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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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Affine $n$-space over a scheme

In an exercise of Eisenbud-Harris The Geometry of Schemes, they ask to prove the following: Let $S$ be any scheme. Let $\mathbb{A}_{\mathbb{Z}}^n = \mathrm{Spec}\mathbb{Z}[x_1, \dots , x_n]$ be ...
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$Spec(R)$ is irreducible if and only if $Spec(R[T])$ is irreducible

I want to show that for an arbitrary ring $R$ the following equivalence holds: $Spec(R)$ is irreducible if and only if $Spec(R[T])$ is irreducible. I have tried to show this by using the ...
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Scheme-theoretic Preimage and Fibered Product of Schemes

Following Eisenbud-Harris The Geometry of Schemes, and I'm having trouble understanding a specific part of their proof that fibered products exist in the category of schemes. The affine case is okay,...
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Intuition and technique for (strict) Henselization of nodal cubic at node

Consider the union of the axes $\frac{\Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin. ...
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29 views

(Krull) dimension of dense open subset of finite type algebra over a domain

Let $D\to A$ be a finite type algebra with $D$ a domain. Suppose $V\subset \operatorname{Spec}A$ is open and dense. Is it true that $\dim V=\dim A$? I know that if $X\to \operatorname{Spec}\Bbbk$ is ...
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Geometric interpretation of subalgebras as non-linear coordinate change? (context of Noether normalization)

Consider a $k$-algebra morphism $k[x_1,\dots ,x_d]\overset{g}{\longrightarrow }\frac{k[x_1,\dots ,x_n]}{(f_1,\dots ,f_m)}$ defined by $x_i\mapsto g_i$. In a nice setting, e.g what $k$ is a field and $...
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Find one-dimensional $P,Q$ such that $PQ = P \cap Q$ and $P,Q$ not coprime

I am looking for an example of the following situation: Let $R = k[x,y,z]$ be the polynomial ring in three indeterminates where $k$ is a field. I want to find two prime ideals $P$ and $Q$ of $R$ ...
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Affine Varieties over separably closed fields

Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$. On the ...
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61 views

When is a map of local rings finite?

Let $A,B$ be Noetherian local rings, and let $A \to B$ be a ring homomorphism such that the induced map $\operatorname{Spec} B \to \operatorname{Spec} A$ is surjective and quasifinite (of finite type ...
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The big étale and Zariski topoi are generated by small sites

Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same ...
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Geometry of the subalgebra $\Bbbk[x^2-1]\leq \Bbbk[x]$ (intuition for integral elements)

Given a field $\Bbbk$ consider the subalgebra $\Bbbk[x^2-1]\leq \Bbbk[x]$. This is an integral extension of algebras. Write $\mathfrak q= (x-1)\vartriangleleft \Bbbk[x]$ and $\mathfrak p=\Bbbk [x^2-1]\...
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Is constancy of fiber degree analytic-local on the source in a finite flat family?

Let $X \subset \mathbb{A}_{\mathbb{C}}^2$ be a closed subscheme, and let $\pi \colon X \to \mathbb{A}_{\mathbb{C}}^1$ be a finite flat map. For a point $p \in X(\mathbb{C})$, is it true that we can ...
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When does a quasifinite surjective flat morphism have constant fiber multiplicity near a point?

Let $V \subset \mathbb{A}_{\mathbb{C}}^2 = \operatorname{Spec}\mathbb{C}[x,t]$ be a closed subscheme containing the point $x = t = 0$, and suppose we have a quasifinite flat surjective morphism $\pi \...
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proposition II.5.9 in Hartshorne

I have a question about the 1 to 1 correspondence between quasi-coherent sheaf of ideals and the closed subschemes. suppose $(i,i^×):Y\subset X$ is a closed subscheme,where $i^×:O_X\rightarrow i_*...
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Action of finite group of ring automorphisms

Let $A$ be commutative ring and $G$ a finite group of ring automorphisms of $A$. We have a finite ring extension $A^G \to A$ that induces a surjective map $\phi: Spec(A) \to Spec(A^G)$. I'd like to ...
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Zero-dimensional Hilbert scheme of two points and Hilbert-Chow morphism

I would like to understand what happens to the Hilbert scheme of two points on a scheme if the scheme is zero-dimensional. The background for this question is just general curiosity, Hilbert schemes ...
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Is the homomorphism $g_P^\sharp: \mathcal O_{F,g(P)}\to \mathcal O_{\operatorname{Spec} B,P}$ a monomorphism?

Given a morphism of affine schemes $f: \operatorname{Spec}B\to \operatorname{Spec}A$, denote the scheme theoretic image of $f$ by $F$, then $f$ can be factorized as $ \operatorname{Spec}B\stackrel{g}\...
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Question about Hartshorne Ch. II. 2 Proposition 2.2a (and basis for topology of Spec(A))

I'm a bit confused about a proof of the following proposition in Chapter II.2 of Hartshorne's Algebraic Geometry. Prop. 2.2.a: Let $A$ be a ring and $(Spec(A), \mathcal{O})$ its spectrum. For any $...
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Kernel of a morphism of $k$-algebras

Assume you have two affine varieties $X$ and $Y$. A morphism $\phi$ between them induces a morphism between the k-algebras of regular functions (functions on them that are locally the quotients of ...
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What is the tangent space of $\mathbb{Z}$ at $\mathfrak{p}=(5)$?

What is the tangent space of the scheme $\text{Spec}\,\mathbb{Z}$ at the point $\mathfrak{p}= (5)$ ? the local ring of $\mathbb{Z}$ at $(5)$ is $\mathcal{O}_{\mathbb{Z},5}=\mathbb{Z}_5$ the 5-adic ...
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Normalization of an isomorphism

Let $R$ be a normal ring, and let $A$ and $B$ be normal $R$-algebras. Consider a morphism of $\text{Frac}(R)$-vector spaces $f:A\otimes_R \text{Frac}(R)\rightarrow B\otimes_R\text{Frac}(R)$. Its ...
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A lemma about standard open sets of a scheme.

I came across the following lemma in the Stacks project: Link. I have two questions: How do we obtain $D(g)=D(g_A)$? How can we assume that $U\subset V$?
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Functor of points of the affine line with double origin

If $X$ is a scheme then the functor of points is the map $\hom(-, X)$ from the category of schemes to the category of sets. If we restrict this to affine schemes and apply Yoneda then the functor of ...
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Proof check for closed sub scheme of an affine scheme is affine

Basically I was solving an exercise of Hartshorne. So let $X=Spec A$ be an affine scheme and $i : Y \rightarrow X$ is a closed immersion. I have to show that $Y$ is affine. This is how I proceed. ...
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Are $u_1,u_2,\cdots,u_n$ independent in $M$?

Suppose $X=\operatorname{Spec}A$ and $A$ is Noetherian, $M$ is a $A$-module and the $\mathcal O_X$-module $\widetilde M$ is coherent. For some $x\in X$, if $\widetilde{M}_{x}$ is free of rank $n$ on $\...
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Is $D(f)$ the smallest open set of $\operatorname{Spec}B$ such that $D_+(f)\subset D(f)$?

Let $B$ be a graded ring and $\rho:\operatorname{Proj}B\to \operatorname{Spec}B$ the canonical injection, that is, $\forall \mathfrak p\in \operatorname{Proj}B$, $\rho(\mathfrak p)=\mathfrak p$. For ...
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Question about meaning behind a map of schemes

I saw the following map of schemes in an example, $${A}^1_\mathbb{C} \rightarrow \operatorname{Spec}(\mathbb{C}[x,y]/(y^2-x^3)),$$ $$t\mapsto (t^2, t^3).$$ Equivalently, this can be written as, $$ ...
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Does there exist an ideal sheaf $\mathcal F$ on some affine scheme $X$ such that $\mathcal F$ is not quasi-coherent?

Please give an example such that $X$ is an affine scheme, and there exists an ideal sheaf $\mathcal F$ on $X$ such that $\mathcal F$ is not quasi-coherent.
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Finite maps and jacobian condition

Let $k$ be an algebraically closed field and take $f_{1}, ..., f_{n} \in k[X_{1},..., X_{n}]$ with the jacobian condition: $\det J_{f} = 1$. Let $A:= k[X_{1},...,X_{n}]/(f_{1},...,f_{n})$ and ...
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Have you seen the symbol $D(g)$ with $g\in \mathcal O_X(X)$ where $X$ is a scheme?

Have you seen the symbol $D(g)$ with $g\in \mathcal O_X(X)$ where $X$ is a scheme?
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Criteria for Ideal of a Hopf Algebra to Yield Closed Subgroup

Let $k$ be a commutative ring with unity, and let $A$ be the $k$-algebra of an affine group scheme over $k,$ endowed with its usual structure as a Hopf algebra over $k.$ In Waterhouse's textbook, An ...
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Is an affine formal scheme quasi-compact?

It is well known that an affine scheme $X=\mathrm{Spec}(A)$ is quasi compact. In analogy, what can we say about an affine formal scheme $\mathrm{Spf}(A)$ (here $A$ should be an adic ring and $\mathrm{...
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What is an $R$-valued point on a scheme over $R$?

An $R$-valued point on an arbitrary scheme $X$ is defined to be a morphism $\mathrm{Spec}(R) \to X$. (1) What if $X$ itself is a scheme over $R$? In this case, what is the difference between the set ...
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A naive question about the scheme theory: regard $\mathbb C^n$ as a scheme

Note that $\mathbb C$ can be regarded as the set of closed points of $\mathrm{Spec} ~\mathbb C[T]$. And, $\mathbb C^n$ should be regarded as that of $\mathrm{Spec} ~\mathbb C[T_1,\cdots,T_n]$ What if ...
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60 views

What's an $\mathcal O_X$-algebra when $X= \operatorname{Spec} R$?

Take $X= \operatorname{Spec}R$, $R$ a commutative ring with unit. What is an $\mathcal O_X$-algebra in that case? Is there more than just ordinary $R$-algebras? Thank you in advance.
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Morphisms on affine schemes and induced morphisms on rings

Let $X=$ Spec$A$ and $Y=$ Spec$B$ and let $f : X \rightarrow Y$. What requirements on $f$ mean that the induced map $A \rightarrow B$ makes $B$ a finitely generated $A$-algebra? I've seen things about ...
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Notion of inner automorphisms for group schemes

Let $G$ be a finite group scheme over the field $k$ ( you can assume $k$ is algebraically closed). I've seen the term "inner automorphisms" for $G$ at many places but I still don't understand what the ...
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1answer
61 views

Underlying variety of $\mathbb{G}_m$

I try to understand what is meant by the expression that $\mathbb{A}^1 \setminus \{0\}$ is the underlying variety of the multiplicative group $\mathbb{G}_m$ over the field $K$ . I know that $\...
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Is $X=(x^2-yx, y^2-xz, z^2-x^2y) $ isomorphic to $ \mathbb A_{K}^1$? [closed]

Is the algebraic affine set $X=(x^2-yx, y^2-xz, z^2-x^2y) \subset \mathbb A_{K}^3$ isomorphic to $ \mathbb A_{K}^1$? Any hints and comments would be highly appreciated.
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Closed and $\mathbb{C}$-rational points of $\operatorname{Spec}(\mathbb{R}[x,y]/(f))$

I want to understand the definition of an $K$-rational point of an affine scheme with this example. Let $f \in R[x,y]$ be a polynomial such that $f=0$ has no solutions in $\mathbb{R}$ $X=\...
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1answer
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Puzzling with Kahler differentials

There is some big mistake i'm doing that I can find. Take an affine scheme $X= Spec A$ smooth over $k$ of dimension 1, and consider $\Omega_X$, kahler differentials. I want to understand 'how many' ...
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How an Affine hull can be represented by an Affine set? [closed]

Please see the image for detail of the question.
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if $f$ and $g$ compose with a dominant morphism are the same, show that they agrees on a open dense subset as morphisms of schemes

To be specific about my question, let $A$ be a valuation ring (so local and integral) and $K$ be the fraction field of $A$. The injection $A\rightarrow K$ induces $i: \mathrm{Spec} \ K\rightarrow\...
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63 views

How does the closed schemes between $X$ and $X_{red}$ look like?

Consider locally Noetherian schemes over an algebraically closed base field $k$. Given a scheme $X$, the scheme $X_{red}$ is the smallest closed subscheme of $X$ with the same underline topological ...
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Computing tensor product $k[t]/(t^p) \otimes_{k[x,y]/(x^p,y^p)} k[t]/(t^p)$

Let $k$ be a field of chracteristic $p>0$. I want to show that $M=k[t]/(t^p) \otimes_{k[x,y]/(x^p,y^p)} k[t]/(t^p)$ is not isomorphic to $k[t]/(t^p)$ as $k$-algebra. Here the map from $k[x,y]/(x^p,...
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1answer
158 views

two intersection of affine open subsets

let $ X $ be a separated scheme over an affine scheme $ S$. let $U$ and $V$ be open affine subsets of $X$.Then $U\cap V$ is also affine.and Give an example to to show that this is fails if $X$ is not ...
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What is connection between Modules and Fields in mathematics ? Is maybe quotient of the action / space of coinvariants?

Why this question? I try to understand connection between Module & Field more deeply to try to build a particular class of 'fixed points' of non-trivial affine space to 'extract' an affine ...
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1answer
73 views

characterization of affine open sets in an affine scheme

Let $R$ be a commutative ring with unity. Consider the affine scheme $\operatorname{Spec} R$. For an ideal $I$ of $R$ show that $D(I)$ is affine iff $I$ generates the unit ideal in $\Gamma (D(I), \...
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Dimension of the tangent cone of $V(\mathfrak{a})$

Let $\mathfrak{a}$ be prime in $k[T_1,\dots,T_n]$ contained in $(T_1,\dots,T_n)$, with $k$ a field. Define the tangent cone $\text{TC}_x(X)$ of $X = V(\mathfrak{a})$ at $x=0 \in V(\mathfrak{a})$ to ...
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(Connected) non-contractible schemes

This question is motivated by this other question and this answer, which show that irreducible algebraic varieties and more generally integral schemes are contractible as topological spaces. What are ...