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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

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How can we have he similar decomposition of $ \mathbb{Q}_p$ and $\mathbb{Q}_p(\zeta_p)$?

We have, $\mathbb{Q}_p=$p-adic field, $\mathbb{Z}_p=$ring of p-adic integers, $\mathbb{Z}_p^{\times}=$multiplicative group of units in $\mathbb{Z}_p$. We have the following decompositions: $\...
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Dimension of affine affine algebras as a module

Suppose that $A\cong \mathbb{R}[f_1,\dots,f_d]$ is a (commutative) affine $\mathbb{R}$-algebra (with identity). When is $A$ a finite-dimensional $\mathbb{R}$-module? Some examples are Simple $\...
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1answer
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Why is this a filtered category? Localization of rings.

I was reading this post I am trying to show that $S^{-1}R=\operatorname{colim}F(s)$, where $S$ is a multiplicative closed set in a commutative ring $R$ and $F$ is a functor from a filtered category ...
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Derivations of a formally smooth artinian ring

All rings are assumed to be commutative. Let $A$ be a ring, and $B$ an $A$-algebra which is Artinian local with the maximal ideal $\mathfrak m$. Suppose $B$ is formally smooth over $A$. My question ...
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$K[X^2,X^3]\subset K[X]$ is a Noetherian domain and all its prime ideals are maximal

Consider $K$ field and consider the ring $R=K[X^2,X^3]\subset K[X]$. It is clear that $R$ is not a Dedekind domain, since with the element $X$ one immediately see that it is not integrally closed. But ...
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Sufficient conditions on $M$ such that the intersection of infinitely many $\mathfrak{m} M$, $\mathfrak{m}$ maximal ideal, is zero?

Let $R$ be a one-dimensional, noetherian and reduced ring. Let $M$ be a finitely generated and torsion-free $R$-module. I am looking for sufficient conditions on $M$ such that $$\bigcap_{\mathfrak{m}}...
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1answer
44 views

Relation between the Zariski topology on $k^n$ and $\operatorname{MaxSpec}k[x_1,\dots,x_n]$

There is the Zariski topology on the set $k^n$. There is also the Zariski topology on the set of prime ideals of $k[x_1,\dots,x_n]$. I was wondering if anyone could explain, on a basic level, without ...
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1answer
26 views

Free generators for the localization of a module

Let $A$ a commutative ring with 1 and $M$ a finitely generated $A$-module. Assume the localization $M_p$ is a free $A_p$-module for some prime ideal $p$. Is it true that there is some set of ...
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1answer
46 views

Divisibility of a certain two variable polynomial by $x$

Let $F,G \in \mathbb{C}[x,y]$. (We do not know if $F$ and $G$ have a common zero or not). Assume that there exists $H(x,y) \in \mathbb{C}[x,y]$ with $H(0,0)=0$, such that $x$ divides $H(Fy, Gy)$. ...
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When can I have “short” projective resolution?

When we use universal coefficient theorem, we only need to compute Tor$(A,B)$. But suddenly, later on, like in more advanced homological algebra. We start to construct Tor$_n$, Ext$_n$... Can't we ...
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1answer
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Certain algebraic dependence of two polynomials

Let $u,v,d \in \mathbb{C}[t]$ be three polynomials such that $\deg(u) \geq 1,\deg(v) \geq 1,\deg(d) \geq 2$. Assume that $\gcd(u,v)=\gcd(u,d)=\gcd(v,d)=1$, and denote $U=du$ and $V=dv$. Further ...
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1answer
51 views

How to understand the notation $S(-3)$?

I asked a question about linear maps between free modules. I have a related question. Let $S=k[x_1, \ldots, x_n]$ where $k$ is a field. Then $S$ is a graded ring with usual grading given by $\deg x_i ...
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32 views

Constructive Krull dimension (singular sequence in localization)

In this question a constructive approach to Krull dimension is mentioned following this short note. Since the paper is very short and probably well known, I am not copying its contents. I hope this ...
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1answer
22 views

Minimal set of generators of an ideal

Let $R$ be a commutative local ring with maximal ideal $\mathfrak{m}$. Let $I$ be an ideal and $x\in R$ such that $x$ is not a zero divisor on $R/I$. Then a minimal set of generators for $I$ is sent ...
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1answer
64 views

The open sets in the Zariski topology are the complements of finite sets

What does "that is, on maximal ideals of $k[x]$" mean? Just before that remark it was said that we should work in the Zariski topology on $A^1(k)$, so the remark following is confusing. What exactly ...
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Quasi-compactness of $\mathrm{Spec}(A)$

I've been reading this question Compactness of $\operatorname{Spec}(A)$ and I don't quite understand what for all that job with complements was done. We have the following lemma: $\mathrm{Spec}(A)=\...
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Saturation and associated primes

Let $I\subset k[x_0, \ldots, x_n]$ a saturated ideal. Is true that $\mathrm{Ass}(R/I)$ doesn't contain any maximal ideal?
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1answer
61 views

Finitely generated extension of integral domains [on hold]

I have come across the following problem that I have not been able to solve. Especially, the implication (i) to (ii) remains elusive to me. Hints and/or solutions would be greatly appreciated! Let $...
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1answer
54 views

Exercise VIII.3.9 in Hungerford's Algebra

I am working on the following exercise (Exercise VIII.3.9 in Hungerford's Algebra). Let $R$ be Noetherian and let $B$ be an $R$-module. If $P$ is a prime ideal such that $P=\text{ann }x$ for ...
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Is $\langle x_{1}\cdots x_{n}-1 \rangle$ a prime ideal in $\mathbb{C}[x_{1},\dots, x_{n}]$?

I've heard from Google that the algebraic torus, the zero locus of $x_{1}\cdots x_{n}-1=0$, is an affine variety, which means $x_{1}\cdots x_{n}-1$ is an irreducible polynomial, which means $\langle ...
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1answer
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Which primes intersect the submonoid $f^\mathbb{N}(1+fA)\subset A$?

Let $A$ be a commutative ring and $f\in A,I\vartriangleleft A$. We have obvious submonoids $f^\mathbb{N},1+I\subset A$. The submonoid $f^\mathbb{N}\subset A$ is saturated, and its complement is the ...
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1answer
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(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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What is the meaning of the matrices over the arrows in a free resolution?

I am reading the lecture notes. I have a simple question about free resolution. What is the meaning of the matrices over the arrows in a free resolution? For example, the matrices in the following ...
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Check if the following ring homomorphism is flat?

Let $k$ be a field. Consider the ring morphism $f: k[x,y,x^{-1}, y^{-1}] \to k[t,t^{-1}]$ where $x \to t$ and $y \to t$. How do we know if $f$ is flat or not? Now let $R=k[x,y,x^{-1}, y^{-1}]$. Then ...
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2answers
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Do the complements of this chain of submodules form a chain?

I have two questions about the answer here: In a left noetherian ring, does having a left inverse for an element guarantee the existence of right inverse for that element? First question: they say ...
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1answer
33 views

Can two different multiplicative systems give same localisation?

Can we have two different multiplicative systems $S_1$ and $S_2$ in $\mathbb{Z}$ having same localisations $\mathbb{Z}_{S_1}=\mathbb{Z}_{S_2}$? I have a trivial solution by taking two different ...
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Functorial proof of Cayley-Hamilton using exterior powers

Let $V$ be a rank $n$ free module over a commutative ring $R$. Let $\dagger$ denote the adjoint with respect to the natural perfect pairing given by the wedge product $$\Lambda ^k\otimes \Lambda ^{n-k}...
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When $R$ is Noetherian, $\prod M_i$ is injective implies $M_i$ is injective.

Let $R$ be a ring and $\{M_i | i\in \mathcal{I}\}$ a family of $R$-modules. Show that in case $R$ is Noetherian, $\bigotimes\limits_{i\in\mathcal{I}}M_i$ is injective if and only if $M_i$ is injective ...
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1answer
49 views

Veronese embedding as ring isomorphism [duplicate]

At the level of rings, the Veronese map corresponds to isomorphisms like $$k[w,x,y,z]/(wz - xy, wy-x^2, xz - y^2) \cong k[a^3, a^2 b, a b^2, b^3].$$ This isomorphism is the statement that the ...
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1answer
43 views

Proof of a stronger form of Chinese remainder theorem (12.3) in Neukirch

Let $\mathcal O$ be an order in a number field and ${\mathfrak a} \neq 0$ an ideal in $\mathcal O$. Then the theorem shows ${\mathcal O}/{\mathfrak a} = \oplus_{{\mathfrak p} \supseteq {\mathfrak a}} {...
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If $\dim(A/\mathfrak p)=\dim(A)-1$, then is $\mathfrak p$ principal?

Let $A$ be a commutative ring, and let $\mathfrak p$ be a prime ideal in $A$. When is it true that if $\dim(A/\mathfrak p)=\dim(A)-1$ then $\mathfrak p$ is a principal ideal in $A$? I'm pretty sure $...
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1answer
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If two ideals agree on $K\lhd R$ and yield the same quotients $\pi_K(I)=\pi_K(J)$, are they equal?

Inspired by my lectures on the basics of commutative algebra, I tried to investigate whether the noetherian property still holds when taking extensions (here meaning: if $I\lhd R$ and $R/I$ both ...
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What are the prime ideals in the ring $\mathbb{Z}[i](\epsilon)/(\epsilon^2) $?

Let $\mathbb{Z}[i,\epsilon] \simeq \mathbb{Z}[i](\epsilon)/(\epsilon^2)\simeq \mathbb{Z}[x,\epsilon]/(x^2 +1, \epsilon^2)$ be the Gaussian integers with an infinitesimal number $\epsilon^2 = 0$. What ...
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57 views

Hilbert Nullstellensatz, Eisenbud's proof

I am trying to understand the proof on Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. The theorem I am trying to prove is: Let $k$ be an algebraically closed field. If $I \...
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1answer
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points vs functions in prime spectra

(Commutative algebra and algebraic geometry - Bosch - page 16) I am struggling to reconcile the highlighted items in the following extract: I can understand (2) : if Ring element $f$ is a member of ...
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1answer
25 views

$G$-Action on a Ring extends to a Module

Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$. Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well ...
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Finite ring extensions and finite field extensions

Let $R\subset S$ be two finitely generated integral domains over an algebraically closed field $k$. If $S$ is finite as $R$-module then $[L:K]<+\infty$, where $L$ and $K$ are the quotient fields of ...
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Showing that $F$ is not representable [closed]

As I'm trying to find (counter)examples of representable functors, I tried looking up some instructive examples. One of the counterexamples I'm having trouble with, is the following: Show that the ...
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1answer
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What does this symbol mean in commutative algebra?

(Algebraic Geometry and commutative algebra - Bosch - page 16) I could not find the symbol in the glossary at the end of the book. What does it mean ? I cannot see it in my reference manual either ...
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Definition of $going-up$ map

The exercise 5.10 of Atiyah's Commutative Algebra gives the definition of $going-up$ map: A ring homomorphism $f:A\rightarrow B$ is said to have the $going-up$ (resp. the $going-down$ property) if ...
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Generalization of Atiyah-Macdonald Proposition 5.7

The Proposition is Let $A\subseteq B$ be integral domains, $B$ integral over $A$. Then $B$ is a field iff $A$ is a field. The proof is easy. I want to generalize this proposition. I want to prove ...
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The intersection of the associated primes of a reduced ring

Let $R $ be a commutative ring with identity. Recall that a prime ideal is called associated prime ideal whenever it is the annihilator of a nonzero element. Also a ring is called reduced whenever has ...
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Localization of self dual algebra over a ring

Let $R$ be a commutative ring with identity and $A$ an $R$-algebra that is finitely generated, projective $R$-module and has identity. Furthermore, assume that $A$ and $Hom_R(A,R)$ are isomorphic as $...
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Is there an isomorphism between $ \text{Spec}(R_{\mathfrak{p}}) $ and the prime ideals of $ R $ which are contained in $ \mathfrak{p}$?

Suppose $ R $ is a ring, and $ \mathfrak{p} \in \text{Spec}(R). $ I have been told that $ \text{Spec}(R_{\mathfrak{p}}) \cong \lbrace \mathfrak{q} \in \text{Spec}(R)\;| \mathfrak{q} \subset \...
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Is $\operatorname{coker}\colon\mathcal C^{[1]}\to\mathcal C$ a faithful functor?

Let $\mathcal C$ be an abelian category; e.g., $\mathcal C=\operatorname{\mathit{A}-Mod}$. Let $\mathcal C^{[1]}$ denote the category of morphisms in $\mathcal C$, where morphisms $\mathcal C^{[1]}$ ...
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quotients by extended and contracted ideals as tensor products?

Let $f:A\to B$ be a morphism of commutative rings and let $I\vartriangleleft A,J\vartriangleleft B$ be ideals. The left square below is a pushout. Question 1. When is the right square a pushout as ...
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1answer
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Flatness of Residue Field

My question refers to a step of in the proof of Corollary 8.5.17 in Bosch's "Commutative Algebra and Algebraic Geometry"; see page 395 See the red tagged line below: We consider the exact sequence ...
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1answer
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Relation between $k[a,b]$ and $k[c,d]$, given that $(a,b)=(c,d)$.

Let $k$ be a field of characteristic zero, $R$ a commutative $k$-algebra, $a,b,c,d \in R$ such that the ideal generated by $a$ and $b$, $I_{a,b}=(a,b)$, equals the ideal generated by $c$ and $d$, $I_{...
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'Solutions' to a specific equation in $K[t]$

Let $K$ be a field of characteristic zero. Let $f,g,h \in K[t]$ be three separable polynomials, with no common zeros. Denote $\deg(f)=a,\deg(g)=b,\deg(h)=c$, $a \geq 2, b \geq 1, c \geq 1$, and denote ...
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1answer
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A question about initial ideals.

Let $R = k[x_1, \dots, x_m]$ be a polynomial ring over a field $k$ and $I, J$ be ideals of $R$. Further assume that $J$ is generated by the polynomials $f_1, \dots, f_r$. Fix a monomial order $<$ ...