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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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 $G$. ...
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1answer
115 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 ...
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57 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 ...
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105 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 ...
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72 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 ...
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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 ...
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52 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 ...
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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 ...
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48 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 ...
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43 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 ...
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1answer
39 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 ...
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1answer
35 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} ...
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1answer
97 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 ...
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2answers
32 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} ...
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21 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 ...
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671 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 ...
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1answer
65 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 ...
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1answer
77 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 ...
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1answer
83 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 ...
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1answer
56 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 ...
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1answer
178 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 ...
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47 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 ...
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31 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 ...
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1answer
43 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 $ ...
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74 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 $\mathsf{hilb}^{2t+1}_{3}$, ...
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4answers
91 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 $Y$? ...
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1answer
42 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
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1answer
99 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
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1answer
76 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) ...
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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
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1answer
75 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 ...
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1answer
443 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 ...
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2answers
95 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 ...
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1answer
111 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
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1answer
179 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 ...
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2answers
301 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
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1answer
51 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
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0answers
62 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
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2answers
303 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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1answer
125 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\} = ...
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0answers
50 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
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0answers
148 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 $$\{(x,\iota) \, | \, x \in X, \, \iota:\kappa(x) \hookrightarrow k\},$$ ...
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73 views

Noetherian schemes and varieties

What types of varieties (e.g. projective, affine,...) over a field $k$ (char = $0$) are Noetherian schemes?
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1answer
106 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
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1answer
79 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
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0answers
149 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
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1answer
61 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
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1answer
284 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
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1answer
100 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
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1answer
65 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 ...