# Tagged Questions

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### In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
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### what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
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### Non-Noetherian ring $R$ with Spec($R$) a Noetherian Scheme

In looking at the examples of Non-Noetherian rings I knew/found I wasn't able to find one where I could conclude that Spec($R$) was a Noetherian scheme (not just merely a Noetherian topological ...
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### Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
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### Geometric reducedness (integral) versus reducedness (integral)

All the schemes here are over $\mathbb{C}$. Suppose $X \to Y$ is a morphism of varieties, then the geometric reducedness (integral) of the generic fibre implies the geometric reducedness (integral) ...
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### Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
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### Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
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### For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
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### Hartshorne's definition of structure sheaf

Hartshorne at page $70$ defines the structure sheaf on Spec $A$. The elements of $\mathcal O_{\textrm{Spec}A}(U)$ are particular functions $s:U\longrightarrow\coprod_{p\in U}A_p$. With the symbol ...
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### Relationship between maps and maps of rings

I take this exercise from Ravi Vakil's book. Consider the map of complex manifolds sending $\mathbb{C} \rightarrow \mathbb{C}$ via $x \mapsto y=x^2$. We interpret $\mathbb{C}$ as the $x$-line and ...
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### How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...
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### Generic point of a curve in affine plane

Consider the affine plane over $k$, i.e. Spec $k[x,y]$. There are three kinds of prime ideals: $(0)$, $(x-a,y-b)$, and $(f(x,y))$, for $f$ irreducible. Let the ideal $(f(x,y))$ correspond to the point ...
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### Proj construction and fibered products

How to show, that $Proj \, A[x_0,...,x_n] = Proj \, \mathbb{Z}[x_0,...,x_n] \times_\mathbb{Z} Spec \, A$? It is used in Hartshorne, Algebraic geometry, section 2.7.
Let $A$ be a ring and let $\mathcal{F}$ be the inductive system of subrings of $A$ which are of finite type over $\mathbb{Z}$: $$\mathcal{F} = \{ \mathbb{Z}[a_1,\dots,a_n] \subseteq A \mid n \geq 0, ... 1answer 123 views ### Ring homomorphism and affine scheme How to describe all ring homomorphisms f: A \rightarrow B, such that corresponding affine scheme morphism f: Spec \, B \rightarrow Spec \, A is open immersion? 1answer 106 views ### Functional sheaf (Hartshorne, Cartier divisors) In Hartshorne there is the following description of the sheaf K on the scheme. For each open U = Spec \, A we define K(U) = S^{-1} A, where S is the set of non-zero-divisors. Why is it a ... 1answer 102 views ### If f an isomorphism of ringed spaces, is f necessarily an isomorphism of locally ringed spaces? I'm not sure that this is generally true, but Harthorne p73 seems to suggest it. If it is true could someone give me a hint for the proof? 1answer 84 views ### Relation between DVR's of a local domain and localizations of its integral closure. \textbf{1.}\,\,\,\,\,\,\,\, Let (A,\mathfrak m_A) be a one dimensional local domain and let B be its integral closure in the fraction field L=\textrm{Frac}\,A. Assume that B is finitely ... 0answers 108 views ### Is reducedness an open condition? If X is a (general) scheme and X is reduced at p, i.e. \mathscr{O}_{X,p} is reduced, does there necessarily exist an open neighborhood of p on which X is reduced, i.e. \mathscr{O}_X(U) ... 1answer 557 views ### Affine scheme X with \dim(X)=0 but infinitely many points As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ... 1answer 364 views ### Prop. 2.3 Hartshorne: \varphi:A\to B induces a morphism \operatorname{Spec}(B)\to\operatorname{Spec}(A) I don't fully understand a step in the proof of the above-mentioned Proposition; more precisely, in part (b): If \varphi:A\to B is a homomorphism of rings, X=\operatorname{Spec}(A), ... 1answer 301 views ### The spectrum of a product of rings Let A be the product of a family (A_i)_{i\in I} of commutative rings, and c the canonical continuous map from the disjoint union U of the spectra of the A_i to the spectrum of A:$$ ...
Let $X$ and $Y$ be schemes over a field $k$ and $p,q$ the projections of $X \times Y$ on $X$ and $Y$. Let $M$ and $N$ be modules on $X$ and $Y$. Then the exterior product $M \boxtimes N$ is defined ...