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

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42 views

Tangent space of an algebraic variety $ X $ in a point $ x \in X $.

Let $X$ be an algebraic variety, and $ \mathcal{O}_{X,x} $ its local ring at a point $x \in X$, and $ \mathfrak{m}_x $ its maximal ideal. Let set $ k_x = \mathcal{O}_{X,x} / \mathfrak{m}_x $ the ...
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1answer
69 views

When is “number of points in the fiber” semicontinuous?

Let $f:X \rightarrow Y$ be a finite morphism of schemes. For $y \in Y$ let $n(y)$ be the cardinality of the set-theoretic fiber $f^-1(y)$. This is finite since $f$ is finite. I think that the ...
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29 views

Global section on product of sheafifications

Say I have two presheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules on a scheme $X$. Let $\mathcal F^+\!, \mathcal G^+\!$ be their respective sheafifications. We then have a commitative ...
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38 views

Simple question about closed immersions

I would like to know if I understand closed immersions correctly. Let $f:X \rightarrow Y$ be a morphism of schemes (or locally ringed spaces in general). The morphism $f^\sharp:\mathscr{O}_Y ...
2
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1answer
70 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
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52 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ ...
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1answer
57 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
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1answer
39 views

Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...
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1answer
48 views

Lifting of partial section

Does someone know if the following is true, and if so could you provide a reference. Let $X$ be a smooth $S$-scheme, let $Z$ be a closed subscheme of $S$, and let $t : Z \to X$ be an $S$-morphism. ...
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85 views

Uniqueness of the structure sheaf

Given a ring $A$ and his Spectrum $X=Spec(A)$ seen as a topological space with the Zariski topology, it's possible to build a sheaf on $X$ satisfying the conditions $O_X(X)=A$ $O_X(X_f)=A_f$ where ...
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28 views

Elementary transformation of vector bundles - equivalent definition

The background is as follows: Let $S$ be a locally noetherian scheme, $E$ a vector bundle over $S$ of rank $N+1$. Let $X=\mathbb{P}(E)$ be the projective bundle and $\pi:X\longrightarrow S$ be the ...
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52 views

Irreducible component of a scheme over a DVR

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a DVR (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
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49 views

Are Prevarieties irreducible?

In Goertz-Wedhorn, a prevariety is defined to be a connected space with functions that locally is an affine variety (were an affine variety is a space with functions that is isomorphic to the space ...
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2answers
457 views

What *is* affine space?

In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$: $\mathbb{A}_k^n$ is $k^n$ 'without ...
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2answers
87 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
3
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1answer
43 views

Zero-section as homomorphism of rings

Let $s : X \to E$ be the zero section of a vector bundle $E$ over a scheme $X$. Zariski-locally this corresponds to a homomorphism $Sym_A(M) \to A$ of $A$-algebras where $M$ is a finitely generated ...
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35 views

Discriminant ideal and short exact sequence of finite group schemes

Let $0 \to G' \to G \to G'' \to 0$ be a short exact sequence of finite flat commutative group schemes over a Dedekind domain $\mathcal O$ with field of fractions of characteristic zero. Let ...
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46 views

Birational map between reduced schemes.

I am reading Ravi Vakil's notes "Foundation of Algebraic Geometry" and in Proposition 6.5.5., it states the following: Suppose $X$ and $Y$ are reduced schemes. Then $X$ and $Y$ are birational if and ...
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52 views

Gluing schemes: Tips and tricks.

Like many other people I have talked to, I always find checking the cocycle condition quite hard and messy. I notice that most books avoid showing explicitly that the cocycle condition is satisfied, ...
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29 views

Invertible Prime in a Noetherian Scheme

What means the following: Let $\ell$ be a prime, and $X$ a Noetherian Scheme, on wich the prime $\ell$ is invertible. The reference is the appendix of "Weil Conjectures, Perverse Sheaves and l’adic ...
3
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1answer
71 views

Translation from schemes to varieties

At the moment, I know very little algebraic geometry (sadly!) so I apologise for the silliness/stupidity of these questions. Set up: Let $k$ be a field (not necessarily algebraically closed). Take ...
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39 views

Intersection of affine open subschemes

If X is separated scheme, then for any affine opens U, V, the intersection is affine. If X is quasi-separated, then the intersection can be covered by finitely many affine opens. Is there some ...
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1answer
54 views

Ideal sheaf is aquasi-coherent sheaf of ideals.

Let $X$ be a scheme, For any closed subscheme $Y$ of $X$, the corresponding ideal $I_Y$ given by the kernel of the morphism $i^{\#}:\mathcal{O}_x\rightarrow i_{*}\mathcal{O}_y$ is quasi-coherent sheaf ...
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51 views

Kodaira dimensions

I am learning about schemes, specifically abstrac varieties over $\mathbb{C}$ and I want to read a good book or reference where explain the kodaira dimensions, somebody can help me?
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1answer
63 views

Two definitions of coherent sheaf

There are at least two ways to define a (quasi)coherent sheaf on the scheme $(X,\mathcal{O}_X)$, namely the one given in Hartshorne's "Algebraic geometry" (I guess you know it: it is given in terms of ...
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1answer
56 views

Confusion about affine schemes

Let $\varphi:A \rightarrow B$ be a ring morphism. Consider the corresponding map of schemes, $f: Spec\ B \rightarrow Spec\ A$. This map $f$ is given by $\varphi ^{-1},$ since prime ideals pull back ...
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1answer
56 views

Hartshorne generically finite morphisms (II, 3.7)

I have a question concerning one of the exercises of Hartshorne, Ch. II. Namely: Exercise 3.7 about gerneically finite morphisms. A morphism $f: X \rightarrow Y$ with Y irreducible and $\eta$ ...
2
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1answer
38 views

Closed subscheme of an affine scheme.

This is a portion of the exercise 2.3.11 (b) of Hartshorne. Suppose $Y$ is a closed subscheme of $X=\operatorname{Spec}A$. I would like to show that there are $f_{i}\in A$ so $D(f_{i})$ cover $X$ and ...
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0answers
52 views

Computing the cotangent complex: what's the ring?

As far as I understand, deformation theory of schemes may be calculated via the cotangent complex. I have read that in general the cotangent complex may be difficult to compute. However, I have a ...
4
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54 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
5
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1answer
49 views

Quick question: could someone clarify me the notion of “$k$-points” of a scheme?

Let $X$ be a scheme, and $k$ be a field. What does it mean by $X(k)$? The reason I am asking this is that there seems to be several definition(or convention) on it, and I can't see why they lead to ...
3
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1answer
37 views

$\overline{\mathbb{Q}}$-valued points of a $\mathbb{C}$-scheme

I've just read the definition of a $T$-valued point on an $S$-scheme from Ravi Vakil's notes (i.e. the morphisms $T \to S$), and something about the definition doesn't seem to make sense intuitively. ...
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57 views

Example 2.3.4 in Hartshorne algebraic geometry

I am reading Hartshorne algebraic geometry. Example 2.3.4. Let $k$ be an algebraically closed field, and consider the affine plane over $k$, defined as $A^2_k = \text {Spec} k[x,y]$ . The closed ...
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0answers
26 views

Classifying all $E$-torsors up to isomorphism?

If $0\to E\to F\xrightarrow{\phi}\mathbb{A}^1_X\to 0$ is an exact sequence of bundles over the scheme $X$, then it is known that if $s\colon X\to \mathbb{A}^1_X$ is the constant section sending ...
3
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1answer
85 views

definition of singular locus of a variety

Given a variety $X$ over $k$, we can consider which points are regular, and we can define the singular locus $\operatorname{Sing}(X)$ as the complement of the regular points in $X$. My question is, ...
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1answer
45 views

Finite covering by finitely generated B-algebras

While I was working on the first properties of schemes on Hartshorne's Books, I needed the following result that I couldn't prove it: Let $X=Spec(A)$ be an affine scheme. Suppose that there exist an ...
2
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1answer
64 views

Show that $\mathbb{A}_\mathbb{C}^2 \ncong \mathbb{A}_\mathbb{C}^1 \times_{Spec(\mathbb{Z})} \mathbb{A}_\mathbb{C}^1$

Show that $\mathbb{A}_\mathbb{C}^2 \ncong \mathbb{A}_\mathbb{C}^1 \times_{Spec(\mathbb{Z})} \mathbb{A}_\mathbb{C}^1$ Honestly I don't know where to begin... It's the same as proving that ...
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2answers
39 views

If $X=Spec(R)$ and $Y=Spec(S)$ are affine schemes, then the disjoint union $X \sqcup Y$ is an affime scheme with $X \sqcup Y = Spec(R \times S)$

Let $R,S$ be commutative rings with identity. Proving that $X \sqcup Y$ is an affine scheme is the same as proving that $Spec(R) \sqcup Spec(S) = Spec(R \times S)$. I proved that if $R,S$ are rings, ...
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0answers
32 views

Why is this scheme $Y$ an affine bundle?

Suppose you have $F$ a subbundle of a vector bundle $E\to X$ over a scheme $X$. Recall that there is a scheme $\varphi\colon Y\to X$ with the points of $Y$ over the point $x\colon\mathrm{spec}(A)\to ...
4
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2answers
112 views

What are the closed points of $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$?

I am trying to find all the closed points of $\mathbb{A}_{\mathbb{R}}^2$. After a quick google research, I found that $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$ and then all ...
3
votes
1answer
141 views

The assignment $R\mapsto\operatorname{Iso}_{R\text{-alg}}(A\otimes_k R,M_n(R))$ is a scheme?

Let $A$ be a central simple algebra over some field $k$, with degree $n$. There is a functor $F$ defined by the assignment, for a commutative ring $R$, $$ ...
4
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1answer
60 views

Separated scheme stable under base extension.

Given a separated scheme morphism $X\to Y$, and a morphism $Z \to Y$, Hartshorne proves that the extension $X\times_YZ \to Z$ is also separated, as long as the schemes involved are Noetherian. The ...
2
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1answer
60 views

Why are equivariant morphisms of $G$-torsors necessarily isomorphisms?

This was something I read on the Stacks project, but whose proof was omitted. Simply stated, if $f\colon E\to F$ is a $G$-equivariant morphism of $G$-torsors over a scheme $X$, why is $f$ ...
3
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2answers
48 views

Scheme whose points over $x\colon\mathrm{spec}(R)\to X$ are the isomorphisms $x^*(F)$ and $x^*(E)$?

If one has two vector bundles $E\to X$ and $F\to X$ over a scheme $X$, why is there a scheme $S$ over $X$ with points of $S$ over a point $x\colon\mathrm{spec}(R)\to X$ is precisely the set of ...
1
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1answer
36 views

For $0\to F\to E\stackrel{\varphi}{\to} L\to 0$, why is the pullback under $\varphi$ of a constant section an $F$-torsor?

I think the following is used in classifying $F$-torsors. Let's take $0\to F\to E\stackrel{\varphi}{\to} L\to 0$ to be an exact sequence of vector bundles, all over a scheme $S$, where ...
3
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0answers
50 views

Confusion over common criterion for a line bundle on a scheme to be trivial?

Suppose you have a line bundle $L\to X$ on a scheme $X$. I'll denote by $\sigma\colon X\to L$ to be the zero section. Why is it that this bundle is trivial iff there is another section $s\colon X\to ...
2
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1answer
44 views

Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings?

A brief question, if $\pi\colon E\to X$ is a vector bundle over a scheme $X$, is it automatic that any section $s\colon X\to E$ always a closed embedding?
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25 views

Basis for a topology of a scheme

Suppose that $X$ is a proper and connected scheme over an algebraically closed field. Moreover let $\mathcal A$ be a collection of open subsets of $X$ with the following property: For every open ...
2
votes
1answer
88 views

Extending a morphism of schemes

This question is an exercise 2.4 p.96 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves". Let $X$, $Y$ be schemes over a locally Noetherian scheme $S$, with $Y$ of finite type over $S$. ...
2
votes
1answer
49 views

Subscheme of projective space in general position

Let $k$ be a field and let $\mathbb{P}^n(k)$ denote $n$-dimensional projective space over $k$. What is meant by a general linear space in $\mathbb{P}^n(k)$ of codimension $m$, in the language of ...