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

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4
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1answer
49 views

For an ideal $I \subset \mathbb{C} [x_1, … , x_n]$ show an iff about finiteness

For an ideal $I \subset \mathbb{C} [x_1, ... , x_n]$ show that dim$_{\mathbb{C}}R/I$ is finite iff $I$ is contained in only finitely many maximal ideals. Thoughts so far: I'm not sure how to get ...
2
votes
1answer
28 views

Need a counterexample: If the induced homomorphism $M/\mathfrak aM \to N/\mathfrak aN$ is surjective, then $f$ it's surjective.

This problem is from Atiyah and Macdonald, Introduction to Commutative Algebra, Exercise 10, Chapter 2. Let $A$ be a commutative ring with $1 \ne 0$ and let $\mathfrak a$ be an ideal of $A$ ...
1
vote
1answer
16 views

Is a Noetherian normal local domain (universally) catenary?

Let $R$ be a ring. Then $R$ is $\textit{catenary}$ if for a pair of prime ideal $p \subseteq q$, all maximal chains of prime ideals $p = p_0 \subseteq p_1 \subseteq \dots \subseteq p_n = q$ have the ...
4
votes
2answers
49 views

How do ideal quotients behave with respect to localization?

Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When ...
1
vote
3answers
64 views

Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number.

I'm solving some exercises from my class notes on Commutative Algebra,On the following exercise I got stuck: Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number. As far as I ...
0
votes
1answer
58 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
2
votes
0answers
20 views

Which minimal hypothesis is necessary for a matrix on a fairly general ring to have a Jordan-Chevalley decomposition?

When I look at different proofs or expositions of the Jordan-Chevalley decomposition of a matrix, the minimal hypothesis I usually found is about the perfection of the field over which such ...
0
votes
0answers
15 views

Definition of filtration over monoid

I want to know if the following definition is correct. A $\textbf{filtration}$ over a monoid $M$ (operation denoted multiplicativity), is a total order $<$ on $M$ that fulfills $1_M < x$ and if ...
7
votes
1answer
176 views

If $G$ is shellable, then $G \backslash \{x_i\}$ is shellable?

A simplicial complex $\Delta$ on the vertex set $\{x_1,\dots,x_n\}$ is shellable if the facets of $\Delta$ can be ordered, say $F_ 1 , . . . , F _s$, such that for all $1 \leq i < j ...
3
votes
3answers
66 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not ...
2
votes
1answer
50 views

Prove that $ k[x_1,\ldots,x_4]/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay.

Prove that $ k[x_1,\ldots,x_4]/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay. We have $\langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle=\langle x_1,x_3 \rangle \cap \langle ...
2
votes
1answer
39 views

A question about finitely generated projective modules

Let $A$ be a commutative ring with unity and let $P$ be a finitely generated projective $A$-module. For $any$ $A$-module $M$, how does one show that $\operatorname{Hom}_A(P,A) \otimes_A M \simeq ...
2
votes
0answers
58 views
+50

What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $S \leftarrow ...
6
votes
2answers
442 views

Motivation behind the definition of localization

What is the motivation behind definition of localization of rings? From where does the term "localization" come from? Why is the equivalence relation between the ordered pairs $(m,u),(m',u')$ with $ ...
12
votes
5answers
239 views

Checking that a $3$-D diagram is commutative

When proving certain results I need to use commutative diagrams, some of which quite complicated. My question is: Do we need to check every small square all the time to make sure that they are all ...
3
votes
0answers
116 views

Can an element in a Noetherian domain have arbitrarily long factorizations?

I tried to answer this question two days ago. Unfortunately, I said ring, rather than domain, which is what I wanted. So I try again. Let $R$ be a Noetherian commutative domain and let $r\in R$. ...
3
votes
0answers
26 views

Cohomological dimension of an arbitrary module.

In the paper, [P, Schenzel, On formal local cohomology and connectedness, J of Alg, 315 (2007), 894--923], he proves the following statement. (Corollary 2.2) Let $M$ be a finitely generated ...
0
votes
1answer
38 views

An incorrect(?) proof of the Hilbert's Basis Theorem

This is my proof of the Hilber's Basis Theorem. I think it is incorrect. Because it is easier than other proofs. But I can't find out the mistake in my proof. Can anyone help me? Thanks! Claim If ...
7
votes
0answers
169 views

Module of Kähler differentials for a formal power series ring

Let $A$ be a ring and $A[[T]]$ the formal power series over $A$. Then, one can show that $\Omega^1_{A[[T]]/A}$ is not finitely generated over $A[[t]]$. Now, in $\Omega^1_{A[[T]]/A}$ I am trying to ...
0
votes
0answers
45 views

generalized affine scheme

I'm thinking about following theorem. For a finitaly algebraic theory $\mathbb{T}$, $\text{FP}\mathbb{T}$ denotes the full subcategory of $\mathbb{T}\text{-Alg(Set)}$ consisting of finitely presented ...
1
vote
1answer
19 views

Meaning of 3-disjoint

Definition: Two edges $\{x, y\}$ and $\{w, z\}$ of $G$ are said to be 3-disjoint if the induced subgraph of $G$ on $\{x, y, w, z\}$ consists of exactly two disjoint edges. (See page 5 of this file.) ...
0
votes
0answers
28 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
4
votes
0answers
28 views

Which of the algebra isomorphisms hold?

Fix $m, n \ge 1$. Which of the algebra isomorphisms below hold? $k\langle t_1, \dots, t_m\rangle \otimes_k k\langle s_1, \dots, s_n\rangle \cong k\langle t_1, \dots, t_m, s_1, \dots, s_n\rangle$ $k[ ...
7
votes
1answer
288 views

What's the “real” reason a finite map has finite fibers?

This is a soft question. I have encountered two very different proofs of what seems like "basically the same theorem," and I want to understand how they relate and "what the real explanation is." ...
4
votes
1answer
38 views

Converse of “localization at a prime is local”

Suppose $S^{-1}R$ is the localization of a ring R at a multiplicative subset S, and is local. Must S be the complement of a prime ideal?
2
votes
1answer
70 views

Every commutative ring is a quotient of a normal ring?

In the book Étale cohomology by Milne I found on p. 37 (in the context of constructing the henselization of a local ring) the following claim: "Every ring is a quotient of a normal ring". The same is ...
8
votes
2answers
230 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
1
vote
1answer
172 views

Determining the minimal number of generators of the maximal ideal of a local Noetherian ring

Let ($A,m$) be a local, Noetherian ring. If $n$ is the minimal number of generators of the unique maximal ideal $m$, then by Krull's Hauptidealsatz and Nakayama's Lemma, we have the following ...
5
votes
2answers
188 views

Can every ideal have a minimal generating set?

Let $I$ be an ideal of commutative ring $A$ with unity. Does $I$ have a minimal generating set? At times, I am able to compute what they are for specific example, but it seems like it is true in ...
2
votes
0answers
46 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
1
vote
0answers
19 views

The completion of the ring of Laurent polynomials with respect to the augmentation ideal.

Let $A = \langle a\rangle$ be an infinite cyclic group on one generator. I'm trying to understand the completion $\widehat{\mathbb{Q}}A$ of the group algebra $\mathbb{Q}A$ with respect to the ...
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vote
0answers
63 views

Computing a valuation of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are transcendental over $k$. I want to compute the valuation ring of $F$, the field of fractions of the ring $A=k[x,y]/I$, where $I=\langle ...
1
vote
1answer
54 views

Embedding tensor product of integral domains

Let $C$ be a subring of integral domains $A,B$ and let $C',A',B'$ denote their field of fractions respectively. Can we always embed $A\otimes_CB$ in $A'\otimes_{C'}B'$ by $a\otimes b\mapsto ...
7
votes
1answer
144 views

Can an element in a Noetherian ring have arbitrarily long factorizations?

Suppose $R$ is a Noetherian ring. Is it possible that an element $r\in R$ have arbitrarily long factorizations? That is, for all $n>0$, is there a factorization $r=a_{1n}a_{2n}\cdots a_{nn}$ such ...
5
votes
1answer
93 views

Image of point of codimension one has codimension one?

I'm working on the following exercise (7.2.3) from Liu's Algebraic Geometry and Arithmetic Curves: Let $f: X \rightarrow Y$ be a morphism of Noetherian schemes. We suppose that either $f$ is flat ...
1
vote
0answers
37 views

Primitive vectors in $A^n$

Let $A$ be a commutative ring with 1. Let n be a positive integer. I call a vector $(a_1,...,a_n) \in A^n$ primitive, if the ideal generated by $\{a_1,....,a_n\}$ is $A$. Question: Given a primitive ...
0
votes
0answers
58 views

Axiomatization of the equational theory of ideals in a commutative ring

Is there a known axiomatization of the equational theory of ideal operations in a commutative ring? I have in mind the following: Consider a language with operations for ideal intersection, product, ...
13
votes
3answers
524 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
2
votes
1answer
32 views

Vanishing of Ext group and Krull dimension

Suppose $A=k[x_1,..,x_n]_{(x_1,..,x_n)}$, it is a regular local ring of dimension $n$. Let $B=A/I$ be a quotient ring of Krull dimension $r$. How to show $\operatorname{Ext}_A^i(B,A)=0$ for ...
10
votes
3answers
376 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
1
vote
1answer
18 views

Extending valuations and linear disjointness of fields

Let $F$ be a field and let $K$ be a field extension of $F$. Suppose that $K$ can be written as the compositum of field extensions $E$ and $L$ of $F$, linearily disjoint over $F$, thus $K$ can be ...
5
votes
1answer
163 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
0
votes
0answers
17 views

Extending McCoy's theorem to multiple indeterminates [duplicate]

So, working in a commutative ring with unity $R$, I've proven that $f\in R[x]$ is a zero divisor iff there exists $s\in R$ such that $sf=0$. I'm now being asked the followup question to extend ...
3
votes
1answer
55 views

Commutative diagram of algebras in Atiyah and Macdonald.

On page 31 of Atiyah and Macdonald, there is a commutative diagram. It essentially says that if $B$ and $C$ are $A$-algebras with ring morphisms $f:A\to B$ and $g\colon A\to C$, and $D=B\otimes_A C$ ...
0
votes
0answers
15 views

Local Constancy of Rank Function

Recently I asked this question. I believe that I have come up with a solution, but I am unsure, because the proof I have seems too easy to be true, and doesn't make very many assumptions. My ...
2
votes
1answer
38 views

Discrete convolution of two sequences

Let $R$ be a commutative ring with unity. A finite sequence $x=\left< x_0,\dots,x_n\right>$ with elements in $R$ is called to be prime if there exists $a_0,\dots,a_n \in R$ such that ...
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vote
2answers
115 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
5
votes
2answers
88 views

Are these quotient modules isomorphic?

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. For a non-zero ideal $\mathfrak{a}$ of $\mathcal{O}_K$ and an element $c \in \mathcal{O}_K \setminus \{0\}$ I wonder ...
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vote
0answers
26 views

Can the torsion of a formal group have rank going to infinity?

Suppose $F(x,y) \in \mathbb{Z}_p[[x,y]]$ is a formal group over $\mathbb{Z}_p$. I denote with $G(-)$ the corresponding functor, so that $G(K)$ will denote for me the maximal ideal of $O_K$ equipped ...
0
votes
1answer
65 views

Maximal ideals and the projective Nullstellensatz

This is a simple question, but it's one of those things that I've been thinking about so much that I've just kind of lost where I am and need some explicit reference. One of the main corollaries of ...