Questions about commutative rings, their ideals, and their modules.
5
votes
1answer
59 views
Function field question from Silverman's AEC
Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring.
Then he states Proposition 1.7 (the intrinsic characterization of ...
4
votes
0answers
62 views
Maximal ideals in the algebra of continuously differentiable functions on [0,1]
This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
3
votes
0answers
33 views
Computing a rational function at a point in terms of a uniformising parameter
I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
7
votes
0answers
92 views
Class group of $k[x,y,z,w]/(xy-zw)$
I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
2
votes
0answers
48 views
Uniqueness of minimal resolution
Let $R$ be a domain, and $a_1,\dots,a_r$ be a regular sequence of $R$. Let $b_1,\dots,b_r$ be another regular sequence, such that two regular sequences generate the same ideal, i.e. ...
1
vote
1answer
85 views
Artinian ring and faithful module of finite length
Let $A$ be a ring. How can I prove that:
$A$ is an Artinian ring $\Leftrightarrow \exists$ a faithful $A$-module which is of finite length.
I know that if a ring has a faithful $A$-module which ...
6
votes
1answer
119 views
If a maximal ideal $\mathfrak m$ is flat, then $\dim_k (\mathfrak m/\mathfrak m^2) \leq 1$
This is an exercise that bothers me a lot:
Let $R$ be a commutative ring with $1$. Let $\mathfrak{m}$ be a maximal ideal in $R$.
If $\mathfrak m$ is flat as an $R$-module then the vector space ...
0
votes
1answer
44 views
Some question about localization
Let $S$ be a graded ring generated by finite elements of $S_1$ as $S_0$-algebra
and let $M$ be a graded $S$-module. For $m \in M$, if $m=0$ in $M_f$ for all generators $f \in S_1$, then $m=0$?
6
votes
1answer
86 views
Krull dimension of $\mathbb{Z}[x_1,…x_n]$
I'm trying to prove that the krull dimension of $\mathbb{Z}[x_1,\dots x_n]$ is n+1. I know there is a result that says $$dim(A[x_1,\dots x_n])=n+dim(A),$$ when $A$ is a Noetherian ring, but I was ...
11
votes
1answer
113 views
Intersection of powers of maximal ideals
Let $A=\mathbb K[X_1,\ldots,X_n]$ be a polynomial ring over some field $\mathbb K$. Let $\mathfrak p\subseteq A$ be a prime ideal. Let $Z(\mathfrak p)=\{ \mathfrak m\subset A\text{ maximal}\mid ...
2
votes
1answer
31 views
Inequalities among heights
Let $R$ be a Noetherian ring and $I$ a non-zero ideal of $R$. Let $x\notin I$. Could someone provide me a counterexample to the following:
$$\operatorname{ht}(I)\leq \operatorname{ht}(I+(x))\leq ...
13
votes
1answer
256 views
$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic
Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
1
vote
0answers
33 views
Equality of two $k$-algebras
Let $f\in k[X_1,\ldots, X_n]$ and $1-fX_{n+1}\in k[X_1,\ldots, X_{n+1}]$. Moreover $X\subseteq k^n$ is a subset and
$$I(X)=\{g\in k[X_1,\ldots, X_n]\,:\, g(x)=0\,\forall x\in X \}$$
is the ideal of ...
3
votes
1answer
43 views
$S^{-1}B$ and $T^{-1}B$ isomorphic for $T=f(S)$
Let $f:A\to B$ be a homomorphism of rings, $S$ be a multiplicatively closed subset of $A$ and $T=f(S)$. Then $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$-modules.
First we define the ...
1
vote
1answer
31 views
If $R$ is integral over $S$, then $frac(R)/frac(S)$ is finite extension of fields
How to show that:
If $R\supset S$ are rings, $R$ is integral over $S$, $K$ and $L$ the fraction fields of $R$ and $S$ respectively, then $K/L$ is finite extension of fields.
0
votes
0answers
39 views
Are the minimal primes of the same height
Let $R$ be a Cohen-Macaulay ring and let $I$ be an ideal of height $n$. Is it true that height of $P$ is $n$ for all minimal primes over $I$?
If the answer to the above question is negative, then is ...
1
vote
1answer
30 views
An $R/\mathfrak p$-sequence is also an $R$-sequence
Let $R$ be a commutative Noetherian ring with $\mathfrak p$ a nilpotent prime ideal, that is $\mathfrak p^r=0$ for some positive integer. If $\oplus_{i=0}^{r-1}\mathfrak p^i/\mathfrak p^{i+1}$ is a ...
6
votes
1answer
59 views
Showing that a ring is Noetherian
I show the following:
Let $R$ be a commutative ring with unity and $I \subseteq R$ an ideal. Prove: if $R/I$ is a Noetherian ring and $I/I^2$ is a finitely-generated $R$-module, then $R/I^n$ is a ...
3
votes
3answers
139 views
every field of characteristic 0 has a discrete valuation ring?
How can we prove that every field of characteristic 0 has at least one Discrete Valuation Ring?
My effort: Let $K$ be an field of characteristic 0. Then $\mathbb{Z}$ is a subring of $K$. Let $p$ be a ...
1
vote
0answers
43 views
For $R$-modules $M,N$, what are sufficient conditions for $\operatorname{Supp}(M\otimes_R N)\subseteq \operatorname{Supp}(\operatorname{Hom}_R(M,N))$?
Let $R$ be a commutative ring, $M$ and $N$ be finitely generated $R$-modules. What additional conditions will ensure $\operatorname{Supp}(M\otimes_R N)\subseteq ...
1
vote
3answers
66 views
Is the ideal $(X^2-3)$ proper in $\mathbb{F}[[X]]$?
Let $\mathbb{F}$ be a field and $R=\mathbb{F}[[X]]$ be the ring of formal power series over $\mathbb{F}$. Is the ideal $(X^2-3)$ proper in $R$? Does the answer depend upon $\mathbb{F}$?
Clearly ...
0
votes
3answers
71 views
Integral Dependence & Finitely Generated Modules
How to prove $(3)\Rightarrow(1)$ of this theorem:
Let $A\subseteq B$ be commutative rings. The following are equivalent:
$(1)~~x\in B$ is integral over $A$;
$(2)~A[x]$ is a finitely generated ...
4
votes
1answer
86 views
Field of fractions of $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ [duplicate]
This problem goes as follows:
Prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ is an integral domain and that its field of fractions is isomorphic to the ring of rational functions ...
2
votes
1answer
48 views
Proposition 3 in Chapter I.7 (Dimension) of Mumford's Red Book
In Mumford's Red book, chapter I.7 (Dimension), the proof of Proposition 3 (1.) has the step:
If $B=f^{\star -1}(A)$, apply the going-up theorem to $S/B\subset R/A$.
What does the inclusion ...
4
votes
0answers
52 views
A question on an answer on Math Overflow about Artin approximation
I have a question on an answer of this Math Overflow question.
Let $(A,I)$ be a commutative excellent normal local domain. The completion
$$
\hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
5
votes
0answers
63 views
Description of $\mathrm{Ext}^1(R/I,R/J)$
Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$?
What do I mean by a nice description? For example ...
2
votes
1answer
56 views
Noetherian and Artinian modules over subrings
I have a question about whether Noetherian-ness and Artinian-ness of modules are preserved under changes of the base ring. More precisely:
Let $R$ be a commutative ring and $S \subseteq R$ a ...
3
votes
2answers
52 views
Is it true that $\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap \mathbb{Q}$?
I know $\mathbb{Z}_{(p)}\subset \mathbb{Z}_{p}\cap \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at prime ideal $(p)$ and $\mathbb{Z}_p$ is the set of p-adic integers. I ...
2
votes
1answer
37 views
The local cohomology modules are Artinian
Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that
the local cohomology modules $H^i_m(M)$ are Artinian
and that this ...
4
votes
1answer
52 views
A graded abelian group and a graded map
I have an elementary question about a graded abelian group and a graded map. Here is the situation.
Let $A$ be a free abelian group of rank $2$ spanned by $1$ and $x$. Let us make $A$ be a graded ...
2
votes
1answer
33 views
The set of zero divisors is the union of radicals of annihilators
I am trying to figure out why the statement
$$\text{the set of zero divisors }=\bigcup_{0\ne x\in R} \sqrt{\text{Ann}(x)}$$
is true. Here $R$ is a commutative ring, $\text{Ann}(x)=\{r\in R\mid rx=0\}$ ...
6
votes
4answers
88 views
Definition of Jacobson radical
This may be a rather silly question, but I wonder why the definition of the Jacobson radical always is
$$\{x\in R\mid 1-xy \text{ is a unit for all } y\in R\}$$
and not
$$\{x\in R\mid 1+xy \text{ is ...
1
vote
1answer
32 views
If a polynomial ideal can be generated by $k$ elements, can it be generated by $k$ elements of any generating set?
Let $I = (p_1,\ldots, p_k) \subset \mathbb{C}[x_1,\ldots,x_n]$.
If we have a set of $k'$ polynomials such that $(q_1,\ldots,q_{k'}) = I$, can we always find a $k$-member subset such that ...
2
votes
2answers
51 views
Atiyah and MacDonald, Proposition 2.4
Let $M$ be a finitely generated $R$-module, $\mathfrak a \lhd R$ an ideal and $\phi:M\to M$ an $R$-linear map such that $\phi(M)\subseteq \mathfrak a M$. Then $\phi^n+a_1 \phi^{n-1}+\cdots+a_n=0$.
...
8
votes
1answer
97 views
How to show $M_{\mathfrak q}$ is flat over $A$
Let $f:A\rightarrow B$ be a homomorphism of commutative rings, and $M$ a finite $B$-module. If $a\in A$ and $M_a$ is a free $A_a$-module, then for a prime ideal $\mathfrak q$ of $B$ with ...
2
votes
0answers
44 views
Spectrum of a Laurent polynomial ring
I suspect this question is either easy/known or far too general to answer, but I'm finding it difficult to google for so I'd appreciate directions to good resources on the subject.
Can we describe ...
3
votes
0answers
70 views
Finite type ring extension + condition = finite extension?
Is the following true ?
If $A \subset B$ is finite type extension (i.e. $B$ is a finitely generated $A$-algebra) of integral domains such that the set $\{\mathfrak ...
2
votes
1answer
34 views
On regular elements and Maximal Cohen-Macaulay modules
I was reading theorem 3.3.3 in Bruns-Herzog: we have a Cohen-Macaulay local ring $(R,\mathfrak m,k)$, $C$ and $M$ are maximal Cohen-Macaulay modules. (Probably to solve my question some of these ...
3
votes
0answers
55 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
0
votes
1answer
29 views
If $x_i$ generate an $A$-module $M$, why do $1 \otimes x_i$ generate the extension of scalars $B \otimes_A M$?
In the following, let "ring" be a synonym for "commutative ring with identity".
For rings $A, B$ and an $A$-module $M$, let $M_B = B \otimes_A M$ be the $B$-module obtained from $M$ by extension of ...
0
votes
1answer
58 views
Seidenberg's Lemma
I have a problem with the proof of this
Lemma. Let $K$ be a field, $R=K[X_1,\dots,X_n]$, and $I\subset R$ a zero-dimensional ideal. For every $i$ there exists $g_i\in I\cap K[X_i]$, $g_i\neq 0$, ...
3
votes
1answer
64 views
question related to the krull schmidt theorem
Let $M$ be a finitely generated projective $R$-module, where $R$ is an Artinian ring. Then I must show that $M$ is isomorphic to a direct sum of principal indecomposable $R$-modules.
We have a ...
4
votes
1answer
62 views
Why are quotient modules $M / \mathfrak{m}M$ over residue fields $A / \mathfrak{m}$ considered for local rings rather than general rings?
In the following, let "ring" be a synonym for "commutative ring with identity". In the book on Commutative Algebra by Atiyah and MacDonald, I read:
Let $A$ be a local ring, $\mathfrak{m}$ its ...
2
votes
1answer
68 views
zero-dimensional ideals and finite-dimensional algebras
I encountered in the literature the term "zero-dimensional ideal", however i can not find a relevant definition anywhere in Atiyah-MacDonald or Matsumura. In fact, i encounted the statement:
$I$ ...
1
vote
2answers
43 views
A complete set of orthogonal idempotents in a commutative ring
I'm reading David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. At page 13, Chapter $0$, he says: "... if $e_1,\ldots,e_n$ is a complete set of orthogonal idempotents in a ...
0
votes
0answers
77 views
How to show an ideal is zero-dimensional? [duplicate]
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional.
How do I go about showing this?
5
votes
1answer
49 views
For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?
I'm reading the Atiyah-MacDonald book on Commutative Algebra.
At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is:
$G$ = finite group, ...
3
votes
1answer
76 views
The natural map $M \to M \otimes_R K$ is injective iff $M$ is torsion free
I'm reading some lecture notes of Pete L. Clark, and there's one problem that I cannot solve. It's on page 45 of this book: Commutative Algebra. The problem reads as follow:
Exercise 3.42
Let ...
3
votes
1answer
37 views
about the definition of extended ideals
Let $A, B$ be commutative rings with multiplicative identity, let $f: A \rightarrow B$ be a ring homomorphism, let $\mathfrak{a}$ be an ideal in $A$. The extension $\mathfrak{a}^\text{e}$ of ...
5
votes
1answer
198 views
Intuition behind Hilbert's Nullstellensatz
maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof, however I cannot understand the concept in a more ...
