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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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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238 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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$K$-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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106 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 $|f^{-1}(y)|\le n$? ...
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136 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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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 ...
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47 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 ...
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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 ...
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185 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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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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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 ...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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113 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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Example of nonempty scheme with no closed points

I know that when a scheme $X$ is quasicompact, every point has a closed point in it's closure. This of course means that every nonempty quasicompact scheme has a closed point. If we drop the ...
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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 ...
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Groups acting on schemes: the quotient scheme doesn't always exist.

Preliminary notion: Consider the action of a group $G$ on an object $X$ of some category $\mathcal C$. We have a group homomorphism $\rho:G\longrightarrow\operatorname{Aut}(X)$ which sends $g$ in ...
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$K$-rational points and algebraic sets

Notation: If $X$ is a $K$-scheme, then a point $x\in X$ is said $K$-rational if its residue field $k(x)=\frac{\mathcal O_{X,x}}{\mathfrak m_{X,x}}$ is isomorphic to $K$. The set of all $K$-rational ...
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Closed subsets and closed subschemes

Consider a scheme $(X,\mathcal O_X)$; a closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$ such that: $Z$ is a closed subset of $X$ There is a morphism of schemes ...
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Exercise 2.3 from Hartshorne's algebraic Geometry.

2.3) A scheme $(X,\mathcal{O}_X)$ is reduced if for every open set $U\subset X$, the ring $\mathcal{O}_(U)$ has no nilpotent element. b) Let $(X,\mathcal{O}_X)$ be a scheme. Let ...
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44 views

The two projection maps are different?

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..." Essentially my problem is the following ...
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340 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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The projective line is defined over $\mathbb Q$

Notations: A variety $X$ over a field $C$ is an integral $C$-scheme such that the structure morphism $p: X\longrightarrow \textrm{Spec } C$ is separated and of finite type. We say that a variety $X$ ...
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Global generation of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $\mathcal{E}$

I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: ...
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1answer
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Dimension of the closure of points on a scheme

I would like to prove the following fact. Let $X$ be a scheme, and $x\in X$. Show that $\text{dim}(\mathcal{O}_{X,x})=\text{codim}(\bar{\{x\}},X) $, with $\bar{\{x \}}$ the closure of the subset ...
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Reducibility of a Hilbert scheme in projective space

My question concerns the computation of the Hilbert scheme $\mathsf{Hilb}_{3}^{2x+1}$, which parametrizes all curves of degree $2$ and genus $0$ in $\mathbb{P}^{3}_{k}$, with $k$ algebraically closed. ...
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1answer
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Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
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In a quasicompact scheme every nonreduced point has a closed nonreduced point in its closure

Let $x\in X$ be a non reduced point. Then I can find a nilpotent $f\in \Gamma(U, \mathcal O_X)$, where $U$ is some open neighborhood of $x$. I also know that when $X$ is assumed to be quasicompact, ...
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Valuative criteria with varieties

Let $X$ be an algebraic prevariety over an algebraically closed field $k$ (I.e. an integral scheme of finite type over $k$). In the valuative criteria for separatedness and properness of $X$ over ...
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Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
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On a *ringed space*, show that the non vanishing set of $f$ is open, and that it is invertible there

This is an exercise of Ravi Vakil that I solved by a very trivial argument without using the hint. For this reason I'm worried that I might have missed something. If $f$ is a function on a locally ...
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Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
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Example of an affine scheme where closed points aren't dense.

I'm looking for an example of an affine scheme where closed points aren't dense. It's easy to show (using Hilbert's Nullstellensatz) that if $A$ is a finitely generated algebra over a field, then the ...
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1answer
79 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
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Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
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pull-back and push-forward of quasi-coherent sheaves on affine schemes

Let $f:Y\to X$ be a map of affine schemes, where $X=\text{Spec}A$ and $Y=\text{Spec}B$. Let $M,N$ be modules over $A$ and $B$, respectively. I know the following three facts: The functors $f^{*}$ ...
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Associated sheaf to a $\mathbb{C}$-module

In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A. Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. ...
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Is a nonzero divisor locally nonzero divisor?

Suppose $X$ is a scheme (or locally ringed space), $a\in \Gamma (X,O_X)$, not a zero divisor. Can we show $a_x$is also a nonzero divisor in $O_{X,x}$? (It is true if $X$ is an affine scheme)
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Finite fiber- unramified morphisms

I'm in trouble understandig the proof of Proposition 3.2 Chapter 1 of Milne's Book "Étale Cohomology". Let $f:Y\rightarrow X$ be locally of finite-type. The following are equivalent. $(a)$ f is ...
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Two definitions of Scheme theoretic image

Suppose $f:X \to Y $ is a morphism. I saw two definitions of scheme theoretic image. The first one requires $f$ to be quasi-compact and quasi-separated, or quasi-compact, which ensures the kernel ...
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Morphisms from a proper scheme to the affine line over a field must be constant.

Let $k$ be a field. Let $X$ be a (non empty) connected proper $k$-scheme. I would like to prove the maximum principle, that is for any $k$-morphism $\varphi \colon X \to \mathbb A_k^1$, the image of ...
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Normal cone to a closed subscheme

Suppose $S$ is a closed subscheme of a smooth variety $M$ and that its ideal sheaf $\mathscr{I}_S$ factors as $\mathscr{I}_S=\mathscr{I}_{S_1}\cdot \mathscr{I}_{S_2}$, where $S_1$ and $S_2$ are closed ...
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1answer
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Universally closed morphism and closed morphism

If $f:X\longrightarrow Y$ is a universally closed morphism of schemes (that is the map obtained by base extension is closed), then does it imply $f$ is closed? Or, is the assumption of $f$ being ...