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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When is the diagonal a closed immersion with open image?

Let $A$ be a commutative unital $R$-algebra and consider the multiplication map $m:A\otimes_R A\rightarrow A$ given by $m(a\otimes b) = ab$ with kernel $I$. The map $m$ is surjective, hence the image ...
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Every affine k-scheme can be embedded into an affine space?

While reading through Algebraic Geometry I by Görtz and Wedhorn (2nd version) I came across the following remark on p. 352 (between Proposition 12.66 and Corollary 12.67): If $Y$ is an affine $k$-...
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Quotient of an affine scheme under the action of a finite group

I am trying to solve Exercise 2.3.21 from Liu’s book. I have a finite group $G$ acting on an affine scheme $\operatorname{Spec}A$, and I want to show that the quotient scheme $X/G$ exists and is ...
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Affine-local Computation of Scheme-Theoretic Image for Reduced Schemes

The following is part of an argument in Vakil's Rising Sea, p. 238. Let $\pi : X \to \operatorname{Spec}{B}$ be a morphism of schemes with $X$ reduced and $g \in B$. Consider the morphism $$\pi^{\#}_{...
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Let $X$ be an irreducible algebraic set and let $f$ be in $A(X) = \mathcal{k}[x_1,\ldots,x_n]/I(X)$. Why is f a function on $X$? [closed]

I am reading Hartshorne and I am watching Lothar video's on algebraic geometry. In Lothar's video he mentions that $f \in A(X)$ can be thought of as functions on X. Can someone explains that?
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Show that $\mathbb{A}^2_\mathbb{C}\not\simeq \mathbb{A}^1_\mathbb{C} \times_{\operatorname{Spec}(\mathbb{Z})} \mathbb{A}^1_\mathbb{C}$

This is exercise 5.6.10 in Gathmann's note: Show that $\mathbb{A}^2_\mathbb{C}\not\simeq \mathbb{A}^1_\mathbb{C} \times_{\operatorname{Spec}(\mathbb{Z})} \mathbb{A}^1_\mathbb{C}$. The problem is ...
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Is the global sections functor exact for a SES of O_X-modules on an affine scheme when the middle is quasicoherent?

Let $X$ be an affine scheme. Suppose we have an exact sequence of $\mathcal{O}_{X}$-modules \begin{equation*} 0 \to \mathcal{F}_{1} \xrightarrow{\phi} \mathcal{F}_{2} \xrightarrow{\psi} \mathcal{F}_{...
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Hartshorne Ex. II.3.21: computing height of an ideal.

Consider a DVR $R$ with the maximal ideal $(u)$ containing its residue field $k=R/(u)$. The exercise claims that the dimension of $\operatorname{Spec}(R[t])$ is not equal to the Krull dimension of all ...
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When is the glueing of affine schemes again affine?

Let $(R_i)_{i \in I}$ be a family of rings and suppose that we have compatible isomorphisms of subschemes of the spectra of the $R_i$ so that we are able to glue the corresponding affine schemes $\...
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Questions about intersection of algebraic subgroups

I have a question related to Proposition 1.49 of the book Algebraic Group of Milne. Here is the excerpt: My question is the last line. I don't really understand the reasoning in that line. From what ...
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Generalisations belong to any affine open subset of a scheme

I was reading the proof of Lemma 26.13.2. here and it seems they use the following fact, but I haven't been able to find a proof of it. Could anyone provide a reference/proof? Let $X$ be a scheme and ...
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Scheme isomorphic to the affine line

I am asked to prove that if $R=\mathbb Z[x,y]/(x^2+y^2-5)$ and $p$ is a prime $\equiv3\pmod4$, then $\text{Spec }(R\otimes_{\mathbb Z}\mathbb F_{p^2})$ is isomorphic to the affine line $\mathbb A_{\...
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Is the intersection of two affine open sets affine in a separated k-scheme?

Let me make a very begginer question: Let $X\rightarrow Spec(k)$ be a scheme morphism. This (I believe) is named a $k$-scheme. I also have read that some authors define a $k$-algebraic variety like a ...
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Sheaf of a scheme

I'm reading Algebraic and Arithmetic Curves written by Qing Liu. I have a question about the properties of a scheme. Let $X$ be a scheme and $x\in X$ an arbitrary point, we can find an open subset of $...
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Does every scheme admit an affine open subset whose sections are the ring global sections?

The title basically says it all. Given an arbitrary scheme $X$ with global sections $\mathcal{O}_X(X)$, is there an affine open subset $U$ of $X$ such that $\mathcal{O}_X(U) = \mathcal{O}_X(X)$? ...
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Frobenius on affine space is a bijection

Similar questions have been asked in various different settings, but I am not satisfied with the array of answers which have been received. If something truly is a duplicate on the nose, I will be ...
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Is an affine scheme compact?

This is probably a very stupid question, because it is very well known that the spectrum of a ring is compact (quasi-compact, if you want). But an affine scheme $(X,\mathcal{O}_X)$, if I am right, is ...
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Non-quasi-compact open subscheme of an affine scheme

Question 1: Is my proof that $\operatorname{Spec}k[T_1,T_2,\dots]-V((T_1,T_2,\dots))$ isn't quasi-compact correct? Here is my attempt: Let $X$ denote said scheme. Consider the sequence of ideals $$(0)\...
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Is it true that $\mathscr{O}_{X,x}=\mathscr{O}_{\operatorname{Spec} A,x}$ for any $\operatorname{Spec} A$ with $x\in\operatorname{Spec} A$?

Let $X$ be a scheme and $\operatorname{Spec} A\subset X$ be an affine with $x\in\operatorname{Spec} A$, Is it true that $\mathscr{O}_{X,x}=\mathscr{O}_{\operatorname{Spec} A,x}$? I ask this question ...
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Grothendieck point of view of algebraic geometry

Given a ring $R$ and $I\subseteq R[x_1,\dots ,x_n]$ an ideal, define the functor $V_R(I):\operatorname{Alg}_R\to \operatorname{Sets}$, that sends a $R$-algebra $A$ in the subset of points $\mathbf a \...
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Does every affine open of a closed subscheme come from an affine open of the ambient scheme?

Let $(\iota,\iota^{\#}):(Y,\mathcal{O}_Y)\to (X,\mathcal{O}_X)$ be a closed immersion of schemes. Is it true that for every affine open $V\subseteq Y$, there exists an affine open $U\subseteq X$ with $...
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How to construct a closed subscheme from an ideal sheaf $\mathcal{J}$ such that $\mathcal{J}(U)=I_U$?

I'm following Vakils book 'Foundations of Algebraic Geometry', and I'm currently in section 8.1 on closed subschemes. More precisely, I'm trying to do Exercise 8.1.H, which is the following: Let $(Y,\...
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Two questions about integral scheme

Following are examples of 3.0.1 in Hartshorne's textbook If $X=\operatorname{Spec}A $ is an affine scheme, then i) $X$ is reduced if and only if $\operatorname{Nil}A=0$ $(\Rightarrow)$ for every open ...
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How to describe the intersection of all ideals in the image of the canonical morphism $(X,\mathcal{O}_X)\to\operatorname{Spec}\mathcal{O}_X(X)$?

For $(X,\mathcal{O}_X)$ a general scheme, we have a morphism $(\pi_X,\pi_X^{\#}):(X,\mathcal{O}_X)\to\operatorname{Spec}\mathcal{O}_X(X)$ induced by the identity of the corresponding rings of global ...
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How to glue the integral closures of the affine pieces in an integral scheme (Hartshorne II.3.8)?

I'm doing Exercise 3.8 in Chapter II of Hartshorne, on the construction of the normalization of an integral scheme. Let $X$ be an integral scheme and let $\{U_i=\operatorname{Spec}(A_i)\}_{i\in I}$ ...
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Higher partial derivatives of a regular function

Let $X$ be a smooth algebraic variety over a field $k$, and let $x\in X$. Let $(x_1,\dots,x_n)$ be a regular system of parameters at $x$, so that $\Omega^1_{X/k}$ is locally free at $x$ with basis $...
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Exercise 8.3.E in Vakil FOAG : equivalence of union of $\operatorname{Spec} (k(\mathcal{O}_{Y,p}))$ with smallest closed subscheme containing $X$

As explained in 8.3.9 of http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf, we consider $X^{set}$ a closed subset of $Y$ and attempt to define a canonical scheme structure $X$ on $X^{set}$....
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How to glue the sheaf of differentials on a scheme

I have been trying to really understand the sheaf of differentials defined on a scheme but am having a lot of trouble. Let $X$ be a scheme separated over a field $k$. Suppose we have an open affine ...
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Why are finite intersection of affine open sets of a scheme not always affine?

I seem to have confused myself about the notion of separatedness. I am not even sure how it makes sense to say that the intersection of affine open subsets is affine when a scheme is separated, since ...
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open generic points of affine scheme?

Let $S=\text{Spec} A$ be an affine scheme, we assume $A$ is not a field, then we know irreducible components of $S$ correspond to all minimal prime ideals of $A$, in fact, these prime ideals are ...
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Exercise 6.6.A in Foundations of Algebraic Geometry

The exercise is "Suppose $X$ is a $\mathbb{C}$-scheme. Verify that there is a natural bijection between maps $X → \mathbb{A}_C^1$ in the category of $\mathbb{C}$-schemes and functions on $X$.&...
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Integral extension of a discrete valuation ring

$X$ and $Y$ are integral noether schemes over $\mathbb{C}$, and $F:X\rightarrow Y$ is a surjective morphism. Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its fraction field $K$, and $...
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Characterizing pairs of affine schemes $X$ and closed subschemes $Y$ so that $X \setminus Y$ is affine

Let $R$ be a ring and let $X = \text{Spec}(A)$ be an affine $R$-scheme. I will from now on not mention the base $R$ anymore and just say affine scheme or algebra instead (it is fine for me to assume ...
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A ring homomorphism defines a morphism of locally ringed spaces (Hartshorne proof)

I was reading proposition 2.3 and I could not prove the following Given a homomorphism of rings $\phi \colon A \to B$, we define a map $f \colon \operatorname{Spec} B \to \operatorname{Spec} A$ by $f(...
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Showing that $\Gamma(X,\mathcal{O}_X)=k$

Let $X$ be an integral scheme, proper over an algebraically closed field $k$. It is known that $\Gamma(X,\mathcal{O}_X)=k$. In the book that I am reading, the author claims that this fact follows from ...
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Proof that $\mathbb{P}^1$ is not affine.

I am confused by the last line in the proof that $\mathbb{P}^1$ is not affine, as presented in Ravi Vakil's algebraic geometry notes. First, he computes the ring of global sections. It turns out that $...
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Question about recovering the ring of an affine scheme

I am confused about a line in Vakil's algebraic geometry notes (November 2017 version, page 136) right after he gives the definition of a scheme. Suppose we have an affine scheme $(X,\mathcal{O}_X)$. ...
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Weil Restriction and Distinguished Opens

I have a pair of related questions about Weil Restriction. Let $E/F$ be a field extension, and let $A$ be an $E-$algebra. Assume that all relevant restrictions of scalars exist. We have a norm map $n: ...
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fibre with only one point is isomorphic to the spec of a field

Let $R$ and $T$ be commutative rings with unity. Let $Q$ be a prime ideal of $R$ and $\phi:R \to T$. Suppose $T \otimes_R (R_Q/Q R_Q)$ has only one prime ideal. Then I would like to prove that the ...
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Can the structure sheaf of the spectrum of a ring be defined by taking appropriate localizations on every open set?

I recently learned that the structure sheaf on the spectrum of a ring $\mathrm{Spec}(R)$ is first defined on the distinguished open subsets $D_f$ for $f\in R$ so that $\mathcal{O}(D_f)=R_f$ where $R_f$...
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Exercise 11.1.G Vakil FOAG

I am trying to solve Ex 11.1.G from Ravi Vakil's FOAG. It says if $X$ is an affine scheme over $k$, a field and $K|_k$ is an algebraic field extension, then $X$ is of pure dimension $n$ iff $X_K:=X\...
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differentials and tangent space of a fibre

The setup I have is as follows: Let $f: X \to Y$ be a morphism of non-singular $n$-dimensional varieties (separated reduced irreducible scheme of finite type over $k$) over $k$ an algebraically closed ...
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1answer
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How to define a morphism from the Spec of the completion of $O_{Y,y}$ to $Y$?

Let $Y$ be a noetherian scheme and $y \in Y$. We denote by $\hat{O}_{Y,y}$ the completion of the local ring $O_{Y,y}$. I want to define a morphism $$ \operatorname{Spec} \hat{O}_{Y,y} \to Y $$ which ...
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On global section of Hom-functor on quasi-coherent sheaves on a quasi-affine scheme

Let $R$ be a commutative Noetherian ring and $U$ be an open subscheme of the affine- scheme $X=\text{Spec}(R)$ such that $\Gamma_U(\mathcal O_U)\cong R$. If $\mathcal E, \mathcal F$ are quasi-coherent ...
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fibre of a scheme of finite type over a field being quasi-compact

Let $X,Y$ be schemes of finite type over a field $k$. In particular, they are quasi-compact. Let $f: X \to Y$ be a morphism of finite type (for all open affines $U \subset Y$, $f^{-1}(U)$ is quasi-...
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110 views

Spec of $k(X)$ is the fibre of $f$ over the generic point of $Y$ given $f: X \to Y$

I am trying to understand a part of proof of Corollary 2 in Section III. 5 in Mumford's Red book. Suppose I have an etale dominant morphism $f: X \to Y$ where $X$ and $Y$ are separated irreducible ...
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$X$ is locally the locus of roots of $n$ equatinos in $Y \times \mathbb{A}^n$ given etale $f: X \to Y$

We have $f: X \to Y$ an etale morphism of varieties (separated, irreducible, reduced scheme of finite type over $k$) over $k$, an algebraically closed field. Why is $X$ locally the locus of roots of $...
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111 views

If $\operatorname{Spec} A=\operatorname{Spec} B$, then $A$ is isomorphic to $B$

Proposition II.$3.2$ in Hartshorne Regarding to this question, I wonder why $\operatorname{Spec} A=\operatorname{Spec} B$, then $A$ is isomorphic to $B$. To be more precise, Let $A,B$ are rings, and ...
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38 views

Geometric fibres of a spec of a field

Suppose I have a morphism of schemes $f: X \to Y$, and the fibre $f^{-1}(y)$ as a scheme over $k(y)$, the residue field of $y$, is a union of finite set of points $\operatorname{Spec} k_i$, $k_i$ ...
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Definition of étale morphism in Mumford

I am trying to understand the definition of étale morphism in Mumford Chapter III Section 5, which I find confusing. I would appreciate any clarifications. A morphism $f: X \to Y$ of finite type is ...

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