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

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1answer
48 views

Could you please explain the detail of the proof

Proposition: Proof: Question: Why it's isomorphism?
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0answers
52 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
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2answers
63 views

Problem related to prime ideals of B and A where B is integral over A

Let $ A $ be an entire ring, integrally closed. Let $ B $ be entire, integral over $A$. Let $ Q_1, Q_2$ be prime ideals of $B$ with $Q_1 \supseteq Q_2$ but $Q_1 \neq Q_2$. Let $P_i=Q_i \bigcap A$. ...
2
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1answer
82 views

Why is $f'(x)$ the annihilator of $dx$?

Let $B=A[x]$ be an integral extension of a Dedekind ring $A$ where $x$ has minimal (monic) polynomial $f(x)$. Then the module of Kahler differentials $\Omega_A^1 (B)$ is generated by $dx$. Why is its ...
4
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5answers
523 views

Every module over a field is free. Is every commutative ring whose modules are all free a field?

Let $A$ denote a commutative ring. Then if $A$ is a field, we may deduce that every $A$-module is free. Does the converse hold? i.e. If every $A$-module is free, can we deduce that $A$ is a field?
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0answers
51 views

Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
1
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1answer
82 views

Basis of a subset of finitely generated torsion free module

Based on the comments of rschwieb's answer in this question asked recently: Can we contruct a basis in a finitely generated module. If $M=\langle e_1,\ldots,e_n\rangle$ is a finitely generated ...
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2answers
41 views

Can we contruct a basis in a finitely generated module

Let $M=\langle e_1,\ldots,e_n\rangle$ be a finitely generated $R$-module. My question is can we construct a free submodule $F$, i.e, isomorphic to $R^s$ for some $s$, finding a subset ...
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1answer
55 views

a “strange” depth inequality

The following question arises in the context of the proof of Proposition 3.3.18 in Bruns and Herzog, Cohen-Macaulay Rings. Let $R$ be a CM ring, not necessarily local, and suppose that $R$ admits a ...
4
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1answer
130 views

What is the normalization of the ring $\mathbb C[x,y,t]/(t^3-x^3y)$?

I would like to compute the normalization of the ring $A=\mathbb C[x,y,t]/(t^3-x^3y)$, but I do not know how to proceed. I am not an expert in normalizations, and the only examples I saw were ...
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1answer
60 views

Could you please show some hints about the proof

I even don't know how to start.
4
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1answer
288 views

Mayer-Vietoris sequence for local cohomology

Update 7:35pm UTC 3/23/14: I've reposted this quesion on MathOverflow here. As an assignment in my commutative algebra class, I need to prove the Mayer-Vietoris sequence for local cohomology: Let ...
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3answers
99 views

Show that $A \cong \mathbb{C}^n$ with A a commutative algebra [duplicate]

Let A be a commutative algebra of finite dimension, and if $A$ has no nilpotent elements other than $0$, is true that $A \cong \mathbb{C}^n$ ? The question emerge to my mind, I thought that the ...
3
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1answer
79 views

Corollary 3.3.15 in Bruns and Herzog, Cohen-Macaulay Rings (self-contained question)

Theorem 3.3.14 [Bruns and Herzog, CMR]: Let $(R,m)$ be a CM local ring and $(R,m) \rightarrow (S,n)$ a flat local homomorphism. Then (a) If $\omega_R$ exists (this is the canonical module of $R$) and ...
2
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1answer
96 views

Is always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?

Let $A$ be a commutative ring. Let $f \in A$. Let $A_f= A\left [ \frac{1}{f}\right ]$. Let $\hat{A}$ the $f$-adic completion of $A$. Is it always (even when $A$ is not noetherian) true that $$\hat{A} ...
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0answers
90 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
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0answers
73 views

Localization at prime ideal $(x_1, x_2, \dots)$ of infinitely many variables

I'm trying to solve a homework exercise which starts off: Let $k$ be a field, and let $A = k[x_1, x_2, \dots]$ be a polynomial ring in infinitely many variables. Let $\mathfrak p \subseteq A$ be ...
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2answers
230 views

In a noetherian integral domain every non invertible element is a product of ireducible elements

I want to prove that in a noetherian ring $R$ which is also an integral domain, every non invertible element can be expressed as product of ireducible elements. I really do not know where to ...
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1answer
57 views

Coherent and Cauchy Sequences.

I'm reading chapter 10 of A-M in Completions, and I'm trying to understand how coherent sequences give rise to Cauchy sequences. This seems to be pretty clear, according to the text, but it eludes me. ...
2
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1answer
68 views

contracting an associated prime in a local ring homomorphism

Let $\phi: (R,m) \rightarrow (S,n)$ be a morphism of local Noetherian rings. Let $M$ be an $S$-module which is finitely generated as $R$-module. Let $p \in \operatorname{Ass}_S M$ and let $x \in M$ be ...
2
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1answer
91 views

$\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = \{ \mathfrak{p}A_\mathfrak{p}\} $

Let $k$ be a field, $A = k[X_1,X_2,...]$, $\mathfrak{p} = (X_1,X_2,...)$, $I = (X_1^2-X_1,X_2^2-X_2,...)$, $M= A/I$. I am trying to show that $\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = ...
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0answers
91 views

Is every module a direct limit of cyclic modules?

I want to show that $M$ is $A$-flat is equivalent to $Tor_1^A(M,A/I)=0$ for every finitely generated ideal $I$. I want to show $Tor^A_1(M,N)=0$ for any $A$-module $N$. Is every module a direct ...
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1answer
72 views

Homomorphisms from the base change of a module

Let $A, B$ be commutative rings with one and let $M$ be an $A$-module, $f: A \rightarrow B$ a ring homomorphism. Consider the (right) $B$-module $M \otimes_A B$. What can we say about ...
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1answer
74 views

Construct a counterexample of a primary ideal which …

Let $A$ be a Noetherian local ring of dimension $d$, $\mathfrak{m}$ its maximal ideal. Suppose $\mathfrak{q}=(x_1,\ldots,x_d)$ is an $\mathfrak{m}$-primary ideal. Suppose $f(t_1,\ldots,t_d)\in ...
2
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1answer
69 views

Cohen-Macaulay ring and saturated ideal

Let $A=\mathbb{C}[x_0,x_1,\dots,x_n]$ and I ideal of $A$. Is there any connection between "$A/I$: Cohen-Macaulay ring" and "$I$: saturated ideal"? Does one of them imply another? Please give me any ...
4
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1answer
128 views

Maximal ideals in the ring of eventually constant sequences of real numbers

For homework I am studying the ring $R$ of eventually constant sequences of real numbers (with multiplication and addition defined componentwise). What are the maximal ideals of $R$? By looking at ...
2
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3answers
116 views

What is $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$?

I would like to know what $\mathbb{Z}/n\mathbb{Z}\otimes_\mathbb{Z} m\mathbb{Z}$ is isomorphic to, where $n,m\in\mathbb{N}$. Of course there will likely be cases depending on coprimeness and whatnot; ...
3
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1answer
61 views

Is there a topological characterisation of non-Archimedean local fields?

A local field is a locally compact field with a non-discrete topology. They classify as: Archimedean, Char=0 : The Real line, or the Complex plane Non-Archimedean, Char=0: Finite extensions of the ...
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0answers
88 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
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2answers
120 views

How can I complete the proof of Noetherianity of I. S. Cohen?

Theorem (I. S. Cohen). If $R$ is an unital commutative ring, and for each ideal prime $\mathfrak{p}\in Spec(R)$ we know $\mathfrak{p}$ is finitely-generated as $R$-mod then $R$ is Noetherian. ...
0
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1answer
68 views

If $M$ is a maximal ideal of a commutative ring R, then $R/M \cong R_M/M_M$. Where $R_M$ is the localization of $R$ at $M$.

If $M$ is a maximal ideal of a commutative ring R, then $R/M \cong R_M/M_M$. Where $R_M$ is the localization of $R$ at $M$. We know that localization commutes with taking quotients, so $R_M/M_M ...
1
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1answer
116 views

Why is this ring Cohen-Macaulay?

Let $q_1, q_2$ be quadratic homogeneous polynomials in $x_0,x_1,x_2,x_3,x_4$ over $\mathbb C$ and let $X_i:=V(q_i)=\{(a_0,\dots,a_4)\in \mathbb{P}_{\mathbb{C}}^4\mid q_i(a_0,\dots,a_4)=0\}$. If ...
2
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0answers
170 views

Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
2
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1answer
46 views

Basic question in the radical of an ideal

Does the radical of an ideal $\sqrt{I}$ always contain $I$? I think the question boils down to if we are given $\sqrt{I}$, does it contain all the elements such that raised to ANY power? So $x \in ...
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1answer
41 views

Reference for $(N_1\cap N_2)\otimes_A M = (N_1\otimes_A M)\cap ( N_2\otimes_A M)$

Where can I find a canonical proof of the following statement? If $M$ is a flat $A$-module and $N$ is an $A$-module with submodules $N_1, N_2$, then $$(N_1\cap N_2)\otimes_A M = (N_1\otimes_A ...
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2answers
58 views

$\mathbb{Z}/n\mathbb{Z}$ is not flat

On the flat module Wikipedia page, it's stated that $\mathbb{Z}/n\mathbb{Z}$ is not flat over $\mathbb{Z}$. But I don't understand their explanation of why. It is said that ...
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1answer
61 views

Why are $(X_1), (X_1,X_2), \ldots$ prime ideals?

I was looking at the proof of the dimension of the polynomial ring $R[X_1,\ldots,X_n]$ and I had a question: Why are $(X_1), (X_1,X_2), (X_1,X_2,X_3),\ldots, (X_1,\ldots,X_n)$ prime ideals in this ...
2
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0answers
100 views

Ideal in $k[x,y]$ generated by two elements

Suppose $k$ is a finite field of order $q$. Let $f = \prod_{1 \leq i \leq s} (x + b_i y)$ and let $g = \prod_{1 \leq i \leq t} (x + c_i y)$, where $b_i, c_j \in k.$ I am interested in finding out ...
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0answers
47 views

Eisenbud 3.11(d) - A Uniform Bound on the Length of Certain Modules

I am trying to solve this exercise from Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. There is a hint or possibly a solution in the back, but I want to try to get some more ...
2
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1answer
76 views

Localization of an integer quotient is a field

Let $R:=\mathbb{Z}/24\mathbb{Z}$ be our ring, $f: \mathbb{Z}\to R$ be the canonical quotient map (i.e. $f$ sends an element to its equivalence class) and $q$ be the ideal generated by $f(3 ...
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0answers
170 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
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1answer
445 views

A module over an algebra. Is it a vector space?

Let $A$ be an algebra over a field $k$. I would like to know if my understanding of the following correct or not. What I want to clarify is the definition of a module $M$ over $A$. I know the ...
5
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1answer
191 views

Height unmixed ideal and a non-zero divisor

Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal? Background: A ...
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2answers
69 views

How does one find a minimal primary decomposition?

What exactly does it mean for a primary decomposition to be "minimal" and is the a general method to obtain such decompositions? I've tried looking at some examples but they all give very little ...
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1answer
72 views

Let $(R,m)$ be * local and $R_m$ regular. Is R regular?

Let $(R,m)$ be *local and $R_m$ regular. Is $R$ regular?
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1answer
43 views

Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
2
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1answer
96 views

Showing that if $f,g \in k[x,y]$ are irreducible and not associates then $(f,g) \cap k[x] \ne 0$

There is a part of example 10.25.3 at http://stacks.math.columbia.edu/tag/00EX that I'm having trouble understanding. Here, $k$ is a field and $f,g \in k[x,y]$ are irreducible and are not associates. ...
1
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1answer
58 views

Localization of rings and integral closures

I looking at localizations of rings and I have the following problem: Let $R=k[x]$ be a polynomial ring ($k$ a field) and $R'=k[x^2])$ (note $R$ is integral over $R'$. Then if I look at the ideal ...
6
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1answer
282 views

Is there an example of commutative ring with exactly three prime ideals for which this property holds?

Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have ...
0
votes
1answer
49 views

existence of a finite-length maximal regular sequence

Theorem 16.7 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a Noetherian ring, $I$ an ideal of $A$ and $M$ a finite $A$-module such that $IM \neq M$; then the length of a maximal ...