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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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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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 ...
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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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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 ...
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50 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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Show that $X$ and $Y$ are isomorphic as schemes

Let $X = \{(x,y,z) \in \mathbb{C}^3 : xy=xz=yz=0\}$ be the union of the three coordinate lines in $\mathbb{C}^3$. Let $Y = \{(x,y) \in \mathbb{C}^2 : xy(x-y)=0\}$ be the union of three concurrent ...
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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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25 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 ...
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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 ...
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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 $R$, $$ ...
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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 ...
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29 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 $f$ ...
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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 ...
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1answer
30 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 ...
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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 ...
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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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22 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 ...
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1answer
64 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 $S$. ...
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45 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 ...
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74 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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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 $S$? ...
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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 $L, L' \subset \mathbb{P}^3$. ...
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143 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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65 views

Does $L_1\oplus\mathbb{A}^1_X\cong L_2\oplus\mathbb{A}^1_X$ imply that $L_1\cong L_2$?

Suppose $L_1,L_2$ are line bundles over a scheme $X$. If one knows that $L_1\oplus\mathbb{A}^1_X$ and $L_2\oplus\mathbb{A}^1_X$ are isomorhpic, is that enough to conclude that $L_1$ and $L_2$ are ...
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20 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
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1answer
44 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 ...
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103 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 ...
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1answer
50 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 ...
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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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59 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 $ ...
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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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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 ...
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1answer
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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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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 ...
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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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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 ...
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1answer
53 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 ...
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1answer
52 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 ...
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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 ...
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119 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 $Y$ ...
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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 ...
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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 ...
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Is the push-forwad 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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20 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 ...
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1answer
39 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 ...
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1answer
69 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 ...
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1answer
38 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 ...
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1answer
75 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 ...
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1answer
138 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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2answers
83 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?