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:
$\...
2
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0answers
16 views
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
20 views
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 ...
2
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0answers
31 views
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 ...
3
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2answers
83 views
$K[X^2,X^3]\subset K[X]$ 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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0answers
19 views
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 ...
2
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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 ...
1
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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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1answer
40 views
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
16 views
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 ...
2
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1answer
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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0answers
44 views
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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0answers
21 views
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?
1
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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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2answers
88 views
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
40 views
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 ...
1
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1answer
25 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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0answers
40 views
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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0answers
40 views
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 ...
2
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2answers
37 views
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 ...
2
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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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33 views
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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0answers
82 views
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
40 views
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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1answer
38 views
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 ...
2
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0answers
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
40 views
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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0answers
16 views
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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1answer
96 views
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
50 views
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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0answers
29 views
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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1answer
32 views
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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2answers
91 views
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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0answers
14 views
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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1answer
42 views
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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0answers
14 views
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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0answers
23 views
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 ...
2
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1answer
47 views
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
25 views
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_{...
0
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0answers
27 views
'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 ...
0
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1answer
39 views
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 $<$ ...
