# Questions tagged [affine-schemes]

The spectrum of a commutative ring with unit is the set of prime ideals endowed with the Zariski topology. One can define a sheaf of rings on this space : to each Zariski-open set is assigned a commutative ring, thought of as the ring of "polynomial functions" defined on that set. This topological space endowed with this sheaf is called the spectrum of the ring. Every locally ringed space isomorphic to such a spectrum is called an affine scheme.

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### definition of closed immersion of schemes

The definition that I found on books for definition of closed immersion of schemes is the following: A closed immersion $i:Z\hookrightarrow X$ is a morphism which satifies: (1) The underlying ...
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### rank of an $\mathcal{O}_X$-module being constant

Given a finitely generated projective module $M$ over a ring $R$ with exactly two idempotents $0,1$ ($X=\operatorname{Spec} R$ is connected). We have a coherent $\mathcal{O}_X$-module $\widetilde{M}$ ...
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### Proof criticize: rank $n$ vector bundle on $\mathbb{A}^1_k$ is trivial

The problem stated in the title comes from Vakil's AG notes 14.3.C. I am aware that there are few similar questions answered here. But I wish to give my own proof that I am not so sure if is valid. ...
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### Algebraic Torus is a group scheme

I am taking a course on toric varieties this semester, and I am a little confused by how the algebraic torus is a group scheme, as we didn't really define what a group scheme is. I was given the ...
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### Question related to Etale morphism

These questions appeared while studying Vakil's AG notes July 3123 version. This section really confuses me. Let me give the definition in this context first: Definition (Smooth of relative dimension ...
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### Definition of being smooth of relative dimension in Vakil

In the July3123 version of Algebraic geometry book by Vakil, he defined the notion of smooth of relative dimension in 13.6. I will write down the definition first: Definition. A morphism $\pi:X\to Y$ ...
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### Krull dimension of affine open subscheme of Noetherian scheme containing the generic points of all irreducible components

Let $X$ be a reduced Noetherian scheme. It has finitely many irreducible components $\{X_j \}_{j=1}^n$. Let $x_j$ be the generic point of $X_j$. If $U$ is an affine open subset of $X$ containing every ...
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### More examples of morphisms of ringed spaces that aren't local?

$\def\Spec{\operatorname{Spec}}$All questions and answers that I've found in MSE regarding a morphism of ringed spaces between affine schemes that isn't a morphism of locally ringed spaces are the ...
### Geometrical interpretation of $\operatorname{Spec}(\mathbb{R}[x,y]/(x^2+y^2))$
$\def\Spec{\operatorname{Spec}}$If am not mistaken, the prime spectrum of $\Spec(\mathbb{R}[x,y]/(x^2+y^2))$ consists of the points $(\overline{x},\overline{y})$, $(\overline{x}-a)$, $(\overline{y}-b)$...