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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 if noetherian and the reduced is affine

Prove that if a scheme $X$ is noetherian and $X_{red}$ is affine then $X$ is affine.
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37 views

Gluing functions from irreducible components of a reduced curve if they agree on the intersection points

The question is: What is the algebraic machinery/reasoning behind the following intuition? Given a reduced curve over some field $k$ and a regular function on each of its components such that those ...
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Structure sheaf and residue field

Hartshorne defines the structure sheaf of $X=$ Spec$A$ as, for $ U \subset X$, $$ O_X(U)=\{s: U \rightarrow \dot\cup A_{\mathfrak{p}} \}$$ with 2 additional requirements, i.e. $s({\mathfrak{p}}) \in ...
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50 views

Hartshorne Exercise III 3.2: $X$ is affine iff every component is affine

I'm trying to solve the following exercise frome Hartshorne's Algebraic Geometry: Exercise III 3.2. Let $X$ be a reduced noetherian scheme. Show that $X$ is affine if and only if each irreducible ...
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1answer
67 views

Zero-Dimensional Subschemes of Degree 21

I'm working on the following problem from Eisenbud and Harris' Geometry of Schemes. Consider zero dimensional subschemes of $\mathbb{A}^4_K$ of degree 21 such that $$V(\mathfrak{m}^3)\subset \Gamma \...
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29 views

Can every maximal ideal of Dedekind domain be principal after restricting to a small enough distinguished open subset?

Let $S=\textrm{Spec }R$ where $R$ is a Dedekind domain, let $\mathfrak{p}$ be a maximal ideal of $R$, which is a closed point of $S$, can we find an distinguished open affine subset of $S$, say $\...
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1answer
40 views

What is a $k$-scheme isomorphism?

In Example 2.4.2 of Chapter IV 2 of Hartshorne's Algebraic Geometry, it gives $\pi:X=Spec(k[t])\to Spec(k)$ a scheme over $k$ and $F:X\to X$ and $Spec(k)\to Spec(k)$ the morphism with identity on the ...
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1answer
47 views

Characterization of torsion free sheaves

In "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn a torsion free sheaf is defined as coherent sheaf $E$ on an integral Noetherian scheme $X$ s.t. for every $x\in X$ and every non-...
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29 views

Cech cohomology of a quasi-coherent sheaf on an affine scheme and Leray acyclicity Theorem.

Let $X$ be an affine scheme, $\mathcal{F}$ a quasi-coherent sheaf on $X$. Let $\mathcal{U}=\{U_i\}_{i \in I}$ be an affine covering of $U$ (not necessarely made up of principal open subsets). Moreover,...
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1answer
26 views

Statement in stack project on Relative Gluing

This is a screenshot of a statement from stacksproject. I have a number of confusion of this statement: What is a scheme $f_U:X_U \rightarrow U$ over $U$? Does this means that $X_U$ is ...
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34 views

Showing the Affine functor $\underline{\mathbb{A}}^r$ is representable by Affine Space $\mathbb{A}^r := Spec(\mathbb{Z}[X_1, \dots, X_r])$

Let $\underline{\mathbb{A}}^r$ be the functor from $\textbf{Schemes}$ to $\textbf{Sets}$ which associates to each scheme $S$ the set of morphisms $\bigoplus_{k=1}^rO_S \to O_S$ ($O_S$ is the structure ...
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If $G$ is a smooth scheme over $S$ of characteristic $p$, is the relative Frobenius morphism $F_{G/S}$ faithfully flat?

Let $G$ be a smooth scheme over $S$ of characteristic $p$, do we have that the reltaive Frobenius morphism $F_{G/S}$ is faithfully flat? There is an excersice in Liu's book saying that this is true ...
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1answer
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If $Y$ is an affine scheme, and $G$ is an $O_Y$-module, then why is $\operatorname{Hom}_{\operatorname{Mod}(Y)}(O_Y, G)\cong G(Y)$?

Question is entirely captured by the title. This has just been stated in a proof, and despite writing things out on paper I have not been able to see exactly why it should always be true. Potentially ...
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1answer
67 views

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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1answer
58 views

$Spec(R)$ 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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45 views

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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37 views

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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1answer
37 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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37 views

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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1answer
59 views

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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47 views

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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1answer
80 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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1answer
57 views

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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1answer
117 views

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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32 views

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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1answer
46 views

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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1answer
114 views

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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25 views

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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1answer
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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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1answer
73 views

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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1answer
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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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38 views

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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1answer
45 views

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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1answer
100 views

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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63 views

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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1answer
41 views

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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2answers
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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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108 views

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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150 views

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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60 views

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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1answer
49 views

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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25 views

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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1answer
120 views

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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1answer
98 views

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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1answer
64 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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55 views

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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0answers
38 views

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 ...