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

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A regular sequence in a determinantal ring

Let $S=K[X_{ij}\colon 1\le i\le m, 1\le j \le n, m\le n]$ be a ring of polynomials with coefficients in a field, $X=(X_{ij})$ a matrix of indeterminates, $I$ the ideal of maximal minors and $R=S/I$. ...
1
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1answer
32 views

Is the dimension of a finitely generated $K$-subalgebra of $K[X_1,\ldots,X_n]$ bounded above by $n$?

Let $K$ be a field. Is there an example of a finitely generated $K$-subalgebra $$ A\subseteq K[X_1,\ldots, X_n] $$ of Krull dimension $\dim A>n$? If yes, is there such an example for $n=1$?
3
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1answer
71 views

Reconciling two different definitions of constructible sets

This question is really about sets and topology, but it is motivated from commutative algebra, hence the tag. Setup: Let $X$ be a set and let $\{U_\lambda\}_{\lambda\in\Lambda}\subset 2^X$ be a ...
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1answer
39 views

Projective dimension of monomial ideal

Definition. The support of a monomial $x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ is defined to be the set $\operatorname{supp}(x_1^{\alpha_1}\cdots x_n^{\alpha_n})=\{x_i: \alpha_i >0\}$. Let ...
2
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1answer
46 views

Calculating the coordinate ring and irreducible components

Consider the graded ring $S=(R/I)\oplus (I/I^2)\oplus (I^2/I^3)\oplus\cdots$ Take $R=k[X,Y],I=(X^2Y,XY^2)$. Then $S=k[X,Y]/(X^2Y,XY^2)\oplus(X^2Y,XY^2)/(X^2Y,XY^2)^2\oplus\cdots$. I am not sure ...
2
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1answer
55 views

an example of a module that is not injective

I know that since $\mathbb Z$ is a PID hence every free module is projective and conversely. Hence since $\mathbb Q$ is not free as a $\mathbb Z-$ module then it is not projective. But is $\mathbb ...
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0answers
46 views

How bad must be a ring to allow cyclic artinian modules that are not noetherian?

I've been studying the relations between artinian and noetherian modules over commutative rings. One can prove two interesting results for the commutative case. Theorem Every commutative artinian ...
4
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2answers
75 views

Dimension of the affine variety associated to $\langle zw-y^2, xy-z^3 \rangle $

Find the dimension of the affine variety $V(I)$, where $I=\left\langle zw-y^2,xy-z^3\right\rangle \subseteq k[x,y,z,w]$, with $k$ algebraicaly closed field. I tried to solve the system $zw-y^2=0$, ...
4
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2answers
91 views

Integral closure of $\mathbb{Z}$ in $\mathbb{C}$ is not finitely generated as a $\mathbb{Z}$-module?

Let $$ \mathbb{Z}^{'}_{\mathbb{C}}=\{ z \in \mathbb{C} | \exists f \in \mathbb{Z}[X] \text{ monic such that } f(z)=0\} $$ be the integral closure of $ \mathbb{Z} $ in $ \mathbb{C} $. Prove that ...
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1answer
69 views

$R/I$ satisfies $S_2$ conditions

Let $R=k[x_1,...,x_{n},y_1,...,y_n]$ be a ring over $k$ and $I=\langle \{x_iy_j|$ for some $i,j \in\{1,...,n\}\}\rangle$ be ideal of $R$ and there are $r,s\in\{1,...,n\}$ such that $x_ry_s\notin I$. ...
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1answer
26 views

Inverting a nonzerodivisor of a module

I'm reading the Paper "What makes a complex exact?" by Eisenbud and Buchsbaum. On page 266 it says: Thus we may assume $0 \neq \operatorname{rank}(\phi_n,L) < \operatorname{rank}(F_n)$ and ...
1
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0answers
59 views

Quotient field - base change

For my master thesis, I need to examine the following statement: $Frac(R) \otimes_{k} L \cong Frac(R \otimes_{k} L)$, where $R$ is an integral domain over the perfect field $k$ and $L$ is a finite ...
4
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49 views

Rank of tensor product of morphisms

Let $R$ be a commutative, noetherian, unital ring, $F$ and $G$ two projective $R$ modules, $\phi: F\to G$ a module morphism and $M$ a finitely generated $R$ module such that $$\phi \otimes M := \phi ...
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1answer
34 views

Radical ideal in $\mathbb{R}[x,y,z]$

In $\mathbb{R}[x,y,z]$ is the ideal $I=\left\langle xz,yz\right\rangle$ radical? If $f \in I$ tried write $f=g.xz+h.yz+ax+by+c$ and conclude that $f^m \notin I$, if $m>0$, but I could not.
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35 views

Connection between local freeness and the rank of matrices

I am reading ch.16 of Eisenbud's Commutative Algebra, more precisely it's the very first paragraph of 16.7, where he wants to prove: Suppose that $\mathcal{J}: R^t \longrightarrow R^r$ is a map of ...
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1answer
43 views

How does one find the Krull dimension of a composite ring?

For example, if the ring is $\mathbb{Z} + X \mathbb{Q}[X]$. Is the dimension $1$?
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1answer
31 views

Finitely generated projective modules over a simple algebraic ring extension of a polynomial ring

The well-known theorem of Quillen-Suslin says that a finitely generated projective module over $k[x_1,\ldots,x_n]$ is free, See ...
4
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2answers
277 views

Example: Krull dimension 1 but not a PID

It's easy to prove that if $A$ is a PID which is not a field then $\dim A= 1$. What is a counterexample to the converse? Thanks for any insight.
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1answer
20 views

Example of noetherian module non-uniquely expressible as sum of indecomposable submodules

If $M$ is a noetherian module then it can be written as a finite sum of indecomposable submodules of $M$. The same can be concluded if we assume instead $M$ to be artinian. If we ask for both $M$ to ...
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1answer
32 views

One dimensional integral domains are Cohen-Macaulay

A $1$-dimensional integral domain is always Cohen-Macaulay (C-M). I know this fact, but I do not know how can I reach at. Maybe one should use, somehow, the fact that $R$ is C-M if and only if each ...
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0answers
64 views

prove/disprove $\Delta$ is strongly connected.

Let $\Delta$ be a simplicial complex and $F_1,...,F_n$ be the facets of $\Delta$. Let $\Delta_1$ be another simplicial complex and $F_1,...,F_{n-1}$ be the facets of $\Delta_1$. Assume $\Delta$ and ...
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0answers
43 views

Matrix of linear forms

I would like to see if the following is true: Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field. Let $M$ be a $(a+n-1)\times a$ matrix of linear forms in $R$. If $I_a(M)$ is $(x_1,\ldots, ...
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0answers
30 views

Reference for the determinant of an endomorphism of a projective module of finite rank

In Bourbaki algèbre commutative first book exercice 9 of paragraph 5 of chapter II (page 174) there is an exercise where they explain how to define the determinant of an endomorphism of a projective ...
2
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1answer
42 views

Show that $\alpha_A^{-1}(I'+J')=\alpha_A^{-1}(I')+\alpha_A^{-1}(J')$, where $I',J'$ are ideals and $\alpha_A$ is a surjective ring homomorphism.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ constant matrix. Let $I',J'$ be ideals in $k[y_1,...,y_n]$. ...
4
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0answers
83 views

When flatness of $B$ over $A$ implies flatness of $B$ over $C$, where $A \subseteq C \subseteq B$?

Assume $A \subseteq C \subseteq B$ are integral domains, with $B$ flat over $A$. Generally, $B$ is not necessarily flat over $C$. For example, see van den Essen's book "Polynomial Automorphisms and ...
2
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1answer
27 views

Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset \left\langle\alpha_A(I)\right\rangle \cap \left\langle\alpha_A(J)\right\rangle $.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ matrix. Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset ...
3
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1answer
137 views

Projectivity of $B$ over $C$, given $A \subset C \subset B$

I have found a result concerning projectivity of a certain ring extension: Lemma 2.64. This says the following: Let $A$ be an integral domain or a noetherian ring, $B$ an $A$-algebra, $C$ an ...
3
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1answer
62 views

Krull dimension of $A[x]/\langle x^2 + 1 \rangle$

Consider any noetherian ring $A$ and the polynomial ring $A[x]$. Consider the quotient ring $A[x]/\langle x^2+1\rangle$. Is the dimension of this quotient ring equal to dimension of $A$ (i.e. ...
1
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1answer
52 views

How can one show that an ideal with some property is zero-dimensional?

Let $\mathfrak{a}$ be an ideal in $\mathbb{k}[x_1, \ldots, x_n]$ and a Gröbner basis of the ideal be $\{g_1, \ldots, g_t\}$. For each $i = 1, \ldots,n$, there exists $j \in \{1, \ldots, t\}$ such that ...
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1answer
65 views

Why is the affine $\Bbbk$-algebra, $ \Bbbk[x]/\langle x^3 \rangle $ zero-dimensional?

Consider the ideal $\mathfrak{a} = \langle x^3 \rangle \subseteq \Bbbk[x]$. The ideal $\langle x + \mathfrak{a} \rangle$ is a prime ideal in $ \Bbbk[x]/\mathfrak{a}$. Then why is the affine algebra, ...
2
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1answer
36 views

Separability implies flatness, in a special case

A nice theorem of Wang, Corollary 9 of A Jacobian criterion for separability, says the following: Let $B=A[z]=A[Z]/(h(Z))$. If $B$ is a separable algebra over $A$, then $B$ is a flat module over ...
2
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1answer
48 views

Ring localization and ideals

I'm trying to solve a couple of problems involving ring localization and I'm not sure if my solutions are right or if I understand the idea of localization correctly. Let $A$ be a commutative ...
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0answers
41 views

Clarification on notation in Siegfried Bosch's Commutative Algebra book about primary decomposition of ideals.

I'm reading through Siegfried Bosch's Commutative Algebra book, and I'm confused on his notation in one his proofs. He uses this notation a lot, so I think I should I understand it. The notation first ...
0
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1answer
55 views

System of parameters for a local ring

I need some help to solve this problem. This is the kind of problem that makes me stuck at the very beginning. Let $K$ be algebraically closed, $X = \{(x,y)\in\mathbb{A}^2_K: \ y^2-x^3=0\}$ an ...
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1answer
58 views

Flatness and normality

I have just read: Direct proof of non-flatness and wondered what is exactly the claim that Alex Youcis is referring to: "...but are you aware of the fact that flatness preserves normality. In your ...
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1answer
20 views

Extending an absolute value over a localization.

Let's consider the definition of (algebraic) absolute value given by Wikipedia (https://en.wikipedia.org/wiki/Absolute_value_%28algebra%29), and focus the attention under the voice "Fields and ...
0
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1answer
52 views

Integral extension of local ring

I suppose this is a classical result, but I'm having problems to prove it. I want to prove that if $R$ is a commutative local ring and $R\subset S$ is an integral extension, then $S$ is also ...
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votes
2answers
215 views

Cohen-Macaulay but not regular

In the Wiki page it is claimed that $K[[t^2,t^3]]$ is a $1$-dimensional Cohen-Macaulay ring which is not regular. Is there anybody who kindly explain to me the above assertion? Thanks in ...
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1answer
41 views

Class number and complex conjugation

Let $h$ be the be the class number of the ring of integers of the $p$th cyclotomic field. Suppose $p\mid h$ and let $I$ be an ideal of order $m$ such that $p \mid m$. Does $p$ divide the order of $I ...
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1answer
107 views

A power series ring over $\mathbb C$

I have two questions around the ring of formal power series $R=\mathbb C[[x^2,x^3]]$. What is the global dimension of $R$? Is it a local regular ring? The global dimension of a ring is the ...
3
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1answer
50 views

When $f(I)S=S$ for each ideal $I$ of $R$?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). Question 1. ...
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1answer
25 views

Prime ideals in a Dedekind domain

If $R$ is a Dedekind domain and $I\subset R$ is a non-zero ideal then by the Noetherian property of $R$, I can show that there are distinct non-zero prime ideals $P_1,...,P_r$ s.t. $P_1^{a_1}\cdots ...
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1answer
34 views

Localization of ideals at all primes

Let $R$ be a commutative ring with $1$ and $I$, $J$ ideals in $R$. For a prime ideal $P$, let $I_P=(R-P)^{-1}I$ be the localization of $I$ at $P$. Question: If $I_P=J_P$ for all prime ideals ...
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3answers
153 views

In a reduced ring the set of zero divisors equals the union of minimal prime ideals.

If $R$ is a reduced commutative ring with identity, why is the set $Z$ of zero divisors the union of minimal prime ideals? I know that $Z$ is a union of associated primes, and that the ...
4
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3answers
87 views

If the intersection of ideals $I_{1},\ldots,I_{n}$ is contained in a prime ideal $P$, then one of them is contained in $P$

Let $A$ be a commutative ring and $I_{1},\ldots, I_{n}$ and $P$ ideals in $A$ with $P$ prime so that $\cap_{i=1} ^{n} I_{i} \subset P $. Show that there's an $i_0 \in \{1,...,n \}$ so that $I_{i_0} ...
3
votes
1answer
44 views

Equivalence relation on regular functions

In this problem, consider $K$ an algebraic closed field and $X\subset\mathbb{A}^n_k$ an irreducible variety. Given an open Zariski $U\subset X$, we say that a function $\phi:U\rightarrow K$ is regular ...
2
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1answer
38 views

Height and minimal number of generators of an ideal.

Can anyone could give me a reference in a book about the proof of the following Let $I$ be an ideal of a ring. We denote with $\operatorname{ht}(I)$ the height of $I$, and by $\mu(I)$ the minimal ...
0
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1answer
29 views

Exactness of Dual Sequence, A Proposition in Atiyah and MacDonald

The proposition 2.9 of Atiyah and Macdonald syas that a sequence of $A$-modules $$M'\xrightarrow u M \xrightarrow v M'' \rightarrow 0$$ is exact iff the dual sequence $$0\rightarrow Hom ...
0
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1answer
25 views

Infinite direct product of C-M rings

A finite direct product of Cohen-Macaulay rings is a Cohen-Macaulay ring. It could be checked by a scrutiny into localization of a finite direct product of rings at a prime ideal of the product. Now, ...
2
votes
1answer
93 views

Flatness of $\Omega_{B/K}$ over $B$.

Let $K$ be a field of characteristic zero. Assume that $K \subset A \subseteq B$ are noetherian integral domains, with $A$ regular (= all its localizations at maximal ideals are regular local rings). ...