The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.

learn more… | top users | synonyms

1
vote
1answer
50 views

Why is : $ \ \ V \times_U W = p^{-1} (V) \bigcap q^{-1} (W) $?

Let $ f: Y \to X $ and $ g : Z \to X $ be two morphisms of schemes. Suppose we know that $ Y \times_X Z $ exists, and let $ p $ dénote its projection on $ Y $, and $ q $ its projection on $ Z $. ...
0
votes
2answers
116 views

Associated points and reduced scheme

1) Let X is a locally Noetherian scheme without embedded point, show that X is reduced iff it is reduced at the generic points. 2) Let X is a locally Noetherian scheme (maybe has some embedded ...
1
vote
1answer
63 views

A question on smooth morphisms and 'pointwise' smooth morphisms

Let $X$ be a scheme, $x\in X$ a point and $f\colon \operatorname{Spec}(k(x))\to X$ the canonical morphism. Is $f$ always a smooth morphism? Now suppose $g\colon X\to Y$ is a scheme over some ...
2
votes
1answer
63 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
1
vote
1answer
77 views

Finite pushforward commute with taking cohomology

Let $f: X \to Y$ be a finite morphism of schemes. How one can show that $f_*H^i(G) \cong H^i(f_* G)$ for any $G \in D(X)$ and any $i \in \mathbb{Z}$? In english, $G$ is a complex of quasi-coherent ...
0
votes
1answer
34 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} ...
14
votes
2answers
591 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
42 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
83 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" ...
3
votes
1answer
157 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
78 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
42 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 ...
0
votes
0answers
62 views

Grothendieck topology

Hello : Here is a small parapgraph that i try to understand : If $ M $ is a smooth manifold, then we can recover the underlying set of $ M $ by considering the set $ \mathrm{Hom} ( \{ \star \} , ...
1
vote
0answers
12 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
89 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
51 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 ...
0
votes
0answers
98 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
59 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
57 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
47 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
32 views

Unipotent Group Scheme

Is this definition correct: An smooth unipotent group scheme $G$ over a perfect field $k$, with $\operatorname{char}(k) >0$ , is isomorpic to the affine scheme $k[x_1,...x_n]$.
0
votes
0answers
16 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
205 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
39 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
37 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
36 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
37 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
32 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
67 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
29 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
17 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
448 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 ...
0
votes
0answers
34 views

A doubt in affine schemes

Consider the affine scheme (Spec($A$),$O_{Spec(A)}$). where $A$ is a commutative ring. Then I know (from hartshrone page 71) that $\Gamma (Spec(A),$ $O_{Spec(A)}$) is isomorphic to $A$ . I also know ...
1
vote
1answer
59 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
70 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 ...
0
votes
1answer
56 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
51 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
151 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
42 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
36 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
60 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
79 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 ...
2
votes
1answer
89 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
69 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) ...
0
votes
0answers
22 views

Connectedness in Unipotent Groups

Are all subgroups of an unipotent group over a finite field (as a scheme) connected?
3
votes
1answer
37 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
70 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
233 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 ...