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

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28 views

Dimension of a module and its dual [on hold]

Let $(R,m)$ be a Noetherian local ring, $E$ be the injective hull of $R/m$ and $N$ be a finite $R$-module (we can assume length of module $N$ over $R$ is finite), then ...
0
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0answers
14 views

Cokernel of a faithfully flat homomorphism

Let $f:A\to B$ be faithfully flat ring homomorphism and $N=\operatorname{Coker}(f)$ the cokernel of $f$. Let $I$ be an ideal of $A$. How can I use the fact that if $B$ is a faithfully flat ...
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3answers
21 views

Faithfully flat ring homomorphism properties

This is from Liu's Algebraic Geometry and Arithmetic Curves exercise 1.2.19 a. Let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that $f$ is injective and that $I\to I\otimes_AB$ ...
0
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1answer
15 views

Faitfully flatness over $B$ and flatness over $A$ equivalence

Let $B$ be an $A$-algebra, and let $E$ be a faithfully flat $B$-module. How can I show that $E$ is flat over $A$ if and only if $B$ is flat over $A$? (Liu, Algebraic Geometry and Arithmetic Curves, ...
4
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0answers
47 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
0
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1answer
17 views

Equivalence of finitely generated of faithfully flat and annihilator

How one can show that a finitely generated flat $A$-module $M$ is faithfully flat if and only if $\operatorname{Ann}(M)=0$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.17.) I tried ...
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1answer
31 views

Let $R$-algebra $A$. If $P⊂A$ is a minimal prime ideal then $p=P \cap R$ consists of zerodivisors for $A$?

We have: Let $R$ be a Noetherian commutative ring. Suppose $P⊂R$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors. But how to prove this? The $R$-algebra $A$ is ...
1
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1answer
18 views

How to determine a primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$

I am trying to prove that the primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$, for a,b,c positive integers, is $(X^a,(X+Y+Z)^c) \cap (Y^b,(X+Y+Z)^c)$. The equality of the ideal and the ...
0
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0answers
13 views

No such prime ideal contains $I_1+I_2\implies I_1 $ and $I_2$ are relatively prime

It's clear to me that if $I_1$ and $I_2$ are two relatively prime ideals of a ring $R$, then there is no such prime ideal containing $I_1+I_2$, since by definition of relatively prime ideals ...
2
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1answer
29 views

Two questions about discrete valuation rings of varieties

Let $X$ be a proper, normal variety over $\mathbb{C}$, and $k(X)$ be its field of rational functions. I think the following two statements are true, but I was unable to give a proof or find the ...
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0answers
20 views

Two ideals that agree in the completion [on hold]

Suppose that $R$ is a Noetherian local ring with maximal ideal $\mathfrak m$, and that $I$ and $J$ are two ideals in $R$ with $\hat{I} = \hat{J}$ in the completion of $R$ at $\mathfrak m$. What can ...
0
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1answer
32 views

Prove that rings are isomorphic.

I read a text recently where it was explained how to compute the Hilbert function of $R=\mathbb{C}[x_0,...,x_n]$, as I was reading the author seemed to assume that for $f_1,...,f_i,f_{i+1} \in R$ we ...
1
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1answer
35 views

Krull dimension of $R[X]/(f(X))$ for $f(X)$ monic

How can I prove that the Krull dimension of $R[X]/(f(X))$, for $R$ a finitely generated noetherian integral domain and $f(X)$ monic, is equal to the Krull dimension of $R$? I don't even know ...
2
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1answer
40 views

Weil divisors fail over singular varieties

Let be $k$ an algebraically closed field. We know that if $X$ is an irreducibile, normal variety, one can associate to every rational function $(f)\in k(X)^*$ a Weil principal divisor $$(f)=\sum_{Y} ...
1
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1answer
36 views

Is there a consensus on the correct way of raising elements of commutative rings to the power of $a/b$?

I'm trying to understand the "correct" way of raising elements of commutative rings to the power of $a/b,$ where $a$ and $b$ are integers, but not having much luck. Suppose $R$ is a commutative ...
2
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0answers
32 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. $\textbf{Shafarevich definition}$ (pg 128) - A variety is $\textit{non-singular in codimension one}$ if the singular locus has ...
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1answer
19 views

Show that $V(\bigcup_{i \in I} E_{i})=\bigcap_{i \in I} V(E_{i})$

This is a part of a problem in Atiyah's Introduction to Commutative Algebra introducing the Zariski Topology. Here we are given that $(E_{i})_{i \in I}$ is a family of subsets of a unital commutative ...
2
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0answers
43 views

Tensoring two short exact sequences

Let $R$ be a commutative ring with $1$ and consider the following short exact sequences of $R$-modules \begin{align} &0 \to M' \to M \stackrel{f}{\to} M'' {\to} 0 \qquad \text{and } \\ &0 \to ...
2
votes
1answer
48 views

Maximal ideal in local ring

The maximal ideal in $\mathbb{Z}_{(2)}$ should be $(2)$, but I don't understand this well. Suppose I take $\frac35\in \mathbb{Z}_{(2)}$. It is not in $(2)$ but in $(3).$ But what is the ideal between ...
0
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3answers
76 views

$\mathbb{Z}_{(2)}$ has one maximal ideal

My lecture notes state that the set $\mathbb{Z}_{(2)}$, defined as $$\mathbb{Z}_{(2)}:=\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{}\gcd(a,b)=1\text{ and } 2\nmid b\right\}$$ has a ...
1
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0answers
26 views

The prime ideals of the ring $K[x]$

I was wondering what the prime ideals of the ring $K[x]$ are, where $K$ is a ring. My guess is that it's any ideal generated by a set of irreducible polynomials over the ring $K$. Have I covered all ...
3
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1answer
54 views

On finite generation of certain $\operatorname{Ext}$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a ...
0
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1answer
32 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
1
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1answer
17 views

Let $K$ be a field, $A \subset K$, and $p \subset A$. Then $\exists$ a valuation ring $R$ satistfying…

I was stuck when reading a proof of the following theorem (Matsumura p. 72-3, Theorem 10.2), Let $K$ be a field, $A \subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a ...
2
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2answers
86 views

What is the Krull dimension of $\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]$?

I am studying commutative algebra and saw the following question in one of the tests: What is the Krull dimension of $R=\mathbb{Q}[x^2+y+z,\ x+y^2+z,\ x+y+z^2,\ x^3+y^3,\ y^4+z^4]?$ I know ...
1
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1answer
57 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to ...
1
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1answer
32 views

What is the injective hull of a polynomial ring?

The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an ...
2
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1answer
36 views

Quotient of ring is flat gives an identity of ideals

I have problem to understand and solve the exercise 1.2.14 on Qing Liu's book "Algebraic Geometry and Arithmetic Curves". It goes as follows: Let $A\to B$ be a ring homomorphism, and let $J$ be an ...
0
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0answers
45 views

Why prime avoidance lemma allows only at most 2 non-prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case $n > 2$, choose $z_i \in E \cap (I_i - \cup_{j ...
2
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0answers
39 views

Integrally closed domain.

Suppose $A$ is a unique factorization domain, $a$ is an element of $A$. Is the ring $A[x,a/x]$ always integrally closed? ($x$ is a variable over $A$) Thanks!
1
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1answer
52 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 char $\not=2$ and $f(X)$ and $g(Y)$ have only simple roots in $k$. Determine the maximal ideals such that the ...
2
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1answer
60 views

Atiyah and Macdonald, exercise 11.7

I am trying to solve the exercise in Atiyah, that $\dim(A[X]) = \dim (A) + 1$ for $A$ noetherian. The very beginning poses a problem, he states in the hint that: for a prime of height $m$ we can ...
2
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0answers
31 views

Extremal Betti numbers of graded ideal with Cohen-Macaulay quotient; Herzog and Hibi, exercise 4.6 [on hold]

Let $I$ be a graded ideal of $S=K[x_{1},...,x_{n}]$ such that $S/I$ is Cohen-Macaulay. Then show that $I$ has only one extremal Betti number. Here, a Betti number $\beta_{i;i+j}\neq 0$ is called ...
3
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0answers
40 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
1
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1answer
28 views

how does Macaulay2 computes analytic spread for non-local rings?

Macaulay2 computes analytic spread for R=QQ[a,b,c,d,e,f] which is not a local ring. In the books like ...
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3answers
45 views

Surjectivity of the induced map of affine algebraic sets

For a morphism $f: X\rightarrow Y$ of affine algebraic sets, I want to show that if the induced map $f^*:k[Y]\rightarrow k[X]$ is surjective then $f(X)$ is closed. I am trying to prove that ...
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0answers
30 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; ...
2
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1answer
27 views

Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for a minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in ...
0
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1answer
37 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
1
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1answer
21 views

Is the (Krull) dimension of a semi-local Jacobson ring equal to zero? [duplicate]

Let $R$ be a commutative ring with identity element. If $R$ is semi-local (number of maximal ideals of $R$ is finite) and a Jacobson ring (this means that every prime ideal of $R$ is equal to the ...
3
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2answers
42 views

Flat algebra over a Dedekind domain

Let $B$ be a flat algebra over a Dedekind domain $A$. Let $f\in B$ be such that for every maximal ideal $\mathfrak m$ of $A$, the image of $f$ in $B/\mathfrak mB$ is not a zero divisor. How can I show ...
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0answers
42 views

Simplicial homology [closed]

Let $\Delta$ be the simplicial complex on vertex set [5] whose Stanley-Reisner ideal is $I_{\Delta}=(x_{1}x_{4},x_{1}x_{5},x_{2}x_{5},x_{1}x_{2}x_{3},x_{3}x_{4}x_{5})$. Write the augmented oriented ...
2
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0answers
41 views

The annihilator numbers of $S/I$

Let $S=K[x_{1},x_{2},...,x_{n}]$ and $I$ be a strongly stable ideal of $S$. Compute the annihilator numbers of $S/I$ with respect to the almost regular sequence $x_{n},x_{n-1},...,x_{1}$. ...
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1answer
60 views

Finding conditions for $\mathbb Z[i][X,Y]/(Y^2 - aX)$, $a \in \mathbb Z[i]$ to be regular

I am trying to find the dimension and the necessary and sufficient conditions under which $A[X,Y]/(Y^2 - aX)$ is regular, that is, the localizations of $A[X,Y]/(Y^2 - aX)$ at all maximal ideals are ...
3
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2answers
67 views

Finding an ideal such that quotient is Cohen-Macaulay

Let $R$ be a commutative local Noetherian ring which is not a domain and not Cohen-Macaulay. Can we find an ideal $I$ in $R$ such that $R/I$ is Cohen-Macaulay, and $\dim R/I=\dim R$?
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1answer
60 views

Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence?

I have just started reading about the concept of $M$-regular sequences on my own and to understand the definition I asked myself the following question: Is $\{3,5,6\}$ a $2\mathbb{Z}$-sequence? ...
1
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1answer
54 views

Example of $I$-adic topology of submodule not matching subspace topology?

I'm reading about the $I$-adic topology on $M$ for $R$ a commutative ring, $I$ an ideal of $R$ and $M$ an $R$-module. The references I'm reading don't provide examples, but they say that if $N$ is a ...
1
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1answer
30 views

Flatness of module over field of fractions

This is from Liu 1.2.9. Let $A$ be an integral domain, and $K$ its field of fractions. Let $M$ be a finitely generated sub-$A$-module of $K$. Why do $M$ is flat if and only if $M_{\mathfrak p}$ is ...
0
votes
0answers
16 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. ...
2
votes
1answer
54 views

Let $f: U \rightarrow W$ be a morphism of affine algebraic sets and $f': k[W] \rightarrow k[U]$ be the k-algebra morphism of coordinate rings.

Prove if $f'$ is surjective then $f$ is a homeomorphism of $U$ onto the closed subset $W$. Well, it's the first time I've seen this word "homeomorphism" but I read online that a map is a ...