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

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2
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1answer
27 views

$(R\oplus R)/I\cong R$ implies $I\cong R$?

In my class of algebraic topology, a friend of mine stated the following: if $R\ne 0$ is a commutative ring with unit and $I\subset R\oplus R$ is a submodule such that $(R\oplus R)/I\cong R$, then ...
1
vote
1answer
28 views

Help with computation and Groebner basis

Hi guys I am learning a new software and a new topic (groebner basis) I have this problem $6-21(x_1x_2+x_1x_3+x_1x_4)=0$ $10-21(x_2x_1+x_2x_3+x_2x_4)=0$ $12-21(x_3x_1+x_3x_2+x_3x_4)=0$ ...
1
vote
0answers
9 views

Applications of module's length

I'm studying some theory about module's length and want to know motivation for this definition. I know that it's useful for intersection theory, but i know only one example from intersection theory: ...
1
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2answers
35 views

Is $(x^2,xy)$ a primary ideal in $k[x,y]$ for $k$ a field?

In Example of Page 52 in Atiyah's Introduction to Commutative Algebra $\mathfrak a = (x^2,xy)$ is not a primary ideal in $A = k[x,y]$ where $k$ is a field. I think, for any $z \in \mathfrak a$, ...
0
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0answers
19 views

Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be an idempotent ideal?
4
votes
1answer
71 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
0
votes
1answer
37 views

Describe the normalization of the cusp.

Show that the normalization of $A = k[x_1,x_2] / (x_2^2 - x_1^3)$ is isomorphic to $k[x]$ and describe (for $k$ algebraically closed) the induced map $Spec(k[x]) \to Spec(A)$ I know that $A$ is a non ...
0
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1answer
28 views

Show that this map has not the going-down property.

Let $A= k[x_1,x_2,y] / (x_2^2-x_1^2(x_1+1))$ and $Spec(A) \to Spec(k[x_1,x_2,y])$ the natural inclusion induced by the projection $k[x_1,x_2,y] \to A$. Consider the map $f : Spec(k[x,y]) \to Spec(A)$ ...
2
votes
3answers
86 views

Example of commutative ring that doesn't satisfy distribution of intersection over addition

I'm trying to find an example of commutative ring $R$ and ideals $\mathfrak a,\mathfrak b,\mathfrak c \in R$ such that $$\mathfrak a \cap (\mathfrak b + \mathfrak c) \neq \mathfrak a \cap ...
3
votes
1answer
69 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
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0answers
16 views

Questions of commutative [on hold]

show that p is minimal among prime ideals containing a if and only if aAp is pAp-primary
3
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0answers
16 views

What is $HC_0(\operatorname{Spec} k[x,y]/(xy))$?

Does anybody know how to compute $HC_0(\operatorname{Spec} k[x,y]/(xy))$? Here $HC_0(-)$ is the zeroth cyclic homology group. I'm curious since $\operatorname{Spec} k[x,y]/(xy)$ can be viewed as the ...
0
votes
1answer
21 views

Annihilators and exact sequence

Let $R$ be a commutative ring and $0\longrightarrow L \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}N\longrightarrow 0$ be an exact sequence of $R$-modules. How to prove ...
1
vote
0answers
17 views

Lifting points of étale group scheme.

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...
1
vote
1answer
30 views

A short exact sequence + exact sequence of opposite direction = split?

Let $0 \to A \to B \to C \to 0$ be a short exact sequence of modules over a commutative ring $R$ containing $1 \ne 0$. Suppose this is also another exact sequence $0 \to C \to B \to A \to 0$. Do ...
1
vote
1answer
202 views

relation of annihilators on exact sequence

Let $0\to M' \to M \rightarrow M^{''} \to 0$ be an exact sequence of modules. I want to show that ${\rm Ann}(M)= {\rm Ann}(M')\cap {\rm Ann}(M^{''})$. The "$\subset$" case I have shown, but I can't ...
4
votes
3answers
139 views

Hilbert function of a monomial ideal generated by degree two square free monomials

Let $R=K[x_1,...,x_n]$ be a polynomial ring over a field $K$ (one can assume $K$ is the field of complex numbers). Let $I=\langle m_1,...,m_l\rangle= \oplus I_j$ (where $I_j$ is $j^{th}$ graded piece ...
2
votes
1answer
33 views

Writing the ideal $m=\langle X, Y \rangle$ in $R=k[X, Y]$ as a countable union of prime ideals

Here's a problem (Exercise 3.21) from "A Term in Commutative Algebra" by Altman & Kleiman: Let $k$ be a field, and $R=k[X, Y]$ be polynomial ring in two variables. Let $\mathfrak{m}=\langle ...
1
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1answer
43 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
13
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2answers
1k views

Video lectures for Commutative Algebra

Are there any good video lectures for learning commutative algebra at level of Atiyah-Macdonald?
10
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1answer
124 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
4
votes
1answer
62 views

Proof of the Artin-Rees lemma

I am struggling to understand a key step in a proof of the Artin-Rees lemma, which I have put in a red box below. I don't really see how we can pass from a finite direct sum to an infinite one. I've ...
3
votes
1answer
105 views

Computing Hilbert polynomial

We have the following condition: For each $i=2,...,m$ multiplication by $f_{i}$ is injective on $S/(f_{1},...,f_{i-1})$, where $S=k[T_{0},...,T_{n}]$, $m \leq n$, and the $f_{i} \in S_{d_{i}}$ are ...
1
vote
1answer
36 views

polynomial grade

Hamilton-Marley in the paper "Non-Noetherian Cohen–Macaulay rings" have I can't understand highlighted part. my attempt is: $$\text{p-grade} ((x')R',R')=\text{p-grade} ...
2
votes
1answer
39 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
7
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0answers
65 views
+100

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\subsetneq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
0
votes
0answers
25 views

associated prime of a module

Let $f: A\rightarrow B$ be a homomorphism of Noetherian rings, and $M$ a $B$-module. Question: Is $^af(Ass_B(M))=Ass_A(M)$? If $q$ is an associated prime of the $B$-module $M$, $p=^af(q)$, then from ...
3
votes
1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
5
votes
3answers
135 views

Basic application of the Nullstellensatz

Background: I have just started learning basic algebraic geometry. My solution to a simple problem involves an application of the Nullstellensatz and I want to know whether this is overkill (or ...
1
vote
1answer
35 views

$X_1,X_2$ disjoint closed in $Spec(R)$ properties

This is a problem in three parts, I managed to prove the first part, but the others I couldn't. Let $R$ be a ring and let $X_1,X_2\subset Spec(R)$ be closed (in Zariski topology) and disjoint such ...
1
vote
1answer
25 views

$F(t)$ as an $F[t]$-algebra and the Weak Nullstellensatz

Sorry if this question has already been answered somewhere, but it's quite hard to find if so, because of the use of the word 'algebra' in the question... In the lead up to a proof of the Weak ...
5
votes
1answer
92 views

Complement of open set is finite in Zariski topology

This problem has two parts: a) Let $M$ be a finitely generated module over a Noetherian ring $A$. Prove that $S=\{ P \in\operatorname{Spec}(A) : M_P \mbox{ is a free }A_P\mbox{-module} \}$ is an ...
3
votes
0answers
51 views

Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
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0answers
33 views

morphisms of curves and discrete valuation rings

Given a dominant morphism $\varphi\colon C\to C'$ of curves, a nonsingular point $Q\in C'$, such that $\varphi^{-1}(Q) = \{P_1,\ldots, P_m\}$ consists of nonsingular points only. Then it is clear to ...
0
votes
1answer
39 views

Flat Module finitely generated when over the residue field finite dimensional? [on hold]

Let $(A, \mathfrak{m})$ be a local ring with residue field $\kappa=A/ \mathfrak{m}$. Let $M$ be a flat $A$-module. Assume that $M \otimes_A \kappa$ is a finite dimensional $\kappa$-vector space. Is it ...
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votes
0answers
78 views

Closedness and going up property: an explicit proof.

Suppose that a ring homomorphism $f: A \to B$ satisfies the going up property (i.e. for any $\mathfrak{q} \in\operatorname{Spec}(B)$ and any $\mathfrak{p}'\in \operatorname{Spec}(A)$ containing ...
0
votes
0answers
41 views

Dimension of irreducible components of variety [closed]

Consider the affine variety $X=\{ (a_1,a_2,a_3,b_1,b_2,b_3) \in \mathbb{C}^6 \mbox{ : }a_1b_2=a_2b_1, a_1b_3=a_3b_1 \}$. Prove that $X$ has two irreducible components, and that both of them are of ...
8
votes
3answers
252 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 ...
5
votes
1answer
47 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
2
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0answers
85 views

Regular subrings of a polynomial ring

Let $R=\mathbb{C}[x,y]$. I have the following situation: $\mathbb{C} \subseteq D \subseteq R$ is affine (=finitely generated as a $\mathbb{C}$-algebra), noetherian, has field of fractions ...
4
votes
1answer
27 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
0
votes
1answer
62 views

A proof for Atiyah-Macdonald Exercise I.21.iii

The following is exercise I.21.(iii) of Atiyah-Macdonald: Let $\phi \colon A \to B$ be a ring homomorphisms. Let $X = \operatorname{Spec} A$ and $Y = \operatorname{Spec} B$ [and let $\phi^\ast ...
1
vote
1answer
32 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
1
vote
1answer
69 views

Regularity of $k[X,Y,Z]/(Z^2 - f(X)g(Y))$

Let $R = k[X,Y,Z]/(Z^2 - f(X)g(Y))$, for an algebraically closed field $k$ with $\operatorname{char} k\not=2$, and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals $M$ ...
4
votes
1answer
750 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 ...
0
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0answers
20 views

Minimal graded free resolution of $R/I\oplus R/J$ in terms of minimal graded free resolution of $R/I$ and $R/J$.

Let $R=k[x_1,...,x_n]$ be a graded ring over a field. Let $I,J$ be homogeneous ideals. Questions: What is the minimal graded free resolution of $R/I\oplus R/J$ (in terms of minimal graded free ...
3
votes
1answer
62 views

Nilpotents after tensoring with a field

Let $A \to B$ be a homomorphism of commutative rings with unit. Let $A_{\text{red}}=A/ \sqrt{(0)}$ and $B_{\text{red}}=B/ \sqrt{(0)}$ be the corresponding reduced rings. Now let $A_{\text{red}} \to K$ ...
3
votes
2answers
52 views

Direct sum of non-zero ideals over an integral domain

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$ I regarded the short exact sequence $0\to I\cap ...
0
votes
3answers
71 views

Finitely Presented Modules Definition

I am a little bit confused with the definition of finitely presented modules. In Lang's Algebra he defines a module $M$ to be finitely presented if and only if there is a exact sequence $F'\to F\to M ...
1
vote
1answer
56 views

$\operatorname{Proj}k[x,y,z]/(xz,yz,z^2)$ isomorphic to $\mathbb{P}^{1}_{k}$

While dealing with the Proj construction, I encountered with this seemingly-simple question, but somehow I can't get the point at this moment. Is the scheme ...