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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Open set of the cuspidal curve is not principal

I'm struggling with this problem: Let $k$ a field and $X=\operatorname{Spec} k[x,y]\big/(x^3-y^2)$ and $U=X\setminus\{(x-1,y-1)\}$. Show that $U$ is not a principal open set in $X$. I tried to show ...
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Why is $H^0(X_n, \mathcal{O}_{X_n})$ a local artin ring.

Let $V$ be regular, proper and of dimension 2 over $S = $ spec $R$, for $R$ a complete discrete valuation ring with uniformizer $t$, maximal ideal $\mathfrak{m}$, and algebraically closed residue ...
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Closed points of $\operatorname{Spec} R$ maximal?

Show that the closed points of $\operatorname{Spec} R$ are exactly the elements of $\operatorname{maxSpec} R$ that is, every closed point of $\operatorname{Spec} R$ is maximal. Proof attempt: Let $\{\...
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Local ring of a scheme whose reduction is the affine line.

Let $k$ be a field and $A=k[X,Y]/\langle Y^2,XY\rangle = k[x,y]$ the algebra corresponding to the well-known non primary ideal $I=\langle Y^2,XY\rangle \subset A$ with embedded associated ideal $M=\...
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Where is the error in this proof that all morphisms of schemes are quasi-compact?

Lemma 1. All preimages of affine open subschemes are affine (Inverse of open affine subscheme is affine). Lemma 2. All affine schemes are quasi-compact (Proof that an affine scheme is quasi compact). ...
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Defining morphism of sheaves [duplicate]

Suppose we have sheaves $\mathcal{F},\mathcal{G}$ on a topological space $X$ where $\mathcal{U}$ is a base of $X$. Then to define a morphism $\varphi:\mathcal{F}\rightarrow \mathcal{G}$, is it enough ...
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Are there criteria for Zariski closure of principal affine open?

Let $A$ be a commutative ring. Let $f\in A$ be a nonzero element. Then $\mathrm{Spec}(A_f)\to\mathrm{Spec}(A)$ is an open immersion of affine schemes. Consider the Zariski closure $\overline{\mathrm{...
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Specific open covering in affine scheme

Theorem 3.42 (Wedhorn and Görtz "Algebraic Geometry") states that, for $X = \operatorname{Spec} A$, there is a bijection between ideals of $A$ and closed subsechemes of $A$ given by $\...
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Easy question regarding open subschemes of closed subschemes

I am trying to follow the proof of the theorem 3.42 of Wedhorn's and Görtz book on algebraic geometry. The step I can't follow is like this: suppose we have $Z \subseteq X = \operatorname{Spec}A$ a ...
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Question about a proof that all étale morphisms are locally standard étale

In chapter 1 of Milne's Étale cohomology book from 1980, Theorem 3.14 states that : If $f: Y\longrightarrow X$ is étale in some open neighbourhood of a point $y\in Y$, then there are affine open ...
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Is there a field k such that $Spec(k)\times \mathbb P^1$ is affine?

Is there a field k such that $Spec(k)\times \mathbb P^1$ is affine? Probably I have no technical background to claim any good idea on this. However, I think the answer should be negative since $Spec(k)...
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Show that $Z_{red}=(Z,\mathcal{O}_{Z_{red}})$ is a scheme and it satisfies the universal property

I am trying to show that, given a scheme $(X,\mathcal{O}_X)$ and a closed subset $Z\subset X$, with the subsheaf $$\mathcal{I}^Z_X(U)=\{f\in\mathcal{O}_X\mid f(z)=0 \forall z\in Z\}.$$ Then define a ...
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Computing residue fields of affine schemes

I have taken a course on schemes, so I am familiar with the basic definitions, but I'm very rusty and I've forgotten how to do this (if I ever knew). Basically, I want to compute the residue fields of ...
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Generic zeros of a polynomial according to Lang's IGA

The concept of generic zero of a prime ideal is defined in Lang's Introduction to Algebraic Geometry at page 27 (1972 ed.). Because the setting is quite weird in my opinion (I have posted another ...
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Picard Group of the spectrum of a Noetherian UFD.

Let $A$ be a Noetherian UFD, how can I compute $\text{Pic}(A):=\text{Pic}(\text{Spec}(A))?$ The Picard group of a scheme $X$ is defined as the group of isomorphism classes of invertible $\mathcal{O}_X$...
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Is a blowing up always surjective?

Consider $A$ a commutative ring, and $I\leq A$ an ideal. Let $X=\mathrm{Spec}(A)$ and $Z=\mathrm{Spec}(A/I)$. Consider the blowing-up algebra \begin{equation} \mathrm{Bl}_I(A):=\bigoplus_{i=0}^\infty ...
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What schemes correspond to varieties in the sense of Weil?

Out of (perhaps morbid) curiosity I am trying to learn the basics of Weil's foundations of algebraic geometry. I tried to ask a question earlier but it turned out I had misunderstood some more basic ...
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Prove that if a ring homomorphism is surjective, the associated spectral map is injective

I recently had a bunch of questions on a problem set that I could not solve and the instructor is not providing solutions. Let $f: A \to B$ be a surjective ring homomorphism. Prove that the induced ...
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Is the "vanishing locus" of a section of a quasi-coherent sheaf closed?

Suppose $A$ is a (possibly noetherian) ring and M a (possibly finitely generated) $A$-module. Consider for $m \in M$ the set $$\{m = 0\} := \{\mathfrak p \subset A \text{ prime ideal} \,|\, m_{\...
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When is the closed subscheme of $\Bbb P^n_{\Bbb C}$ cut out by a homogeneous polynomial $f$ integral?

I am new to algebraic geometry. I am trying to understand whether closed subschemes of $\mathbb P^n_{\mathbb C}$ is an integral scheme. Let $X = \operatorname{Proj} (A/(f))$ where $f$ is an ...
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If $f:\operatorname{Spec} A\to\operatorname{Spec} B$ is a map of schemes, how can I show $V(\varphi^{-1}(I)) = \overline{f(V(I))}$ for $I\subset B$?

If $X = \operatorname {Spec}(A)$ and $Y = \operatorname{Spec}(B)$ are affine schemes and I have a morphism $f : X \to Y$, then I am trying to show $$V(\varphi^{-1}(I)) = \overline{f(V(I))}$$ for any ...
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How to prove Neron model is not affine scheme?

Let $K$ be a local field and $E/K$ is an elliptic curve over it. Neron model $ε$ of $E/K$ is a smooth group scheme over $SpecR$ which satisfies Neron mapping property. Neron model is known to be ...
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Example of scheme which is not an affine scheme

In Hartshorne, On Page 75 Example 2.3.6. I understood that gluing between two schemes is a scheme but how to prove that it is not an affine scheme without using the notions of next Chapters like ...
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Is quotient of group scheme is again group scheme?

Is quotient of group scheme is again group scheme ? The following is the case I'm interested in. Let $K$ be a local field, and $R$ be ring of integers of $K$, then,let ε be a group scheme over Spec$R$ ...
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On closed points in quasi-compact schemes

I'm working on this exercise: Show that if X is a quasicompact scheme, then every point has a closed point in its closure. Show that every nonempty closed subset of X contains a closed point of X. I ...
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3 votes
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Structure sheaf of $\operatorname{Spec} A$ with $A$ an integral domain

Let $A$ be an integral domain, $K = \operatorname{Quot}(A)$ the field of fractions of $A$ and $X=\operatorname{Spec}(A) = \{p\subset A \mid \ p \ \text{prime ideal of }A \}$ the spectrum of $A$ with ...
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Proving that the disjoint union of two affine schemes is isomorphic to an affine scheme

An "Easy Exercise" from Vakil's text: Show that the disjoint union of a finite number of affine schemes is also an affine scheme. To me it's not so easy. Let's do it for two affine schemes ...
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Quasi-coherent sheaves of algebras over affine scheme

Hi there I am study algebraic geometry for the first time and I have a question. If we consider an affine scheme $\operatorname{Spec}(R)$, is the category of quasi-coherent sheaves of $\mathcal{O}_{R}$...
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Equality of two Zariski closures

Let $X$ be an affine scheme, $R=\Gamma(X,\mathcal O_X)$, $R_0$ a subring of $R$, $Y=\mbox{Spec}(R_0)$, and $\phi:X\rightarrow Y$ the morphism induced by the inclusion of $R_0$ in $R$. Let $\{I_i\}$ be ...
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Open subsets of affine schemes (refined question)

We work over the complex numbers, $\mathbb{C}$. Let $X$ be a smooth affine variety. Let $U$ be an open subvariety of $X$. Then we have a natural map $$ i:\operatorname{Spec}\mathbb{C}[U]\to X $$ My ...
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Exercise 8.1.H from Vakil's FOAG: criterion for ideals in affine open sets defining closed subscheme

I'm trying to solve the following exercise, namely Exercise 8.1.H, from Vakil's FOAG: As hints, there are three approaches provided, of which I'm following roughly the first. The way I want to do ...
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Example of a non-affine scheme whose reduction is affine

In Hartshorne Exercise III.3.1, he asks the reader to prove that a noetherian scheme $X$ is affine iff $X_{\bf{red}}$ is affine. I am not sure if Noetherian is needed here (of course I only know the ...
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1 vote
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Induced morphism between affine schemes is injective

$\DeclareMathOperator{\Spec}{Spec}$ Problem Let $X = \Spec A$ be an affine scheme and $\Spec B = Y \subset X$ an affine open subset of $X$. Let $\rho$ be the restriction map of $\mathcal{O}_X$. We ...
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2 answers
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Understanding the morphism of schemes involving nilpotents induced by: $\mathbb C[x] \to \mathbb C[x]/(x^2)$ where $x \mapsto ax \pmod {x^2}$

From Vakil's FOAG: The following imprecise exercise will give you some sense of how to visualize maps of schemes when nilpotents are involved. Suppose $a \in \mathbb C$. Consider the map of rings $\...
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Proving a surjective k-algebra homomorphism to be an isomorphism

I am trying to prove the following statement: Let $f:A\to B$ be a surjective $k$-algebra homomorphism. Let $A$ and $B$ be local Noetherian rings with same finite Krull dimension. Then $f$ is an ...
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Inclusion of open affines implies injection of coordinate rings?

Let $U, V$ be two affine open subsets of a scheme $X$ (Noetherian, over an algebraically closed field of characteristic $0$) and assume that $U \cap V$ is also affine. Is it true that the map $\...
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How to prove the Frobenius induces a homeomorphism?

Suppose $A$ and $B$ are two $\mathbb F_p$-algebras, and denote the Frobenius on $A$ by $\sigma$. Then $\sigma \otimes id$ is an endomorphism on $A\otimes_ {\mathbb F_p} B$. How to prove it induces a ...
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Can we define $\operatorname{Spec}(H^0(X, \mathcal O_X))$ for a scheme $X$?

If $X=\operatorname{Spec} A$, then $H^0(X,\mathcal O_X)=\mathcal O_X(X)=A$ is a ring. (1) For a general scheme $X$, does the space $H^0(X,\mathcal O_X)$ of global sections of structure sheaf still ...
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Definition of the structure sheaf on $\text{Spec} A$

In his book Algebraic Geometry, Hartshorne defines the structure sheaf of $\text{Spec} A$ to be the set of functions $s:U\to\coprod_{p\in U}A_p$ such that $s(p)\in A_p$ and $s$ is locally a quotient ...
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Is a pure coherent sheaf locally pure?

Let $X$ be a Noetherian scheme and $F$ be a pure coherent sheaf of $O_X$-modules, that is any non-zero subsheaf $E\subset F$ satisfies $dim(E)=dim(F)$ (by dimension of a sheaf, I mean the dimension of ...
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How to get such a scheme?

Here is an exercise from the Wedhorn's book. Exercise 3.2. Prove that there exists a scheme which admits a covering by countably many closed subschemes each of which is isomorphic to the affine line $\...
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How to find the lifting in the spectra of tensor product?

Let $A,B$ be two $R-$algebra. (i.e. we have two homomorphisms: $f:R\rightarrow A$ and $g:R\rightarrow B$) We denote two canonical homomorphism $A\rightarrow A\otimes_R B$ and $B\rightarrow A\otimes_R ...
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How to show compact, Hausdorff and totally disconnected space is homeomorphic to the spectra of a boolean ring?

Let $X$ be a compact, Hausdorff and totally disconnected space. By giving the field of two elements $\mathbb F_2$ the discrete topology, let $A$ denote the set of continuous maps from $X$ to $\mathbb ...
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Tensor product of quasi-coherent sheaves on an affine scheme

Given an affine scheme $X = \text{Spec}(A)$, then from an $A$-module $M$ we can form the associated sheaf $\tilde{M}$, where $$ \tilde{M}(D_f) = M_f = M \otimes_A A_f. $$ Now, given $A$-modules $M$ ...
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Alternative definition of sheaf associated to a module?

Given an affine scheme $X = \text{Spec}(A)$, then from an $A$-module $M$ we can form the associated sheaf $\tilde{M}$, where $$ \tilde{M}(D_f) = M_f = M \otimes_A A_f $$ This agrees with the presheaf $...
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Could you provide an example for a nonreduced local ring with reducible Spectrum?

As the title says, could you provide a local ring which is nonreduced and its $Spec$ is reducible? It's one of the exercise in the Wedhorn's book. To get a local ring I only need to do localization at ...
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Resolving a coherent sheaf $\mathcal F$ over $\mathbb P^n_S$ via sheafs of the form $\oplus^p\mathcal O_{\mathbb P^n_S}(n)$, where $S$ is Noeth., aff.

I am studying Nitsure's exposé on the representability of the Hilbert functor. On p. 14, he makes the claim that if $S$ is an affine Noetherian scheme and $\mathcal{F}$ is a coherent sheaf over $\...
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2 votes
0 answers
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About the definition of Schemes

It is said that scheme theory was made to get something that it was not "visible" in classical algebraic geometry, for example as varieties $V(x)=V(x^2)$, in general. If $X$ is an affine ...
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Why is the structure sheaf on an affine scheme defined the way it is, and why is there a difference in the literature?

Let $A$ be a commutative unital ring. I am confused about a subtelty in defining the structure sheaf on $\text{spec}A$. The most conventional way seems to be to define a sheaf on a base, and then show ...
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$\DeclareMathOperator{\Spec}{Spec}$Doubt about Nike's Lemma about the intersection of $\Spec A$ and $\Spec B$

Nike's lemma. Suppose $\Spec A$ and $\Spec B$ are affine open subschemes of a scheme $X$. Then $\Spec A \cap \Spec B$ is the union of open sets that are simultaneously distinguished open subschemes of ...
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