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

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4
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94 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
3
votes
2answers
132 views

On Hilbert's Nullstellensatz Theorem

I was reading Ravi Vakil's notes on his website and he states the Hilbert Nullstellensatz (3.2.5.): If $k$ is any field, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension ...
2
votes
2answers
144 views

Nonintegral element and a homomorphism

Assume $R\subseteq S$ are rings. Choose $x\in S$ nonintegral over $R$. I want to define a homomorphism from $R[x^{-1}]$ to a field which maps $x^{-1}$ to zero. I was trying to show that ...
2
votes
1answer
177 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
1
vote
2answers
117 views

Cohen-Macaulay and regular rings

I know this is a simple question but to make sure...: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ we have $\dim A_{\mathfrak{m}}=\dim A$. Is ...
3
votes
2answers
113 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
2
votes
1answer
164 views

Writing $I= (xz-y^2, yt- z^2)$ as an intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that $I$ is radical, then it suffices to ...
0
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2answers
105 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
1
vote
0answers
98 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
1
vote
1answer
66 views

Definition of degree of commutative ring $d(A) $ based on Hilbert polynomial

I'm studying chapter 11 (Dimension Theory) in Atiyah / Macdonald - Intro to Commutative Algebra. Let $ A $ be a Noetherian local ring with $\mathfrak{m}$-primary ideal $\mathfrak{q}$. The book defines ...
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0answers
93 views

Going Down Theorem, AM

I'm trying to understand the proof of the going down theorem in Introduction to Commutative Algebra by Atiyah and Macdonald. My main confusion is when they say it suffices to show that $B_{\mathfrak ...
1
vote
3answers
132 views

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
3
votes
0answers
93 views

Tensor products over monoids : Element structure

Let $A$ be a (commutative) monoid. Let $M$ be a right $A$-set and let $N$ be a left $A$-set. Then we can construct the tensor product $M \otimes_A N$, which is a set (of even $A$-set when $A$ is ...
0
votes
1answer
96 views

Intuition behind the abstract definition of a node (singularity)

Look at the following definition of an ordinary double point (node). The source is the book:"Freitag, Kieh - Etale Cohomology and the Weil Conjectures:" I don't understand the geometry behind ...
0
votes
1answer
90 views

Canonical ring map

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at ...
2
votes
1answer
110 views

Proposition 5.15 from Atiyah and Macdonald: Integral Closure and Minimal Polynomial

I am having some trouble understanding Proposition 5.15 in Introduction to Commutative Algebra by Atiyah and Macdonald. Let $A\subset B$ be integral domains, $A$ integrally closed, and $x\in B$ ...
1
vote
1answer
56 views

Noetherianity of valuation ring and valuation being discrete

I need a hint for left to right part of the following: Let $K$ be a valued field with $\nu$ and $\mathcal{O}_\nu$ be its valuation ring. Then, $\mathcal{O}_\nu$ is Noetherian if and only if ...
3
votes
1answer
52 views

What is the least integer $k$ such that any $n$ by $n$ integer matrix is a $\mathbb Z$-linear combination of $k$ indempotents?

Let $n$ be an integer $\ge2$. (a) What is the least integer $k$ such that any $n$ by $n$ integer matrix is a $\mathbb Z$-linear combination of $k$ indempotents? (The idempotents are also ...
0
votes
2answers
301 views

Field of fractions of an integral closure.

I am reading through the section of Atiyah-MacDonald on fractional ideals. They describe the group of invertible fractional ideals for an integral domain $A$ in its field of fractions $K$. They then ...
1
vote
2answers
49 views

Finitely generated ideal with special property

Is there a ring with a finitely generated ideal $I$ which has an infinite subset $M\subseteq I$ such that $M$ generates $I$ but no finite subset of $M$ does it? What I found out: If such a rings ...
1
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1answer
87 views

$\dim(R/x) = \dim(R)-1$ for Noetherian integral domains?

Let $R$ be a Noetherian integral domain of finite Krull-dimension and $0 \neq x \in R$ a non-unit. Do we have $\dim(R/x) = \dim(R) -1$ in general? If this is wrong, does it change something if we ...
2
votes
1answer
79 views

Castelnuovo-Mumford regularity and maximal degree of generators

I am reading a few texts on Castelnuovo-Mumford regularity. If I understand correctly, almost all of them say: If $I$ is a homogeneous ideal in $k[X_0,...,X_n]$ where $k$ is algebraically closed ...
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0answers
51 views

maximal chains of graded prime ideals

In Theorem 1.5.8 [Bruns,Herzog - Cohen-Macaulay-Rings] it is proved that for a noetherian graded ring $R$, a finitely generated graded $R$-module $M$ and any chain $\mathfrak{p}_0 \subsetneq ...
0
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1answer
73 views

Example of Tor-Rigid Module

Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ ...
2
votes
1answer
63 views

Is the trace of an idempotent matrix a sum of idempotents?

Let $R$ be a commutative ring, $n$ a positive integer, and $A$ an idempotent $n$ by $n$ matrix with entries in $R$. Is the trace of $A$ necessarily a sum of idempotents of $R$?
2
votes
2answers
102 views

Powers of prime ideals

I was reading through Atiyah-MacDonald and they mention that if a ring $A$ is a Noetherian domain of dimension 1 has the property that every primary ideal is equal to the product of a prime ideal ...
10
votes
2answers
149 views

$AB=z \mathrm{Id}_n$ implies $z^m BA = z^{m+1} \mathrm{Id}_n$ for what $m$?

This question builds on a series of questions looking for elementary proofs that $AB=\mathrm{Id}$ implies $BA=\mathrm{Id}$, for $A$ and $B$ both $n \times n$ matrices over a commutative ring. First ...
3
votes
1answer
152 views

How to characterize all finite commutative local rings with the maximal ideal of fixed order (if the order is small)?

Let $R$ be a finite commutative local ring with the maximal ideal $M$ of order $m$. How to characterize all such finite commutative local rings? For examples, if $m=2$, then $R\cong\mathbb{Z}_4$ ...
4
votes
2answers
285 views

Atiyah and Macdonald, Proposition 2.9

The following simple claim is used without proof in Proposition 2.9 of Atiyah and MacDonald (p.23). Although I believe I can prove it with a fairly involved argument, the claim is treated by the ...
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vote
0answers
70 views

Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in ...
5
votes
1answer
86 views

$(x,y)$-primary ideals

I want to find all ideals $I$ in $\mathbf{C}[x,y]$ with $\sqrt{I}=(x,y)$ and $\dim_{\mathbf{C}}\mathbf{C}[x,y]/I=2$. I have no clue how to about it, I mean I can write down some examples, ...
2
votes
0answers
51 views

G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
4
votes
2answers
124 views

Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
0
votes
1answer
88 views

How to prove this comment of Fulton

I'm trying to understand why this is true in Fulton's Algebraic Curves: Why we add this point $(0,\ldots, 0)$? Why this equality is true? I really need help. Thanks in advance.
2
votes
3answers
151 views

Isomorphic quotient of a module over Noetherian commutative ring

I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
1
vote
1answer
68 views

Injectivity of simple modules

If $R$ is a commutative ring with $1$ having a maximal ideal $m$ such that the local ring $R_m$ is a field, how could one check that $R/m$ is an injective $R$-module? If we want to use Baer Lemma, we ...
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vote
1answer
42 views

Generators for a finitely generated graded ring

Given a Noetherian graded ring (commutative and with 1) $A=\bigoplus_{n=0}^\infty A_n$, that's generated as an $A_0$-algebra by $x_1,\ldots, x_s\in A$. I am having difficulties seeing why there is no ...
2
votes
2answers
101 views

When a finite local ring $R$ has $-1$ as a square in $R^\times$?

Let $R$ be a finite local ring with maximal ideal $M$ such that $|R|/|M|\equiv 1\pmod{4}$. Then $-1$ is a square in $R^\times$ (that is, there exists $u\in R^\times$ such that $u^2=-1$) if and only ...
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vote
1answer
74 views

An injective-injective module problem

I want to prove this problem: For an $R$-module $M$ and an ideal $J⊆R$, let $A=\{m∈M∶mJ=0\}$. If $M$ is an injective $R$-module, show that $A$ is an injective $R/J$-module. We can view $A$ as an ...
1
vote
1answer
69 views

Can a rational map $X\leadsto Y$ be defined as a scheme morphism $Z\to Y$ for some $Z$?

Let $X=\operatorname{Spec}(R)$ be an integral scheme with generic point $\eta$ and let $Y$ be a separated scheme. A rational map $X\leadsto Y$ is a certain equivalence class and it is represented by ...
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0answers
100 views

Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
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votes
1answer
70 views

Does the $I$-torsion functor commute with inverse limit?

Let $I$ be an ideal of a commutative ring with unit. Is $\Gamma_I(\varprojlim M_j)\cong \varprojlim(\Gamma_I M_j)$? Any reference of the proof or a counterexample is appreciated. It seems this ...
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1answer
296 views

Rank of a module when the base ring is not a domain

Suppose $R$ is a commutative Noetherian local ring with $1$, which is not a domain. Let $M$ be a (non-free) finite $R$-module. What is meant by rank of $M$ in this case?
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1answer
60 views

Generating set of the algebra invariants of finite group.

Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of ...
0
votes
1answer
104 views

Lemma 5.3.6 in Bruns and Herzog, Cohen-Macaulay Rings

In the picture we discuss the Stanley-Reisner ring over a simplicial complex $\Delta$. I do not understand the steps "(i) implies" and "(ii) implies", maybe I do not catch how to translate the ...
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votes
1answer
45 views

Does the relation $\pi(S_{i})=S^{-1}R-P_{i}\cdot S^{-1}R$ hold for prime ideals $P_i$ in a commutative ring $R$?

Let $R$ be a commutative ring. Let $P_{i}$, $1\leq i\leq n$ be prime ideals none of which are contained in each other. Let $S=R-(\cup_{i=1}^{n} P_{i})$. Then $S$ is a multiplicatively closed set and ...
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vote
1answer
53 views

Vanishing of local cohomology and primary decomposition

Let $R$ be an $n$-dimensional Noetherian ring with proper ideal $I$. If $I = \mathfrak{a} \cap \mathfrak{b}$ and $H^n _\mathfrak{a}(M) = H^n _\mathfrak{b}(M) = 0$, for some $R$-module $M$, show ...
1
vote
1answer
123 views

$\Gamma_{\mathfrak a}(I)$ is an injective $R$-module for every injective $R$-module $I$

Is there a proof for Proposition 2.1.4 of Local Cohomology book by Brodmann-Sharp not using Artin–Rees Lemma? Proposition 2.1.4: Let $I$ be an injective $R$-module. Then $\Gamma_{\mathfrak a}(I)$ ...
5
votes
1answer
198 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
4
votes
1answer
186 views

In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal

Let $k$ be a field and let $A \subset B$ be two finitely generated $k$-algebras. Prove that the contraction of any maximal ideal of $B$ is a maximal ideal of $A$. thank you very much again!