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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516 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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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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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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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 ...
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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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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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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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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 ...
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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 ...
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'$R$-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 $X(k)$ ...
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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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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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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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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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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 ...
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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, ...
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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 ...
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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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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 ...
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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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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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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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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 ...
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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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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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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? ...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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$K$-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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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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341 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 ...
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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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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 ...
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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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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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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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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 ...
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Exercise on locally ringed spaces

I try to solve exercise 2.19 on page 65 in "Algebraic Geometry I" by U.Görtz/T.Wedhorn. The exercise reads: Let $(X,O_X)$ be a locally ringed space, and $f\in O_X(X)$. Define $X_f:=\{ x\in X; f(x) ...