Questions about commutative rings, their ideals, and their modules.
4
votes
1answer
46 views
rational functions on projective n space
How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
2
votes
1answer
90 views
What does “Hauptidealsatz” mean in “Krull's Hauptidealsatz”?
What does "Hauptidealsatz" mean in "Krull's Hauptidealsatz"? Thank you very much.
0
votes
0answers
35 views
dimension of an ideal (definition)
Let $A$ be a commutative ring and $I$ an ideal. When we refer to the "dimension" of $I$, what exactly do we mean? Is it the Krull dimension of $A/I$? In particular, i am trying to understand the ...
1
vote
1answer
25 views
Annihilators of Modules
I'm stuck trying to prove that for two $R$-modules $M,N$ ($R$ commutative with a 1), then $$Ann(M+N)=Ann(M) \cap Ann (N)$$
I was trying to do double inclusion, and I can prove the RHS is contained in ...
1
vote
2answers
69 views
Integral closure $\tilde{A}$ is flat over $A$, then $A$ is integrally closed
Question. Let $A$ be an integral domain and $\tilde{A}$ be its integral closure in the field of fractions $K$. Assume that $\tilde{A}$ is a finitely generated $A$-module. I want to prove that if ...
3
votes
0answers
40 views
Homogeneous ideals are contained in homogeneous prime ideals
Let $I$ be a homogeneous ideal of a graded ring $S$. I want to show that there exists a homogeneous prime ideal which contains $I$.
I proved the following:
Let $T$ be the set of all homogeneous ...
2
votes
1answer
41 views
Is localization of a prime ideal still a prime ideal?
Im still new to the topic so this question might seem trivial. But I hope if someone can help explaining to me if a prime ideal $P$ of a domain $A$ is still a prime ideal $P_s$ in the localization ...
4
votes
1answer
60 views
Primary decomposition of power of a prime.
Let $R$ be a commutative Noetherian ring with unit. Suppose $P$ is a prime ideal that is not maximal. How can we go about finding a normal (reduced) primary decomposition of the power of $P$, say a ...
0
votes
0answers
31 views
Ideal membership problem for monomial ideals
Hi guys. I'd really appreciate help on understanding the proof for this Lemma above. I'm not sure how we got: "we see that every term on the right side of the equation is divisible by some x^{a(i)}. ...
10
votes
2answers
80 views
What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?
Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find the primary decomposition of $I$.
I tried to draw the graph of the variety of $I$ and get a decomposition of ...
2
votes
1answer
26 views
degree lexicographic monomial ordering
With respect to deglex X>Y, what would the leading monomials of these polynomials be?
$f_1=XY^3-X^2$ and $f_2=-X^3Y^3-4X^2Y^3+3X^2Y$
My understanding is that you prioritise X over Y based on their ...
10
votes
1answer
158 views
Classification of local Artin (commutative) rings which are finite over an algebraically closed field.
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
1
vote
1answer
38 views
Not primary ideal having a prime radical
Find an example of a ring $A$ and an ideal $I$ such that I is not primary but if $fg\in I$, then $\exists n\in \mathbb{N} $ such that $f^{n}\in I $ or $g^{n}\in I$.
-1
votes
1answer
68 views
Irreducible polynomials and affine variety
Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such
that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite.
7
votes
1answer
49 views
When does “second annihilator” of a (principal) ideal equal the ideal itself
Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $\operatorname{Ann}_R(\operatorname{Ann}_R(r))=Rr$ or $\sqrt{\operatorname{Ann}_R(\operatorname{Ann}_R(r))}=\sqrt{Rr}$ hold?
I ...
1
vote
0answers
62 views
A counterexample for $\operatorname{Ass}(M_1+M_2)$ [duplicate]
$\newcommand{\Ass}{\operatorname{Ass}}$
Let $A$ be a Noetherian ring and let $M$ be an $A$-module. Suppose $M=M_{1}+M_{2}$, then we have $\Ass(M)\supset \Ass(M_{1})\cup \Ass(M_{2})$. What is an ...
3
votes
1answer
34 views
square system of polynomial equations having infinite number of solutions
Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
3
votes
1answer
64 views
Small question about a proof of Hilbert's Basis Theorem
I am currently going going through the proof of Hilbert's Basis Theorem:
http://www.maths.usyd.edu.au/u/de/AGR/CommutativeAlgebra/pp806-850.pdf
(it starts on slide 832)
On slide 836-837 he makes the ...
2
votes
0answers
129 views
Finite morphisms of schemes are closed
I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely:
Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
-1
votes
1answer
71 views
Question about local ring (from a sentence in Hartshorne)
$A$ is a noetherian domain with the property that $A_{\mathfrak q}$ is a DVR for any prime $\mathfrak q \subset A$ of height $1$. $K$ is the fraction field of $A$, and $f \in K(t)$ is a nonconstant ...
1
vote
0answers
42 views
$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact, $M''$ flat. Why is $M$ flat $\Leftrightarrow M'$ flat?
Let $A$ be a commutative ring with identity, let
\begin{align}
0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0
\end{align}
be an exact sequence of $A$-modules, let $M''$ be flat.
I want ...
3
votes
1answer
61 views
All the Associated Primes are minimal.
Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question.
1) Suppose $\operatorname{Ass}(R/I)$ are all ...
7
votes
0answers
110 views
Find all maximal subrings of $\mathbb{C}[x]$
Definition: A maximal subring $S$ of $R$ is a subring such that if $S \subseteq T \subseteq R$ then $T=S$ or $T=R$.
Find all maximal subrings of $\mathbb{C}[x]$.
Clearly $\mathbb{C}[x^2,x^3]$ ...
1
vote
1answer
62 views
Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and $\{\text{non-zero-divisors}\}$ (exercise in SICA)
In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written:
So we have rings
$$\begin{array}{l l}
R_1:= K[x,y|x^2\!-\!y^3], & R_4:= K[x,y|xy],\\
R_2:= ...
1
vote
0answers
44 views
Noetherian localizations and extra-condition implies Noetherian
I'm trying to solve this question but I'm stucked:
If a ring $R$ satisfies the following two conditions:
i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is ...
7
votes
1answer
54 views
How does Local Cohomology detect UFD
I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFD's.
I know the basics of local cohomology but I have not seen a ...
1
vote
1answer
47 views
What's stronger: projective or locally free? flat or locally free?
maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each ...
5
votes
1answer
138 views
Grobner Basis and generating set
I have come across the following past exam question...
Define an ideal $J:=(z^2x+y^2-2y,x^3+y^3+z^3,x^2+2z^2) \subseteq \mathbb{Q}[x,y,z].
$
Compute a generating set for $J \cap \mathbb{Q}[y]$.
...
8
votes
2answers
126 views
What about a module of rank $\frac{1}{2}$?
Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
1
vote
2answers
179 views
Zero-dimensional ideals in polynomial rings
I have the following past exam paper question, a similar sort of question seems to come up every year. And I'm completely lost with it...
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated ...
2
votes
1answer
62 views
Show that $M=\bigcap_{\mathfrak{p}\in\operatorname{Spec}(R)}M_\mathfrak{p}=\bigcap_{\mathfrak{m}\in\text{Max}(R)}M_\mathfrak{m}$ for certain $M$.
$\newcommand{\Spec}{\operatorname{Spec}}$
$\newcommand{\mSpec}{\operatorname{Max}}$
This is a homework from my algebra course. I am in a situation where I think I have found a solution, though ...
3
votes
1answer
43 views
Difficulty Understanding Primary Modules
I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
5
votes
2answers
68 views
Intuition behind Direct limits
Let $R$ be a commutative ring and $x\in R$ be a nonzero divisor. Then i know that the direct limit of $R\mapsto R\mapsto R\mapsto\cdots $, where each map is multiplication by $x$ is $R_x$, the ...
3
votes
1answer
57 views
Flatness versus vanishing of Tor-groups for a non-finitely generated module
This is something I should probably know, but it is escaping me at the moment.
Let $A$ be a commutative noetherian ring. The following corollary of Nakayama's lemma is well-known (for instance, this ...
3
votes
0answers
38 views
Injective dimension is locally finite but not globally
Let $R$ be a commutative ring. Could someone provide me an example where $\operatorname{id}_{A_{\mathfrak p}}(M_{\mathfrak p})$ is finite for all $\mathfrak p\in \operatorname{Spec}(R)$, but ...
0
votes
0answers
26 views
robust computation of Groebner basis
I am trying to solve numerically polynomial systems of equations, over the reals. I am coming across the following phenomenon: let's say that i have a system of 7 equations with 7 unknowns. I am using ...
4
votes
1answer
89 views
Vakil 14.2.E: $L\approx O_X(div(s))$ for s a rational section.
I am working through Vakil's Ch 14 (march2313 version) on invertible sheaves and am having trouble on 14.2.E.
The question (in notation to be defined) is this: how do I show that each point in the ...
2
votes
0answers
29 views
constructing a sum of squares modulo an ideal
This question refers to the proof of Theorem 7.3, p. 98 of the pdf http://math.berkeley.edu/~bernd/cbms.pdf.
The statement of the theorem and its proof do not depend on what precedes them.
Let $I$ be ...
3
votes
1answer
95 views
About Artinian Rings
I'm studing commutative algebra by the text of Atiyah and Macdonald, and a doubt come at me and I can not prove neither find a counterexample, the problem is:
If a ring (commutative with identity) ...
5
votes
2answers
72 views
Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$
I'm studying for my algebra quals this fall and keep encountering problems like the following:
List all the ideals of $\mathbb{Z}[x]/(16, x^3)$.
or
List all the primes of ...
1
vote
1answer
41 views
Finding a presentation of $A$-algebra $B$
Find a presentation of the $A$-algebra $B$, where $B=\mathbb{Z}[1/2]\subseteq \mathbb{Q}$ and $A= \mathbb{Z}$.
I want to prove it but I can't understand what want to me! Please describe to me.
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
61 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 ...
