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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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Equivalent ways to define structure sheaf on $Spec(A)$.

How do we show that the following two ways of defining the structure sheaf on $Spec(A)$ are the same? Definition 1 $$\mathcal{O}(U) = \left\{s \in \prod_{\mathfrak{p} \in U} A_{\mathfrak{p}} : s \text{...
Babai's user avatar
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Calculating the fibres of a scheme morphism are proper but the morphism is not proper

$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $f:\mathbb A^1_\mathbb C\rightarrow \mathbb A^1_\mathbb C$ be induced by the ring homomorphism $t\...
Chris's user avatar
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How can I check that for a quasi-compact morphism $f:Z\to X$, the kernel of $\mathcal{O}_X\to f_*\mathcal{O}_Z$ is quasi-coherent?

If $f:Z\rightarrow X$ is a quasi compact morphism of schemes and $\mathcal J :=\ker(\mathcal O_X \rightarrow f_* \mathcal{O}_Z)$, then $(\operatorname{Supp}( \mathcal O_X/ \mathcal J)$,$i^{-1}(\...
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For any open set $U\subset X$, $X$ an integral scheme, $p\in U$, $\mathscr O_{X}(U)\rightarrow \mathscr O_{X,p}$ is injective

In the case that $U$ is affine, the statement is trivial because the stalk at $p$ is the localization of $\mathscr O_{X}(U)$ except for elements in prime ideal $p$ and the map identifies with the map $...
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Equivalence of the Definition of closed immersion and closed subscheme

In wedhorn and Görtz‘s algebraic geometry book Algebraic Geometry I:Schemes page 86 I found the following definition. (1)A closed subscheme of $X$ is given by a closed subset $Z\subset X$ and an ideal ...
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Characterization of reduced closed subschemes of a scheme

I'm proving the fact stated below in italics. I couldn't find a detailed proof anywhere, as it seems that it is a ordinary fact to check; the hint I managed to find suggested to reduce to the affine ...
Ezio Greggio's user avatar
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Showing that $\mathbb A^n_\mathbb C\rightarrow \operatorname{Spec}\mathbb C$ is not proper.

$\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\P}{\mathbb P}\newcommand{\C}{\mathbb C}\newcommand{\A}{\mathbb A}$ Let $\A^n_\C=\Spec \C[x_1,\dots,x_n]$, and let $f:\A^n_\C\rightarrow \Spec \C$ ...
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Regarding Vakil's Exercise $10.7.A$

$\newcommand{\O}{\mathscr{O}}\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\Frac}{\operatorname{Frac}}$ In exercise 10.7.A of Vakil's Rising Sea we are tasked with showing that if $A$ is an ...
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Liu lemma 3.3.17: Proper + affine = finite

This has been asked and answered before here. I'm not satisfied with the answer (probably I don't understand it). Also I don't want a proof of "proper+affine=finite" (which is here) but of ...
user128787's user avatar
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What is the stalk map for a morphism of affine schemes?

$\newcommand{\Spec}{\operatorname{Spec}} \newcommand{\O}{\mathscr{O}}$ Let $X=\Spec A$, and $Y=\Spec B$, and suppose that $f:X\rightarrow Y$ is a morphism coming from the ring homomorphism $\phi:B\...
Chris's user avatar
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I am getting very confused by the definition of a minimal prime ideal

I am trying to prove that in an affine scheme $\operatorname{Spec}A$ that an irreducible component can be written as the vanishing locus $V(\mathfrak p)$ for a minimal prime ideal $\mathfrak p$, but I ...
Chris's user avatar
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Is a surjective closed immersion $Y \to X$ an isomorphism when $Y$ is reduced?

$\require{ams}$ Suppose we have an affine scheme $X = \text{Spec }R$, a closed subset $S \subseteq X$ and a closed immersion $\iota: Y \to X$ with image $\iota(Y) = S$. In particular this means $\iota:...
soggycornflakes's user avatar
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Universal property of global spec of a quasicoherent sheaf of algebra

I am reading Vakil's FOAG (july31, 2023). Now in 17.1.C he gives an exercise: Exercise. Given an $X$-morphism $\gamma:W\to\mathcal{S}pec \mathscr{R}$ where $\mathscr{R}$ is a sheaf of $\mathcal{O}_X$-...
Mizutsuki's user avatar
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Zariski opens of an affine space and (po)sites

There is a very neat presentation of the Zariski open sets in $\operatorname{Spec}(R)$: it is freely generated (under arbitrary unions and finite intersections, which distribute over each other) by ...
Trebor's user avatar
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Sections are determined by their germs at associated points

This comes from the exercise 6.6.W of Vakil's FOAG. I guess it is saying that a function (section) is determined by its germs at associated points. And this sort of motivates the introduction of ...
Mizutsuki's user avatar
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Proof critique - Glueing local isomorphism

I apologize first for such a frequently asked question. I am not confident about my approach of this problem which is possibly related to glueing local isomorphisms. I am aware of that local ...
Mizutsuki's user avatar
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Affine group schemes

I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups. My question Let $k$ be an ...
Tommk's user avatar
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Is there a "correct" $k$ scheme structure to put on $\coprod_{i=1}^n \operatorname{Spec}(k)$?

Let $k$ be an algebraically closed field, with $n\in\mathbb N\subset k$ invertible. I am trying to prove that if $\mathbb G_m=k[t,t^{-1}]$ is the multiplicative group scheme over $k$, and $\mu_n$ is ...
Chris's user avatar
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Finding out the induced map via factoring through canonical map $A \to S^{-1}A$ of localization

I was trying to prove that $\varinjlim_{U \ni x} A(U) \cong A_{\mathfrak{p}}$ where $U$ are basic open sets containing $x = \mathfrak{p} \in \operatorname{Spec} A$, and $A(U) = A_f$ where $U = X_f$. $...
Anthony Lee's user avatar
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Finding the image of a morphism between affine schemes

I am confused about two things, one of my problem sets for a course on Abelian varieties, I was given a morphism $\mathbb A^1_k\rightarrow \mathbb A^2_k$ "in coordinates" as: $$f(t)=(t^2-1,t^...
Chris's user avatar
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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 $\widetilde{M}$ ...
Mizutsuki's user avatar
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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. ...
Mizutsuki's user avatar
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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 ...
Chris's user avatar
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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 ...
Mizutsuki's user avatar
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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 $\pi:X\to Y$ ...
Mizutsuki's user avatar
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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 ...
uno's user avatar
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On free direct summand locus of finitely generated modules over commutative Noetherian rings

Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. Assume that there exists an injective $R$-linear morphism $f: R\to M$. Consider the sets $$U:=\{\mathfrak p\in \text{Spec}...
Alex's user avatar
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Reduced, irreducible and integral schemes

I am studying properties of schemes and I have the following problem. Let $R$ be a ring, $S=Spec(R)$, $n\in\mathbb{N}$ an integer. Show that the following are equivalent: $S$ is reduced (resp ...
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Principal opens of spectrum of a ring are (quasi-)compact

I am doing exercise 17 in Atiyah-MacDonald, in particular I am not confident with my solution for part vi): [Let $A$ be a ring.] For each $f \in A$, let $X_f$ denote the complement of $V(f) [= \{\...
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Picture of $\mathrm{Spec}(\mathbb{R}[x])$

Follow-up to Comparing the prime spectra of $\mathbb{Q}[x],\mathbb{R}[x]$ and $\mathbb{C}[x]$. I'm trying to draw a picture of $\mathrm{Spec}(\mathbb{R}[x])$. I've already drawn a picture of $\mathrm{...
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If $R$ is a local ring, the morphism of schemes $f: \operatorname{Spec} R \to X$ is determined by the image of the closed point? [duplicate]

I’m taking an introductory course on Scheme theory. In one of the proofs of the course, we were considering all morphisms of schemes $f: \operatorname{Spec} R \to X$, where $X$ is a scheme and $R$ is ...
Gokimo's user avatar
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If $f:\operatorname{Spec} R \to X$ is a map with $R$ local and with closed point landing in an open $U\subset X$, then $im(f)\subset U$ too.

I’m taking an introductory course on Scheme theory. In one of the proofs of the course, we were considering a situation where we have a morphism of schemes $f: \operatorname{Spec} R \to X$, where $X$ ...
Gokimo's user avatar
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Locus of the fibered product of two affine lines with doubled origin where projections coincide

This is an exercise (Exercise III-1) in the book Geometry of Schemes: Exercise III-1. a) Let $Y$ be the line with doubled origin over a field $K$ and let $\varphi_1,\varphi_2:\mathbb{A}_K^1\to Y$ be ...
Mizutsuki's user avatar
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Can we explicitly describe the derived pullback $\mathbf L\pi^* \widetilde M$ for a closed immersion $\pi$ of affine schemes?

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi:...
uno's user avatar
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image of Segre embedding is cut out (scheme-theoretically) by vanishing minors

I am doing exercise 10.6.B of Vakil's FOAG (July 31, 2023 version). Basically, I need to show that the image of the Segre embedding is cut out by equations so that the matrix \begin{bmatrix} a_{00}&...
Mizutsuki's user avatar
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Morphism of schemes and closed points [duplicate]

I apologize for the vagueness of my question. In the case of morphisms between schemes (that locally look like an $A$-algebra maybe), the textbook (Vakil's FOAG, July 31, 2023 version) often uses the &...
Mizutsuki's user avatar
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1 answer
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Gluing in the sheaf of meromorphic functions on an affine scheme

Let $X = \operatorname{Spec}(R)$ be an affine Noetherian scheme, and $U_i = D(f_i)$ (with $f_i \in R$) be (finitely many) principal open sets with $X = \bigcup_i U_i$. We have the sheaf of meromorphic ...
Verroq's user avatar
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1 answer
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Open subschemes of open sets not contained in basic open sets

I'm just starting to learn algebraic geometry for the first time, and I had an elementary question concerning some of my intuitions about open subschemes (during this period where I haven't yet learnt ...
legionwhale's user avatar
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global section of a B-scheme defines a morphism to projective $B$-scheme

This is problem 7.3.O in Vakil's AG notes (July 31, 2023 version). I am not sure how to define the map locally. Problem 7.3.O. Let $B$ be a ring and $X$ a $B$-scheme. Suppose $f_0,\cdots,f_n$ are n+1 ...
Mizutsuki's user avatar
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Rising Sea exercise 5.3.H

I wish to prove that if $(f_1, \ldots, f_n) = (1)$, and for each $i$, $B \to A_{f_i}$ is of finite type, then so is $B \to A$. I am following Vakil's hints. Let $r_{ij}$ be the finitely many elements ...
Adelhart's user avatar
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Zeros of a finite support function on elliptic curve are torsion

Let $X$ be a elliptic curve over a field $k$ and $f$ be a holomorphic function on $X$, if $f$ only has two zeros $P$ and $Q$, then I'd like to know how to prove see that $P$ is a torsion point of $X$. ...
Jean's user avatar
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When can $\textbf{SpecMax}(R)$ be a scheme?

Let $R$ be a commutative ring, and let $(\text{Spec(R)},\mathcal{O}_R)$ be the affine scheme associated to $R$. Let $\text{SpecMax}(R)$ the subspace the $\text{Spec(R)}$ of the maximal ideals. My ...
Luis Antonio Sanchez's user avatar
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Proof criticize: restriction maps of integral schemes are injective

Claim. Let $X$ be an integral scheme, and $\emptyset\neq V\subseteq U$ for some open subsets $V,U$ of $X$. Then $ \rho_{UV}:\mathcal{O}_X(U)\to\mathcal{O}_X(V)$ must be injective. Here is my proof of ...
Mizutsuki's user avatar
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1 answer
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Intuition of projective line via glueing

I am reading the construction of projective line (4.4.6) following Prof. Vakil's Algebraic geometry notes FOAG. However, I fail to understand few lines during the discussion. Let $X=\operatorname{Spec}...
Mizutsuki's user avatar
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Relationship between affine lines with coordinate $z$ and coordinate $z^2$

Let $k$ be algebraically closed of characteristic zero; assuming $k=\mathbb{C}$ is fine. Consider the subring inclusion $k[z^2]\to k[z]$ and the induced map $$ f\colon\mathbb{A}^1_k=\mathrm{Spec}k[z]\...
VerySlowlyLearning's user avatar
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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 ...
Elías Guisado Villalgordo's user avatar
3 votes
1 answer
188 views

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 $(\overline{x},\overline{y})$, $(\overline{x}-a)$, $(\overline{y}-b)$...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
158 views

Scheme theoretic fiber vs. Degree of a point of a finite morphism of affine schemes

Let $f: R\to S$ be a finite morphism of Commutative Noetherian rings, so $S$ is module finite over $R$ via $f$. Let $Q \in \text{Spec}(S)$, and set $P=f^{-1}(Q)$, so we have an induced map $R_P \to ...
Alex's user avatar
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1 answer
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Should a closed immersion of schemes be closed map of topological spaces by definition?

In Section 9.1 of his "Foundations of Algebraic Geometry" (version August 2022), Ravi Vakil defines a closed embedding (closed immersion) of schemes to be an affine morphism $ \pi: X\...
Joseph Shtok's user avatar

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