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

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3
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1answer
84 views

Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
1
vote
0answers
63 views

$\operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff f_I(M)\subseteq f_I(N) $?

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. It has proven (here) that if $\operatorname{supp}(M) \subseteq \operatorname{supp}(N)$ then ...
3
votes
2answers
118 views

How to show that $\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD [duplicate]

I am trying to show that $R=\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD. Let $I=(xw-zy)$. Let $X=x+I$, $Y=y+I$, $Z=z+I$, and $W=w+I$. My guess is that $X$ is irreducible and therefore $(X)$ is a ...
-1
votes
1answer
47 views

A Question Related to Cohen's Structure Theorem

It is well known that if $R$ is a ramified complete regular local ring then $R\cong V[[x_1,\ldots , x_n]]/I$, where $V$ is a discrete valuation ring and $n$ is the Krull dimension of $R$. My Question: ...
0
votes
1answer
43 views

Questions regarding a proof of Nakayama's lemma.

I refer to this proof of Nakayama's lemma. What is $\varphi^n$? Is it $\underbrace{\varphi\circ\varphi\circ\dots\circ\varphi}_{\text{$n$ times}}$? What is $\varphi\delta_{ij}$?
2
votes
1answer
134 views

Atiyah-Macdonald Exercise 2.15

I have worked out a solution to exercise 2.15 of Atiyah-Macdonald, which is needed in the solution of 2.3 (see Atiyah-Macdonald 2.3). However, the solution seems overly complicated, and I am not ...
0
votes
1answer
65 views

Weak nullstellansatz in Atiyah-Macdonald 5.17

$\newcommand{\fm}{\mathfrak{m}}$ Problem 17 in the exercises after the 5th chapter of Atiyah-Macdonald is the following (with some references and hints omitted): Let $X$ be an affine algebraic ...
1
vote
1answer
40 views

A question regarding Hilbert's Nullstellensatz.

Let $k$ be an algebraically closed field, and $a$ an ideal of the polynomial ring $k[x_1,x_2,\dots,x_n]$. The strong form of Hilbert's Nullstellensatz says that $I(Z(a))=\sqrt{a}$. Note:- Initially, ...
2
votes
1answer
28 views

On a localized ring tensor with a module

Let $A$ be a commutative ring, $S$ be a multiplicative subset of $A$ and $M$ be an $A$-module. The questions says to "describe a natural isomorphism $(S^{-1}A) \otimes_A M \cong S^{-1}M $ as ...
-1
votes
1answer
52 views

Graded ring, and its homogeneous ideals : $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $

Let $ B = \displaystyle \bigoplus_{n \in \mathbb {Z}} B_n $ be a graded ring. Let $ I $ be an ideal of $ B $. Why is $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $ equivalent to ...
1
vote
1answer
24 views

Equivalent definitions of fractional ideals

Let $R$ be an integral domain and $K$ its field of fractions. The usual definition of fractional ideal $I$ ($I$ is an $R$-submodule of $K$) is that for some nonzero $r\in R$ we have $rI\subset R$, and ...
5
votes
3answers
93 views

Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
0
votes
2answers
114 views

Atiyah-Macdonald 2.3

In solving question 2.3 from Atiyah & Macdonald's commutative algebra textbook, I run into the following difficulty: Let $A$ be a local ring with $k:= A/mA$ its residue field and let $M$ and $N$ ...
0
votes
2answers
53 views

Some questions on Hartshorne I.7: intersections in projective space

I am reading I.7 of Hartshorne, and here are some questions I don't understand. 1) Prop. 7.4. Let $M$ be a finitely generated graded module over a noetherian graded ring $S$. Then there exists a ...
1
vote
1answer
53 views

A Direct Sum of Members of a Certain Class of Modules

Let $S$ be a class of $R$-modules and let an $R$-module $M$ be countably generated. Suppose that, for every direct summand $K$ of $M$, each element of $K$ belongs to a direct summand of $K$ that is ...
4
votes
2answers
141 views

What are local homomorphisms, geometrically?

For want of a better name, let us say that a ring homomorphism $f : A \to B$ is local if it (preserves and) reflects invertibility, i.e. $f (a)$ is invertible in $B$ (if and) only if $a$ is invertible ...
1
vote
1answer
38 views

Is the colimit of finite tensor products a tensor product?

Let $(R_\lambda)_{\lambda\in\Lambda}$ be a family of $A$-algebras. Atiyah & MacDonald defines the "tensor product" of the family as the direct limit of the tensor product of finite subfamilies. ...
1
vote
1answer
62 views

Integral closure of 1-dimensional noetherian local domains

Let $(R,m)$ be a $1$-dimensional noetherian local domain and $S$ its integral closure. Clearly $S$ is $1$-dimensional noetherian semi-local domain. Is $mS=J(S)$, where $J(S)$ is the Jacobson radical ...
1
vote
0answers
73 views

Surjectivity implies injectivity of finitely generated modules, localization?

The following problem is canonical: Suppose $A$ is a commutative unitary ring, and $M$ is a finitely generated module over $A$. If an endomorphism $f\colon M\to M$ is surjective, then it's also ...
1
vote
1answer
56 views

Contracted ideals in number fields

I am trying to translate a section of Wolfgang Krull's report "Idealtheorie". At one point (Section $7$ on Quotient Rings) I believe that he makes something like the following statement: Suppose for ...
8
votes
2answers
147 views

Is every affine scheme the complement of the closed point $x$ of the spectrum of a local ring $A$?

Let $R$ be a commutative ring with identity element and let $\operatorname{Spec}(R)$ be the associated affine scheme. Does for each affine scheme $\operatorname{Spec}(R)$ exist a local ring $A$ ...
2
votes
2answers
78 views

Isomorphism of modules arising from algebraic topology

While studying for a course in algebraic topology, the following question popped out: Let $S,R$ be two commutative rings with unit, $A,B$ two $S$-modules, and assume that $R$ is also an ...
4
votes
1answer
101 views

Question on calculating hypercohomology

I want to compute the algebraic de Rham cohomology of $ \mathbb{C}^* $, and I'm confused. I don't have much background in this, so I was hoping a very concrete example would clear up a lot of this ...
2
votes
1answer
114 views

Factorization of ideals in a coordinate ring (Dedekind domain)

Consider $f \in \mathbb{C}[X,Y]$ an irreducible curve non singular. Let $A = \mathbb{C}[X,Y] / (f)$ be the coordinate ring of $f$ and choose a curve $g \in \mathbb{C}[X,Y]$ with no component in common ...
0
votes
2answers
38 views

A question about one of Hartshorne's propositions

Hartshorne says that for $S_1,S_2\in A[x_1,x_2,\dots,x_n]$, where $A$ is a commutative ring, $Z(S_1)\cup Z(S_2)=Z(S_1S_2)$. Shouldn't it be $Z(S_1)\cup Z(S_2)=Z(S_1\cap S_2)$? We know that ...
2
votes
1answer
53 views

What kind of points are there in a finite type $k$-scheme?

Let $k$ be an arbitrary field and $X$ a $k$-scheme of finite type (i.e. a scheme with a finite cover of spectra of finitely generated $k$-algebras). How can I think of the points $x\in X$? What ...
0
votes
1answer
38 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
2
votes
2answers
102 views

Ring such that $q^2\mid p^2$ does not imply $q\mid p$?

Let $R$ be a commutative ring with $1$ and suppose $q^2\mid p^2,$ for $p,q \in R$. Unless $R$ is a UFD, I don't believe I can conclude that $q\mid p,$ but I would like to know a concrete ...
0
votes
0answers
51 views

Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
0
votes
0answers
46 views

Associated prime ideals and local cohomology [closed]

Let $M$ be an $R$-module such that $\operatorname{Ass}(M/N)$ is a finite set for any submodule $N$ of $M$. Show that 1. $\operatorname{Ass}(M/r M)=\operatorname{Ass}(M/r^n M)$ for each natural $n$; 2. ...
5
votes
4answers
192 views

Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
5
votes
1answer
68 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
1
vote
1answer
56 views

Zorn's lemma and maximal ideals

Let's consider two statements: Zorn's lemma and theorem about existence of maximal ideals in commutative ring with $1$. It's easy to prove that Zorn's lemma implies existence of maximal ideals. I ...
0
votes
1answer
51 views

Submodules and quotients of free modules over Noetherian local rings

Let $R$ be a Noetherian local commutative ring, $F$ a finitely generated free $R$-module and $A,B$ some arbitrary $R$-modules. Consider a short exact sequence $0 \to A \to F \to B \to 0$. In [Bruns, ...
0
votes
0answers
90 views

Difference between algebraic and integral extension

I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below). Now I ...
2
votes
1answer
56 views

Strong approximation theorem for Dedekind Domains

This is a theorem in "Maximal Orders" by Reiner. Page 48 stated without proof. And is said to be an easy consequence of The Chinese remainder Theorem. I am attempting to prove the theorem and need a ...
0
votes
2answers
51 views

Every Artinian ring is isomorphic to a finite direct product of Artinian local rings

I was reading a proof of the above theorem (1.6.7 Theorem) from here, but there was something that confused me. The proof says $R$ has finitely many maximal ideals $M_1, \ldots ,M_r$, and the ...
3
votes
2answers
62 views

Kähler differentials of the cuspidal cubic

I want to compute $\Omega^1_{A,\mathbb{C}}$ for $A = \mathbb{C}[X,Y]/(Y^2 - X^3)$, or more precisely, I want to show that the module of Kähler differentials is free of rank 2 at the origin, and free ...
1
vote
1answer
89 views

Example of strict inclusion for the localization of associated primes

Let $A$ be a commutative ring and $M$ an $A$-module. It is well known that $$\operatorname{Ass} M\cap\operatorname{Spec}S^{-1}A\subset\operatorname{Ass}S^{-1}M,$$ and that equality holds if $A$ (or ...
0
votes
0answers
42 views

Exercise 7.10 Atiyah, $M[x] $ is a noetherian $A[x] $-module [duplicate]

The exercise is: Let $M$ be a noetherian $A$-module. Then $M[x] $ is a noetherian $A[x] $ module. The action of $A[x] $ on $M[x] $ is the obvious one. In a previous exercise it was shown that ...
0
votes
2answers
43 views

$R^{(I)} \cong K \oplus H$ where $R^{(I)}$ is free but $K$ is not free

Let $R$ be a commutative ring with unit. Is there an example of a direct sum of $R$-modules $$R^{(I)} \cong K \oplus H$$ where $R^{(I)}$ is free but $K$ is not free ? Clearly $R$ can't be a PID.
2
votes
1answer
62 views

$k[X,Y]/(f)$ not finitely generated as a module (Exercise 4.10 Reid, UCA)

I have been wrestling with this problem for some time and I still can't find $f$. It seems really simple, which annoys me even more. The problem is as follows (Exercise 4.10 Reid, UCA): Suppose ...
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vote
0answers
89 views

What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = ...
1
vote
1answer
67 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
1
vote
1answer
47 views

Localizations of $ \mathbb{Z}_{p^k}$

Let $S \subseteq \mathbb{Z}_{p^k} $ be a multiplicative subset, where $p$ is a prime number, $k$ an integer. Is it true that $$S^{-1} \mathbb{Z}_{p^k} \cong \mathbb{Z} /n\mathbb{Z} $$ for some ...
2
votes
1answer
35 views

Congruence in localization of rings

Please help me to prove for all maximal ideals $\mathfrak{m}$ of $R$, $(aR/a^2R)_\mathfrak{m}\cong (aR)_\mathfrak{m}/(a^2R)_\mathfrak{m}\cong aR_\mathfrak{m}/a^2R_\mathfrak{m}$, where $R$ is a ...
3
votes
1answer
51 views

Is the unique morphism from the empty scheme $\operatorname{Spec}((0))$ to some other scheme $X$ smooth?

This is a very pedantic question, but Is the unique morphism from the empty scheme $\emptyset = \operatorname{Spec}((0))$ to some other scheme $X$ smooth?
1
vote
1answer
65 views

Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5)

This is a problem about finding the normalisation of a quotient polynomial ring. So I have to find the integral closure of the ring in its field of fractions. The problem statement is as follows: ...
1
vote
1answer
29 views

Two points in a proof of regularity of $R/I$

In the proof of the fact that "if $I$ is an ideal of the regular local ring $(R,m)$ such that $R/I$ is regular then $I$ can be generated by part of a minimal generating set of of $m$", I saw in a ...
3
votes
0answers
29 views

Localization and Direct limit [duplicate]

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...