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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206 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, ...
2
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1answer
48 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
66 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
53 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
33 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
votes
2answers
129 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
142 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
votes
1answer
72 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
votes
1answer
111 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
votes
2answers
51 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
vote
1answer
38 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
53 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
votes
1answer
47 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?
2
votes
1answer
112 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
52 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 ...
4
votes
1answer
86 views

Adjoints functors in scheme theory

What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ ...
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0answers
53 views

What is a smooth family of divisors?

Suppose that $S$ is a smooth complex projective surface ($\mathbb C$-scheme, reduced, irreducible...). What do algebraic geometers usually mean with the term a smooth family of divisors in $S$? ...
3
votes
0answers
83 views

A computation related to Hironaka's Example

My questions are at the very end, first I'll describe the context. Let $f:\mathbb{P}^3\to \mathbb{P}^3$ be an involution whose fixed locus consists of two disjoint lines $L, L' \subset \mathbb{P}^3$. ...
5
votes
2answers
215 views

How is the notion of adjunction of two functors usefull?

Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ? ...
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0answers
25 views

Flat Morphisms of schemes

When we suppose a morphism $f:X\rightarrow Y$ of schemes to be a flat, what are the fundamental consequences that come directly to the spirit ? thanks
0
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1answer
68 views

Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
5
votes
2answers
149 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
1
vote
1answer
58 views

What is the class group of the complement of three lines in the projective plane?

I have a straightforward question : Let $ Y$ be the union of the three lines $ L_1:x=0 , L_2 :y=0$ and $L_3:z=0$ in the Projective plane $\mathbb{P }^2$. What is the Class group of the Complement ...
4
votes
1answer
79 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...
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0answers
42 views

Etale locally free sheaf is always locally free in Zarissky topology.

I'm trying to solve exercise III.10.5 from Hartshorne "Algebraic geometry". Let $\mathcal F$ be a coherent sheaf on a scheme $X$ locally free in 'etale topology, namely for any $x \in X$ there is an ...
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vote
2answers
94 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
2
votes
0answers
46 views

About the construction of Quot-Schemes

I am reading in the paper of Nitsure (link: http://arxiv.org/pdf/math/0504590v1.pdf) about the construction of the Quot-scheme $\mathrm{Quot}_{E/X/S}^{\Phi,L}$. After Lemma 5.4. they reduce to the ...
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0answers
39 views

group scheme of prime order p is killed by p

In the article "Group Schemes of Prime Order" by Tate and Oort (see here) it is proved that a group scheme of prime order $p$ over the base $S$ is killed by $p$ (Theorem 1). The authors state that it ...
2
votes
1answer
101 views

Can we recover étale, fpqc etc. morphisms of schemes from the affine versions?

$\DeclareMathOperator{\Spec}{Spec}$ Consider the following procedure for defining a class of morphisms of schemes: Take a suitable class of homomorphisms of rings (e.g., canonical maps to ...
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0answers
66 views

Why is the scheme associated to a product of quasi-projective varieties naturally isomorphic to the product of the associated schemes?

What I mean by this is, suppose $X$ and $Y$ are quasi-projective varieties over some arbitrary field $k$. Then $X\times Y$ is again a quasi-projective variety. I've seen this a few times, but what is ...
3
votes
0answers
86 views

Hartshorne notation in section III.12

I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes ...
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0answers
44 views

If $X$ is quasi-projective but the scheme $\tilde{X}$ is affine, is $X$ necessarily affine?

I'm curious if the following works as a criterion to determine when a quasi-projective variety is actually affine. If $X$ is a quasi-projective variety, and the scheme $\tilde{X}$ is affine, does ...
1
vote
1answer
57 views

Punctual Hilbert scheme of four points

I am looking at $\text{Hilb}^4(\mathbb{C}^2)$, which is the Hilbert scheme of four points on $\mathbb{C}^2$. In particular, I am just looking at four points collided (at the origin), and want to know ...
0
votes
1answer
87 views

Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$)

I am working on Exercise 6.3.N. on Ravi Vakil's notes (on morphism from $S$ scheme to $\mathbb{P}_B^n$) and I would like some assistance. The exercise states: Let $B$ be a ring. If $X$ is a ...
0
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0answers
25 views

Morphism of schemes determined by their induced maps of $Z$ valued points

I am doing an exercise that states: morphism of schemes $X \rightarrow Y$ is determined by their induced maps of $Z$ valued points, as $Z$ varies over all schemes. I am a bit confused with this ...
5
votes
1answer
155 views

Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
2
votes
1answer
79 views

Did I use axiom of choice in my proof?

I have two different affine open covers for a scheme $X$, say $X = \cup_{i \in I} U_i$ and $X = \cup_{j \in J} V_j$. For each $p \in X$, we know there exist some $i(p)$ and $j(p)$ such that $p \in ...
0
votes
0answers
83 views

Set-theoretic intersection of affine open subschemes.

Let $X$ be a separated scheme, $U,V \subseteq X$ open affine subschemes, $\Delta \colon X \to X \times X$ the diagonal morphism and $\pi_1, \pi_2 \colon X \times X \to X $ the natural projections, so ...
4
votes
1answer
63 views

Is the push-forward of a quasi-coherent sheaf under open immersion still quasi-coherent?

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ ...
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0answers
23 views

Equivalent definitions of noetherian scheme.

A scheme $X$ is locally noetherian iff there is a cover $X = \bigcup_i \text{Spec}(R_i)$ with noetherian $R_i$. When $X$ is also quasicompact, it is called noetherian. Question: Why is (as Hartshorne ...
0
votes
1answer
57 views

Properties of dominant morphism of schemes

I am trying to solve the following exercise 4.11, p.67 from Qing Liu's book "Algebraic geometry and arithmetic curves". Let $f:X\to Y$ be a morphism of irreducible schemes with respective generic ...
0
votes
1answer
93 views

Does dominant morphism of integral schemes is injective on sheaves?

Let $f:X \to Y$ be a dominant morphism of integral schemes. Is it true that it is equivalent to the fact that $\mathcal O_Y \to f_* \mathcal O_X$ is injective? Or does one imply another? It's quite ...
1
vote
1answer
62 views

On a canonical morphism from Spec $O_{X,p} \rightarrow X$

Let $X$ be a scheme. I am doing an exercise: Let $p \in X$. Describe a canonical (choice-free) morphism from Spec $O_{X,p} \rightarrow X$, with hint that says to make sure that the morphism is ...
6
votes
1answer
86 views

How to imagine the difference between the following schemes?

Consider $A=\operatorname{Spec} k[x]_{(x)}[t]$ and $B=\operatorname{Spec} k[x,t]_{(x)}$ for a field $k$ (Vakil, note 11.3.8). For me, both are infinitesimal neighborhoods of an affine line - the ...
5
votes
1answer
159 views

Showing closed immersions are stable under base extension without using that they are affine.

This question is based on question $3.11$ from chapter $2$ of Hartshorne, found on page $92$. Part $a)$ of said question asks to show that closed immersions are stable under base extension. In other ...
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vote
2answers
104 views

If local rings are Noetherian, is scheme locally-Noetherian?

If all the local rings of a scheme are Noetherian, is the scheme locally-Noetherian?
2
votes
1answer
79 views

Equivalent conditions for a closed immersion of schemes

In Hartshorne, a closed immersion of schemes is defined to be a scheme morphism $\Phi \colon Y \to X$ such that $\Phi$ is a homeomorphism onto $\Phi(Y)$, $\Phi(Y)$ is closed in $|X|$ and $$ (*) ...
3
votes
1answer
93 views

Hartshorne III 9.5 confused about base extension.

Hartshorne III Proposition 9.5 states: Let $f:X\to Y$ be a flat morphism of schemes of finite type over a field $k$. For any point $x\in X$ let $y=f(x)$. Then $$\dim_x(X_y)=\dim_x X-\dim_y Y$$ ...
1
vote
0answers
51 views

Morphism between projective schemes induced by injection of graded rings

Let $A$ be a graded ring and $d>0$ be an integer. Define the graded ring $B$ such that $B_i=A_i$ if $d$ divides $i$ and $B_i=0$ otherwise. Is it true that a homomorphism of graded rings ...
2
votes
2answers
48 views

Basic question related to stalk of the structure sheaf of a scheme

Let $X$ and $Y$ be schemes. Suppose I have a morphism of schemes $(\pi, \phi)$, where $\pi: X \rightarrow Y$, thus $\pi$ is continuous, and $\phi: O_Y \rightarrow \pi_*O_X$. Let $p \in X$ and $\pi(p) ...