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

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5
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145 views

Why do people look into modules over Dedekind domains?

It is said in this blog that: The reason this turns out to be useful is that many examples in algebraic/arithmetic geometry require you to look no further than understanding modules over Dedekind ...
2
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0answers
58 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
6
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1answer
138 views

Local parameter of curves in affine n-space

I'm looking for a double answer to this question: a mathematical one (say, if the statement is correct or not) and a philosophical one (say, why we do expect this to be true, or not). Let $k$ be a ...
4
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0answers
75 views

Regular monomorphisms of commutative rings

What are the regular monomorphisms of $\mathsf{CRing}$? Is there a purely algebraic characterization? Since regular monomorphisms coincide here with effective monomorphisms (see Prop. 1. here), the ...
3
votes
1answer
79 views

tensor, symmetric, exterior power of a module over a PID

Let $R$ be a PID and $M\cong R^r\!\oplus\bigoplus_{i=1}^s\!R/Ra_i$. Denote the tensor, symmetric, exterior power of $M$ by $T^nM=\bigotimes_{k=1}^nM$ and $S^nM= T^nM/\langle ...
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2answers
384 views

A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
3
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2answers
112 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
5
votes
1answer
126 views

Proof that $K\otimes_F L$ is not noetherian

Let $F$ be a field and $K$ and $L$ be extension fields of $F$ such that $\mathrm{tr.deg}_F(K) = \infty$ and $\mathrm{tr.deg}_F(L) = \infty$. It seems to be proved that $K\otimes_F L$ is not ...
0
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1answer
74 views

Associated prime preserved under the quotient

Let $(R,m,k)$ be a complete local Noetherian ring and let $E$ be an $R$-module such that $\operatorname{Ass}E=\left\{m\right\}$. Let $N$ be a proper submodule of $E$. Question: Is it true that ...
4
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0answers
140 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
2
votes
1answer
155 views

Annihilators and exact sequences

Let $R$ be a commutative ring. Let $M_1$, $M_2$ and $M_3$ be $R$-modules. Let the following sequence be exact: $$0\longrightarrow M_1 ...
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1answer
83 views

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.r mean radical. I can't prove it,here is what I did. a is a intersection of prime ideals mean ...
2
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2answers
68 views

Can a chain of irreducible subvarieties always be extended to one of maximal length?

I'm interested in computing the dimension of a variety $X$. I can get a lower bound by exhibiting some strictly increasing chain of irreducible subvarieties $$\varnothing =Z_{-1}\subset Z_0\subset ...
8
votes
3answers
181 views

Why is the topology on $\operatorname{Proj} B$ induced from that on $\operatorname{Spec}(B)?$

In the proof of Lemma $3.36$ in Algebraic Geometry and Arithmetic Curves, it is stated that, if $B=\oplus_{d\ge0}B_d$ is a graded algebra over a ring $A,$ and if $I$ is an ideal of $B,$ then ...
0
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1answer
452 views

Difference between Matsumura's Commutative Algebra and Commutative Ring Theory

I am a beginner in more advanced algebra and my question is very simple, I would like to know the difference between these books of the same author, Hideyuki Matsumura Commutative Ring Theory ...
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2answers
122 views

$\mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$ and $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. We define similarly ...
2
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1answer
107 views

Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
5
votes
2answers
226 views

Quotient of a local ring at a point is a finite dimensional vector space

$f,g\in \mathbb{C}[x,y]$ are irreducible polynomials, and the varieties $V_1=V(f)$ and $V_2=V(g)$ are not equal. Is the ring $\mathcal{O}_p/(f,g)$ a finite dimensional vector space over ...
0
votes
2answers
78 views

Nonprincipal prime ideals contain two relatively prime elements

Let $R$ be a principal ideal domain and let $P$ be a nonprincipal prime ideal of $R[x]$. I'm having trouble seeing why $P$ must contain two elements with no common divisor. Can anyone help me? ...
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1answer
91 views

local Noetherian of zero depth implies Artinian?

Let $(R,m,k)$ be a local Noetherian ring such that $\operatorname{depth}R=0$. Question: Is it true that $R$ is Artinian? PS: If it is true then please only say so, as i am still attempting to ...
2
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1answer
108 views

Localization at a maximal ideal and quotients.

If we have a commutative ring $R$ and a maximal ideal $m$, then is $m/m^2$ isomorphic to $m_m/m^2_m$? Thx.
5
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1answer
208 views

Exercise 4.5.E a) in Ravi Vakil's Foundations of Algebraic Geometry.

Hi! I am following the hint given in Exercise 4.5.E in Vakil's Foundations of Algebraic Geometry, but I am stuck trying to prove that if $a_1,a_2 \in Q_i$, then $a_1^2 + 2a_1 a_2 + a_2^2 \in Q_{2i}$. ...
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0answers
52 views

Name of a certain type of rings

What is the name given to (if there exists any) commutative rings $R$ with identity such that $R/(a)$ is finite for every non-zero $a\in R$ Thanks a lot
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1answer
107 views

combinatorial commutative algebra

Is there anyone who can help me with this problem? Any hint to the solution would be appreciated! Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. Show that the h- and f-vectors of $\Delta$ ...
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1answer
57 views

Are there homogeneous elements with two distinct grades?

In a graded ring $B=\bigoplus_{d\ge 0} B_d$, the element $0$ is homogeneous with grade $d$ for every $d\ge 0$, in fact since every $B_d$ is an additive subgroup of $B$, then it must contain $0$. Can ...
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1answer
77 views

Question about some details of a proof of Chinese Remainder Theorem

In the proof of 3rd proposition I can prove the intersection of all ideals is the kernel of the map, but why does it imply this proposition is true?
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1answer
89 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
1
vote
1answer
39 views

A question about a detail of proof

proposition: x∈The Jacobson radical <=> 1-xy is a unit in commutative ring A for all y∈A I have proved (=>) I don't figure out a detail of the proof of (<=). Here is the proof on book: ...
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1answer
208 views

Exercise 1.11 of Eisenbud

I'm doing the exercises from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, and I don't understand part of one of them, ex. 1.11 a): Exercise 1.11 a: Over $\mathbb{C}$, ...
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3answers
164 views

How to prove a simple proposition about local rings and maximal ideals

(The word ring shall mean a commutative ring with an identity element in this question.) Actually, there is a proof about this proposition, but I don't get it, even the first step. Proposition: ...
3
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0answers
117 views

Counterexamples to the Artin-Rees Lemma

This well known Lemma about $I$-stable filtrations asserts: Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module. Let $F$ be a submodule of $E$ and $\{E_i\}$ an ...
2
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0answers
79 views

maximal vs non-maximal order in an algebraic number field

I am trying to determine whether an order in a (cubic) number field is maximal or not. I have picked up two different fields. One has a power basis the other does not have it. 1) Let ...
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0answers
88 views

If A is a finitely generated algebra over the integers and m is a maximal ideal, then A/m is finite

I'm trying to prove the following: Suppose $A=\mathbb{Z}[x_1,\ldots,x_n]/I$ where $I$ is some ideal. Then for all $m \in Specm(A)$ we have $\mid A/m \mid$ is finite. I've seen some proofs of this on ...
4
votes
2answers
280 views

Integral morphism between varieties has finite fiber

I'm looking for a proof/counterexample of the following fact: Theorem Let $X \subseteq k^n$ and $Y \subseteq k^n$ be algebraic varieties over a field $k$ and let $\phi$ be a morphism from $X$ to ...
1
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1answer
53 views

If an $A$-module $M$ is locally finitely presented (resp. related) then $M$ is finitely presented (resp. related)

In this question I want to ask for a better proof than the one I am about to give for the statement with finitely presented, and inquiry if the statement is also true for the notion of finitely ...
4
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2answers
155 views

Exercise from Atiyah-Macdonald, Chapter 1, 2.iv)

Let $A$ be a ring and let $A[x]$ be the ring of polynomials in an indeterminate $x,$ with coefficients in $A.$ Let $f=a_0 + a_1x+\cdots+a_nx^n \in A[x].$ $f$ is said to be primitive if ...
2
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1answer
89 views

Are the two prime ideals containing same idempotents always the same?

If two prime ideals contain the same non trivial idempotents, what can we say about those ideals? Are they equal?
3
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1answer
30 views

What is meant by squeezing a module between two f.g. modules?

The author of the first answer in this thread of mathoverflow concluded that a module $K'$ was finitely generated because it was squeezed between two finitely generated modules. In and of itself, this ...
2
votes
1answer
56 views

Motivation of definition of a ring of fractions

Let $R$ be a commutative ring and $S \subseteq R$ its multiplicative subset. The equivalence relation on $R \times S$ used in the definition of the ring of fractions $RS^{-1}$ is defined as follows: ...
4
votes
1answer
150 views

Determinant of long exact sequence

Let the following be a long exact sequence of free $A$-modules of finite rank: $$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$ I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where ...
3
votes
1answer
75 views

Coherent sheaves of finite length over $\mathbb{P}^n_k$

Let $k$ be an algebraically closed field. Are there any nonzero coherent sheaves on the projective space $\mathbb{P}^n_k$ that are supported at (only) finitely many closed points? If they don't exist, ...
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2answers
385 views

Characterize the commutative rings with trivial group of units

This question suggested me the following: Characterize the commutative unitary rings $R$ with trivial group of units, that is, $R^{\times}=\{1\}$. The local case was solved here long time ago ...
2
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1answer
372 views

Is “being an integral domain” a local property?

I need to show that being an integral domain is a local property. That is, a commutative ring $A$ has no zero divisors iff $A_{\mathfrak p}$ has no zero divisors for every prime ideal $\mathfrak ...
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1answer
173 views

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
2
votes
2answers
125 views

What are the ideals in ${\Bbb C}[x,y]$ that contain $f_1,f_2\in{\Bbb C}[x,y]$?

This question is based on an exercise in Artin's Algebra: Which ideals in the polynomial ring $R:={\Bbb C}[x,y]$ contain $f_1=x^2+y^2-5$ and $f_2=xy-2$? Using Hilbert's (weak) nullstellensatz, ...
3
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2answers
159 views

Is $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$ noetherian?

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Z}_p$ be the $p$-adic completion of $\mathbb{Z}$. Similarly for $\mathbb{Z}_q$. Is $\mathbb{Z}_p\otimes_{\mathbb{Z}} ...
4
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1answer
258 views

Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
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2answers
36 views

Requirements on ring for injective-projectiveness

What requirements could be asked (minimal) of a ring R, so that any module M on R which is injective must also be projective? Is this possible?
3
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1answer
230 views

Jacobson radical of rings

I have some question about the Jacobson radical of rings. What is $J(R)$ when $R$ is a Principal Ideal Domain but not a field? e.g. I know that $\mathbb Z$ is a PID and why is $J(\mathbb Z)=0$ but ...
4
votes
1answer
310 views

Inducing homomorphisms on localizations of rings/modules

I'm trying to work out Exercise 2.6 in Commutative Algebra by Eisenbud, which asks to prove the Chinese Remainder Theorem for commutative rings. Exercise: Let $R$ be a commutative ring, and let ...