Questions tagged [schemes]

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

An affine scheme $X$ is a locally ringed space that is isomorphic to $\mathrm{Spec}R$, which is the spectrum of a commutative ring $R$. That is, for our commutative ring $R$, the closed subsets of $X$ correspond to the ideals of $R$, with the points of $X$ corresponding to prime ideals. Then $X$ being a locally ringed space means that it's equipped with a structure sheaf $\mathcal{O}_X$ that assigns to each open set $U$ the ring of regular functions on $U$.

A scheme then, is a locally ringed space that admits a open covering $\{U_i\}$ such that each $U_i$ is an affine scheme.

2441 questions
21
votes
2 answers

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 an isomorphism from $\mathfrak m/\mathfrak m^2$…
Rodrigo
  • 6,874
  • 18
  • 46
20
votes
2 answers

Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be essentially nonexistent. I think that I understand…
user101616
20
votes
2 answers

Why should faithfully flat descent preserve so many properties?

This question is based on the following proposition (EGA IV, 2.7.1) Let $f: X \rightarrow Y$ be a $S$-morphism of $S$-schemes, $g: S'\rightarrow S$ a faithfully flat and quasi-compact morphism. Denote $X \times_S S'$ by $X'$, and denote $Y \times_S…
only
  • 2,586
  • 16
  • 26
19
votes
2 answers

Closed points of a scheme correspond to maximal ideals in the affines?

Let $X$ be a scheme. If $x\in X$ is a closed point, then it corresponds to a maximal ideal in $\mathcal{O}_X(U)$ for some affine open subset $U\subseteq X$. If I take an arbitrary open affine $\mbox{Spec}~A$ in $X$ and a maximal ideal $x$ in $A$,…
Gregor Botero
  • 3,147
  • 3
  • 21
  • 33
18
votes
2 answers

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 233-413 (July 1969). One can find the article…
Babai
  • 4,726
  • 3
  • 23
  • 58
18
votes
2 answers

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 category $\mathcal{X}$, the association…
gradstudent
  • 3,154
  • 16
  • 32
18
votes
1 answer

Why é​t​a​l​e​?

Background: The notion of an étale morphism has proved itself to be ubiquitous within the realm of algebraic geometry. Apart from carrying a rich intuitive idea, it is the first ingredient in notions and theories such as étale cohomology, a Galois…
user554397
18
votes
3 answers

Learning schemes

Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and arithmetic curves" from Qing Liu. I have had…
Jaska
  • 1,251
  • 10
  • 18
18
votes
1 answer

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what I've been concentrating on recently which has been…
Alex Saad
  • 3,259
  • 11
  • 31
17
votes
2 answers

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 induced map $f^\#:\mathcal{O}_X\longrightarrow f_*…
gradstudent
  • 3,154
  • 16
  • 32
17
votes
1 answer

What does Liu mean by "topological open/closed immersion" in his book "Algebraic Geometry and Arithmetic Curves"?

In his book "Algebraic Geometry and Arithmetic Curves", Liu defines open/closed immersions of locally ringed spaces in terms of topological open/closed immersions: What does he mean by the terms "topological open (resp. closed) immersion"? Does he…
user350031
  • 1,730
  • 6
  • 18
17
votes
2 answers

Where do Chern classes live? $c_1(L)\in \textrm{?}$

If $X$ is a complex manifold, one can define the first Chern class of $L\in \textrm{Pic}\,X$ to be its image in $H^2(X,\textbf Z)$, by using the exponential sequence. So one can write something like $c_1(L)\in H^2(X,\textbf Z)$. But if $X$ is a…
17
votes
2 answers

Varieties as schemes

Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne. Firstly, my main question. I understood that Grothendiecks introduction of schemes revolutionized the subject. Just out of curiosity,…
Joachim
  • 5,055
  • 26
  • 50
17
votes
1 answer

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that satisfies $\mathcal{F}(U) = \underset{{x\in U}}{…
Saal Hardali
  • 4,449
  • 2
  • 28
  • 77
16
votes
3 answers

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 that pouring through Hartshorne hasn't turned up a…
Jim
  • 29,407
  • 2
  • 51
  • 88