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

learn more… | top users | synonyms (1)

3
votes
1answer
114 views

Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
3
votes
1answer
71 views

Galois cover an affine scheme

Let $X = \operatorname{Spec}(A)$ be an affine scheme, with $A$ noetherian (and normal if this is useful). We suppose that $X$ is a finite étale covering of $Y = \operatorname{Spec}(B)$, Galois with ...
6
votes
2answers
292 views

When is a local, reduced, (commutative) ring an integral domain?

Question I am wondering whether or not it is true that if $A$ is a reduced ring, then is it the case that the localization of $A$ at any of its prime ideals is an integral domain? Discussion ...
2
votes
1answer
357 views

Is any UFD also a PID?

Is there any counterexample that will disprove that every unique factorization domain (UFD) is also a principal ideal domain (PID)? I mean, any PID is a UFD, does the converse hold? Thanks in ...
1
vote
1answer
194 views

How many ways are there to represent a monomial order, defined by $>$, by term order via matrices?

During the lecture, my professor brought up the list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the ...
1
vote
1answer
52 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
2
votes
0answers
44 views

Under what conditions are the resolutions of two modules subcomplexes of the resolution of the tensor product?

I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing ...
3
votes
1answer
74 views

Wikipedia definition of an order (ring theory)

Wikipedia defines an order $\mathcal O$ of a finite type $\Bbb Q$-algebra $A$ to be a subring of $A$ satisfying the following properties. Here, by finite type $\Bbb Q$-algebra, I mean that $A=\Bbb ...
4
votes
0answers
112 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
5
votes
1answer
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
votes
0answers
55 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
votes
1answer
134 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
votes
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 ...
3
votes
2answers
356 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
votes
2answers
110 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
125 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
votes
1answer
71 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
votes
0answers
138 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 ...
0
votes
1answer
81 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 ...
1
vote
2answers
66 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
176 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
votes
1answer
417 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 ...
5
votes
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
votes
1answer
106 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
220 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
77 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? ...
1
vote
1answer
85 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
votes
1answer
104 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
votes
1answer
204 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 $a_1^2 + 2a_1 a_2 + a_2^2 \in Q_{2i}$. ...
1
vote
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
0
votes
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 $\Delta$ ...
1
vote
1answer
55 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 ...
2
votes
1answer
76 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?
1
vote
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: ...
7
votes
1answer
203 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 $\mathbb{C}$, ...
0
votes
3answers
150 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
votes
0answers
115 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
votes
0answers
78 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 ...
1
vote
0answers
85 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
273 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
vote
1answer
52 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
votes
2answers
154 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
votes
1answer
88 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
votes
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
55 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
142 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
74 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, ...