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.

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Diophantine applications of Spec?

Let $f(\bar x)$ be a multivariable polynomial with integer coefficients. The zeros of that polynomial are in bijection with the homomorphisms $\mathbb Z[\bar x] \rightarrow \mathbb Z$ that factor ...
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1answer
558 views

Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
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943 views

Intuition for étale morphisms

Currently working on algebraic surfaces over the complex numbers. I did a course on schemes but at the moment just work in the language of varieties. Now i encounter the term "étale morphism" every ...
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414 views

Can an integral scheme have closed points of both positive and zero characteristic?

Background Recall that an integral scheme $X$ is a scheme which is both irreducible and reduced; equivalently, its ring of functions is an integral domain on every open subset. Given any point $p$, ...
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Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f: X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where ...
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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 ...
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What are normal schemes intuitively?

A ring is called integrally closed if it is an integral domain and is equal to its integral closure in its field of fractions. A scheme is called normal if every stalk is integrally closed. Some ...
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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 ...
12
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409 views

Why do we need noetherianness (or something like it) for Serre's criterion for affineness?

Serre's criterion for affineness (Hartshorne III.3.7) states that: Let $X$ be a noetherian scheme. Suppose $H^1(X, \mathcal{F})= 0$ for every quasi-coherent sheaf on $X$. Then $X$ is affine. ...
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460 views

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 ...
10
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141 views

Is the product of non-separated schemes non-separated?

My question is the title, but let me be more specific: for schemes $X$ and $Y$ over $S$, with at least one non-separated over $S$, is it true that the fibered product $X\times_S Y$ is also not ...
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253 views

Fibres in algebraic geometry: multiplicity

Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory. My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X ...
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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 ...
9
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92 views

'$R$-rational points,' where $R$ is an arbitrary ring

On page 49 of Liu's Algebraic Geometry and Arithmetic Curves, we find Example 3.32. In it, he shows that if $k$ is a field and $X=k[T_1,\dots,T_n]/I$ is an affine scheme over $k$, the sections $X(k)$ ...
9
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1answer
271 views

Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq ...
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1answer
291 views

Problem about Complete Intersection in $\textbf P^n$ (from Hartshorne).

I am in trouble with Exercise 8.4 in Hartshorne's Chapter II; I am doing part (a). It is about (global) complete intersection in $\textbf P^n$. For those without Hartshorne' book at hand, I describe ...
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Classification of automorphisms of projective space

Let $k$ be a field, n a positive integer. Vakil's notes, 17.4.B: Show that all the automorphisms of the projective scheme $P_k^n$ correspond to $(n+1)\times(n+1)$ invertible matrices over k, modulo ...
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669 views

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. ...
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490 views

Why are locally closed subschemes not open subschemes of closed subschemes?

Ravi Vakil gives the following argument for why open subschemes of closed subschemes are locally closed: "Clearly an open subscheme U of a closed subscheme V of X can be interpreted as a closed ...
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3answers
256 views

Krull Dimension of a scheme

Can someone give a hint or a solution for showing that a scheme has Krull dimension $d$ if and only if there is an affine open cover of the scheme such that the Krull dimension of each affine scheme ...
8
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1answer
549 views

Affine scheme $X$ with $\dim(X)=0$ but infinitely many points

As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ...
8
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165 views

Hom between 2 schemes

Why is the set $Hom(X,Y)$ between 2 schemes $X$ and $Y$ a scheme as well? Where can I read the construction? For example, $Hom(\mathbb{A}^1,\mathbb{A}^1)$ is the set of all polynomial, and what is the ...
8
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1answer
360 views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
8
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1answer
293 views

The spectrum of a product of rings

Let $A$ be the product of a family $(A_i)_{i\in I}$ of commutative rings, and $c$ the canonical continuous map from the disjoint union $U$ of the spectra of the $A_i$ to the spectrum of $A$: $$ ...
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Computing Kähler differentials on $\mathbb P^1_A$ (What did Liu intend?)

Let $A$ be a ring and $X=\mathbb P^1_A$. On $D_+(T_0)\cap D_+(T_1)$, we have $$T^2_0d(T_1/T_0)= -T^2_1d(T_0/T_1).$$ How does one formally compute this transition? Of course this is ...
8
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137 views

Deducing that a locally ringed space is a scheme

I would like to know if the following is true: Let $\mathscr{F}$ and $\mathscr{G}$ be two sheaves on a topological space $X$ and let $\varphi:\mathscr{F}\rightarrow\mathscr{G}$ be a morphism such that ...
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211 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
8
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173 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
7
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423 views

Why is $\mathbb{A}^2$ isomorphic to $\operatorname{Spec}k[x,y]$ as ringed spaces

Suppose that $k$ is an algebraically closed field, and let $\mathbb{A}^2$ denote the affine $2$-space $k^2$. An affine scheme is defined to be a locally ringed space $(X, \mathcal{O}_X )$ which is ...
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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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Scheme of finite type over a field $K$ v.s. $K$-scheme

I'm lost in some definitions about schemes. I have some trouble about two definitions of a scheme of finite type over $K$, for an alg.closed field $K$. Version 1 (Hartshorn) : a scheme of finite type ...
7
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208 views

Inverse of open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that ...
7
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1answer
189 views

Comparison between etale and singular cohomology for a singular variety

In his Lectures on Etale Cohomology Milne proves in Theorem 21.1, that for all $r\geq 0$ $$ H^r_{\acute{e}t}(X,\Lambda)\cong H^r(X_{cx},\Lambda) $$ with $X$ a nonsingular $\mathbb{C}$-variety and ...
7
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Quasicoherent ideal sheaves on open subschemes

Let $X$ be an open subscheme of an affine scheme $\operatorname{Spec} A$, let $f : X \to \operatorname{Spec} A$ be the inclusion, and let $\mathscr{I}$ be a quasicoherent ideal sheaf on $X$. Since ...
7
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1answer
125 views

Definition of prime ideal sheaf

In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf": Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let ...
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Irreducibility of the space of divisors on a curve

Let $X$ be a smooth projective and irreducible curve over a field $k$. Further, define $$ X_d = \{ \text{ Effective Cartier divisors of degree } d \text{ on } X \;\} $$ and $$ W_d = \{ \text{ Line ...
7
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1answer
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Restricting a Weil divisor to a local scheme

Apologies if the title is garbage. I couldn't think up anything more pertinent, as it's really just a notational question. In Hartshorne, p.141, Proposition 6.11, it is shown that Weil divisors are ...
7
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1answer
250 views

divisor class group of a product of schemes

The first part of this question is quite general: let $X$ and $Y$ be noetherian integral separated schemes which are regular in codimension one. Is there any relationship between the divisor class ...
7
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244 views

Spectral sequence for Ext

If I have a morphism of schemes $f:X\rightarrow Y$ and sheaves $\mathcal F,G$ on $X$, then is there a spectral sequence which relates the Ext-groups $\mathrm{Ext}(f_* \mathcal F, f_*\mathcal G)$ on ...
6
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306 views

Why is every Noetherian zero-dimensional scheme finite discrete?

In the book The geometry of schemes by Eisenbud and Harris, at page 27 we find the exercise asserting that Exercise I.XXXVI. The underlying space of a zero-dimensional scheme is discrete; if ...
6
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1answer
638 views

Why does the definition of an open subscheme / open immersion of schemes allow for an “extra” isomorphism?

After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time: ...
6
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2answers
69 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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401 views

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 ...
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How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
6
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1answer
183 views

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if ...
6
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461 views

on the generic points of a scheme

This question may be a little bit metaphysical:are there any important properties about the generic points on a scheme?Or rather,why do we introduce the concept of generic point?I am not very clear ...
6
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1answer
83 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 ...
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198 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 ...
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257 views

bijective morphism of affine schemes

The following question occurred to me while doing exercises in Hartshorne. If $A \to B$ is a homomorphism of (commutative, unital) rings and $f : \text{Spec } B \to \text{Spec } A$ is the ...
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1answer
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Have I got the right definition of formal smoothness?

I'm trying to work out a basic example where formal smoothness should fail. I'm considering $\mathbb{R} \to \mathbb{R}[x,y]/(x^2-y^2)$. The idea is that not every $\mathbb{R}$-homomorphism ...