant sheaf on $G$ with the action the ...
3
votes
2answers
211 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
2
votes
0answers
46 views

Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
2
votes
0answers
42 views

Question on morphism locally of finite type

The exercise 3.1 in GTM 52 by Hartshorne require to prove that $f:X \longrightarrow Y$ is locally of finite type iff for every open affine subset $V=\text{Spec}B$, $f^{-1}(V)$ can be covered by open ...
2
votes
1answer
38 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
0
votes
0answers
39 views

Generic points and base change

Let $X$ be a scheme over a field $k$. Let $K$ be a field extending $k$. What can we say about the generic points of $X_K$ with respect to the generic points of $X$? In particular I suspect that doing ...
1
vote
1answer
34 views

Graph of a morphism between two $K$-schemes: an open covering

Consider a separated morpshism $f:X\longrightarrow Y$ between two $K$-schemes ($K$ is a field). The graph of $f$ is the image of the morphism $(Id_X,f):X\longrightarrow X\times_{\operatorname {spec}K} ...
4
votes
1answer
85 views

Closed points of a fibred product of k-schemes

This question comes from Shafarevich, Chapter V.4, Let $X$ and $Y$ be schemes over an algebraically closed field $k$. Show that the correspondence $ u \to (p_x(u),p_y(u)) $ establishes a 1-1 ...
1
vote
2answers
31 views

Affine open sets of a scheme after a base change

Consider a $k$-scheme $X$ where $k$ is a generic field. If we have a field extension $k\subseteq K$, then we can construct the fibered product $X^K:=X\times_{\operatorname{Spec}k}\operatorname{Spec} ...
0
votes
0answers
18 views

Is the image of Spec$k$ closed under these conditions?

Let $k$ be an algebraically closed field. Let $\phi:\textrm{Spec}\ k\longrightarrow X$ be a morphism of schemes of finite type. Let us denote the only point of $\textrm{Spec }k$ by $\zeta$, and the ...
7
votes
2answers
575 views

Functor of points and category theory

I am trying to read the section on Functor of points from Eisenbud - Harris (and I also referred to Mumford's book). They all motivate functor of points this way : In general, for any object $Z$ of a ...
1
vote
1answer
63 views

Projective varieties are the common zeros of some homogeneous polynomials

definition: A projective variety over a field $k$ is a closed subscheme (over $k$) of $\mathbb P^n_k=\operatorname{Proj}(k[T_0,\ldots,T_n])$. Now it can be proved (I have done it) that if ...
4
votes
1answer
75 views

Nonexistence of Morphisms between Schemes of Differing Characteristic

So I'm new to this whole scheme theory business. I'm working my way through Gortz and I produced a solution to an exercise but it seemed too easy. I'm hoping someone can either tell me that I am ...
1
vote
1answer
75 views

Weak Kodaira Vanishing - Hartshorne III.7.1

In the Serre Duality section of Algebraic Geometry by Robin Hartshorne, the following exercise is posed: If $X$ is an integral projective scheme over a field $k$, prove that an ample invertible sheaf ...
1
vote
1answer
54 views

Quotient of group schemes and its rational points.

At the moment I have some difficulties in understanding the quotient of group schemes and so exact sequences. I am aware that precise answers would be difficult to be given without speaking of sheaves ...
4
votes
1answer
166 views

Is the fiber product of the connected component of a group scheme connected?

Let $G$ be a group scheme over a field $k$. Let $G^0$ be the connected component containing the identity. Is it true that $G^0\times_k G^0$ is connected? I know that this is true if $G^0$ is ...
1
vote
0answers
44 views

Check a non-projective morphism

The following is from the wiki: http://en.wikipedia.org/wiki/Proper_morphism Projective morphisms are proper, but not all proper morphisms are projective. For example, it can be shown that the ...
1
vote
0answers
28 views

The Chow variety of conics in $\mathbb{P}^{3}_{k}$

Consider the family of all conics in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Such curves all have degree $2$ and genus $0$, and they can be uniquely defined to be all curves in ...
3
votes
1answer
39 views

Counting the dimension of a component of $\mathsf{hilb}^{2t+1}_{3}$

Consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, parametrizing varieties of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Consider the component $ ...
3
votes
0answers
65 views

Hilbert Scheme and Chow variety in the case of Conics in $\mathbb{P}^{3}$

My question concerns the relationship between chow varieties and hilbert schemes in the case of conics in $\mathbb{P}^{3}_{k}$. More precisely, consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, ...
4
votes
4answers
86 views

Morphism between two $K$-schemes restricted to an affine subscheme

Suppose that $f:X\longrightarrow Y$ is a morphism between two $K$-schemes. If $U\subseteq X$ is an affine open set, then can we conclude that $f(U)$ is contained is some affine open subset of $Y$? ...
3
votes
1answer
41 views

Complexifications of degree 3 subschemes in $\mathbb A^2_{\mathbb R}$

I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes ...
3
votes
1answer
96 views

Motivating examples of Spec(R) where R is not a finitely generated k-algebra

A scheme is a ringed space locally isomorphic to an affine scheme. An affine scheme is the spectrum of a commutative unital ring together with a structure sheaf. What are examples of "interesting" ...
2
votes
1answer
73 views

Geometric reducedness (integral) versus reducedness (integral)

All the schemes here are over $\mathbb{C}$. Suppose $X \to Y$ is a morphism of varieties, then the geometric reducedness (integral) of the generic fibre implies the geometric reducedness (integral) ...
3
votes
1answer
39 views

A scheme with not-numerable affine covering.

This question maybe finds the answer in the definition. Let $X$ be a scheme with not hypothesis on it (not noetherian, not affine,... ). Can I find a scheme that has a not numerable affine covering? ...
3
votes
1answer
72 views

Confused with $\mathbb Q$-rational points

Look at the following enlightened part of a proof extracted from a paper (here $C$ is an algebraically closed field of characteristic $0$): Clearly $\mathbb Q\subset C$ and $\mathbb P^1_C$ is ...
2
votes
1answer
367 views

global sections of structure sheaf of a projective scheme X over a field k?

Some may have already asked this question. What are the global sections of structure sheaf of a projective scheme $X$ over a field $k$? By Hartshorne page 18, Chapter 1, Theorem 3.4, global sections ...
6
votes
2answers
87 views

$K$-schemes as varieties: the importance of the structural morphism

Consider a variety $p:X\longrightarrow\operatorname{Spec K}$ where $X$ is an integral scheme and $p$ is a separated morphism of finite type. Now chose an element ...
2
votes
1answer
110 views

Morphisms between schemes such that every point in the codomain has at most $n$ preimages.

Consider a finite morphism $f:X\longrightarrow Y$ between two integral and Noetherian schemes. If $\operatorname {deg}(f)=[K(X):K(Y)]=n$, is it true that for every $y\in Y$ then $|f^{-1}(y)|\le n$? ...
3
votes
1answer
157 views

What is a field of definition of a morphism?

An algebraic variety $X$ over a field $K$ is a $K$-scheme integral separated and of finite type over $K$. Definition: If $L$ is a subfield of $K$ we say that $X$ is defined over $L$ if there ...
5
votes
2answers
252 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
3
votes
1answer
50 views

Group Scheme - equivalent ways of defining it

I know that a group scheme is an $S$-scheme $G$ equipped with one of the equivalent sets of data a triple of morphisms $μ$: $G$ ×S $G$ → $G$, $e$: $S$ → $G$, and $ι$: $G$ → $G$, satisfying the ...
3
votes
0answers
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
252 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 ...
1
vote
1answer
115 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
42 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 ...