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

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51 views

What is the Krull dimension of $A[x,y,z]/\langle xy,xz \rangle$

What is the Krull dimension of $B=A[x,y,z]/\langle xy,xz \rangle$, given $A$ is a Noetherian, commutative ring? I am thinking can it be shown to be an integral extension of another ring?
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3answers
54 views

Domain strictly contained in the intersection of localizations at the primes of height one

If $R$ is a normal domain, then it is equal to the intersections of localizations at height one primes. Maybe Noetherian is required here, I'm more than willing to assume it. What is an example ...
5
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2answers
74 views

Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?

I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in ...
1
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0answers
29 views

Minimal Free Resolutions

Matsumura, Commutative Ring Theory, Chapter 7 p. 153-4: Let $(A, \mathfrak{m}, k)$ be a local ring. An exact sequence $$(*) \cdots \rightarrow L_i \xrightarrow{d_i} L_{i-1} ...
12
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2answers
488 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
4
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1answer
69 views

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.

If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
2
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0answers
31 views

If $G$ is shellable, then $G \backslash \{x_i\}$ is shellable?

A simplicial complex $\triangle$ is shellable on the vertex set $\{x_1,\ldots,x_n\}$, if the facets of $\triangle$ can be ordered, say $F_ 1 , . . . , F _s$ , such that for all $1 ...
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0answers
38 views

What is $\overline{D(f)}$?

Let $A$ be a ring, $f\in A$. If $A$ is Noetherian, $\text{Spec}(A)$ has finitely many irreducible components, let us call them $\{Z_i\}_{i=1}^n$. So we write $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
3
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0answers
45 views

What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
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1answer
17 views

$R\subset S$ rings conditions implying that there is at most $t$ maximal ideals in $S$ lying over any maximal ideal in $R$

The problem is the following. Let $R,S$ be rings such that $R\neq 0 $, $R\subset S$ and $S$ is finitely generated as a $R$-module, with $t$ generators. Let $\mathfrak{m}$ be an maximal ideal in $R$. ...
3
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1answer
32 views

Decomposition of a homogeneous polynomial

Let $k$ be a field. Suppose I have a homogeneous polynomial $f$ in $k[x,y,z]$. If $f$ is reducible, does it always decompose as a product of homogeneous polynomials? Thanks!
2
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1answer
27 views

Computation of permanents of general matrices

In the following paper http://www.stat.uchicago.edu/~pmcc/reports/permanent.pdf it is stated that: "Exact computation of permanents of general matrices is a #P (sharp P) complete problem, so no ...
3
votes
1answer
58 views

when is the cokernel of a map of free modules free?

Let $R$ be a commutative ring (noetherian if needed) and $n,m$ be two nonnegative integers. Consider a map $\varphi: R^n\rightarrow R^m$ Is there a characterisation, e.g. in terms of the matrix ...
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0answers
7 views

How to simulate Permanental Point Process

I have simulated a determinantal point process in a square grid using Gaussian Kernel. The Gaussain matrix is decomposed into its eigenvectors and eigenvalues. In core implementation, the elementary ...
3
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1answer
43 views

Non-finitely generated, non-projective flat module, over a polynomial ring

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat ...
0
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1answer
22 views

Non-finitely generated, non-divisible, non-projective, flat module, over a polynomial ring

(1) Let $R=k[x_1,\ldots,x_n]$. I wish to find an example of a non-finitely generated, non-divisible, non-projective, flat $R$-module. Notice that $k(x_1,\ldots,x_n)$ is NOT an example of what I am ...
3
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1answer
35 views

Global Dimension of a Ring and its Localizations

Why is the following true? The global dimension of a noetherian ring $A$ is the supremum of the global dimension at its localizations at its maximal ideals: ...
2
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0answers
64 views

Fields of Research in Algebra [on hold]

I'm a last-year student in mathematics and I'm looking for a master degree in algebra. So I'm trying to understand what are the most interesting fields of research in algebra all around the world. ...
3
votes
2answers
55 views

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
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1answer
40 views
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2answers
49 views

Systems of Parameters are exactly $R$-sequences

If $(R,m)$ is a local Cohen-Macaulay ring, it is well-known that each system of parameters is an $R$-sequence. Is any $R$-sequence (in a Cohen-Macaulay ring) a system of parameters? I am aware ...
4
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2answers
53 views

Finding the kernel of maps between (polynomial) rings

If I have a map between rings like $f\colon k[x_1,x_2]\to k[t],x_1\mapsto t^2-1,x_2\mapsto t^3-t$, how can I prove that the kernel is $\mathfrak{a}=(x_2^2-x_1^2(x_1+1))$? I see that ...
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1answer
90 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$ is an integral domain?

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $$I = (x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$$ is an integral domain. In other words I want to show ...
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2answers
131 views

Can you determine from the minors if the presented module is free?

Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function ...
3
votes
1answer
32 views

Is radical of finitely generated ideal finitely generated?

Let $R$ be non-noetherian commutative ring with identity and $I$ be a finitely generated ideal of $R$; say $I = (a_1, \cdots, a_n)$. Question.1 Is $\sqrt I$ necessarily finitely generated? ...
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1answer
63 views

If $A$ is a maximal ideal, then $\mathbb{F}_p[x,y]/A$ is a finite field [on hold]

Let $A$ be a maximal ideal of $\mathbb{F}_p[x,y]$. Then $\mathbb{F}_p[x,y]/A$ is a finite field. PD: I cannot show that it is necessarily finite.
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2answers
65 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
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1answer
68 views

Prove that in the ring $F[t,t^{-1}]$ we have $x=t^n \Leftrightarrow x \mid 1$ and $t-1 \mid x-1$

I want to prove the following lemma: For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and ...
4
votes
1answer
808 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 ...
4
votes
1answer
116 views

Is there an example of a non-noetherian one-dimensional UFD?

Or in the contrapositive form Is every one-dimensional UFD noetherian? I know how to construct a non-noetherian UFD (polynomials in infinite number of variables over a field) and I know that it ...
6
votes
2answers
135 views

Does there exist such an invertible matrix?

Let $n \geq 1$ and $A = \mathbb{k}[x]$, where $\mathbb{k}$ is a field. Let $a_1, \dots, a_n \in A$ be such that $$Aa_1 + \dots + Aa_n = A.$$ Does there exist an invertible matrix $\|r_{ij}\| \in ...
1
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1answer
27 views

A question about fields and separability in Serre's “Local Fields”

On page 14 of the English edition of Serre's "Local Fields", that is chapter 1, section 4, I am confused by the following; there is talk of fields $B/\mathfrak P$ and $A/\mathfrak p$ for prime ideals ...
2
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1answer
73 views

Is direct limit of local rings a local ring?

Let $\{R_i\}_{i\in A}$ be a directed set of commutative local rings with directed index set $A$, and let $R$ be the direct limit of this set. I want to know if $R$ is a local ring (we know that $R$ is ...
3
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2answers
53 views

Constructing DVR's from arbitrary UFD's

Is the following statement true? Let $A$ be an UFD and $p\in A$ prime, then $A_{(p)}$ is a discrete valuation ring. I think yes: For every element $x$ of $Q(A_{(p)})=Q(A)$, there is a unique ...
10
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1answer
127 views

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

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
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1answer
57 views

Module of constant rank over noetherian reduced ring

Let $A$ be a reduced noetherian commutative ring and $M$ be a finitely-generated $A$-module such that for all prime ideals $\mathfrak p$, $M_{\mathfrak p}/\mathfrak pM_{\mathfrak p}$ is an ...
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0answers
38 views

When is the Zariski closure of subset connected [closed]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is an indecomposable ring, how can we show that the ...
2
votes
1answer
37 views

Why do we need injectivity in the definition of integral dependence?

Let $f: A \rightarrow B$ a ring morphism of commutative rings, then one has on $B$ a multiplication by elements of $A$ defined by $b*a \doteq b.f(a)$ (where . is the multiplication in the ring $B$). ...
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0answers
87 views

On a theorem of Akizuki concerning the minimal number of generators of an ideal

I am looking for a theorem of Akizuki I was told by my professor. He said me that Akizuki showed in his paper "Zur Idealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz" (1938) a result ...
2
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2answers
45 views

If $A\otimes_k l$ is a normal integral domain then $K(A)\otimes_k l$ is a field.

I am trying to solve Ex. 5.4.M in Vakil's notes. Quoting the text: Suppose $A$ is a $k$-algebra, and $l/k$ is a finite extension of fields. (Most likely your proof will not use finiteness; this ...
0
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0answers
64 views

When will $A[x_1, \ldots, x_n]$ satisfy the dimension formula?

What property should $A$ satisfy so that $A[x_1, \ldots, x_n]$ satisfies the dimension formula, $\mathrm{dim}(A[x_1, \ldots, x_n]) = \mathrm{dim}(A[x_1, \ldots, x_n]/\mathfrak{p}) + ...
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1answer
43 views

Is the ring $A[x_1, \ldots, x_n]$ Cohen-Macaulay? Does the dimension formula hold?

Let $A[x_1, \ldots, x_n]$ be a polynomial ring over a Noetherian, commutative ring, $A$. Is the polynomial ring Cohen-Macaulay? If not, does it follow the dimension formula, $ \mathrm{dim} (A[x_1, ...
4
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0answers
69 views

What properties $R \subseteq S$ should have in order that every prime ideal of $S$ is extended?

My question is almost the same as In what conditions every ideal is an extension ideal?; I allow myself to ask this question, since there is no answer to the above question. My question: Given ...
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0answers
39 views

Question about Poincare series

Let $R=\mathbb Q[x,y]_{(x,y)}$ and $I=(x^{10},x^8y,xy^4,y^5)$. Then how can we calculate the Poincare series of $I$ by Macaulay 2?
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votes
3answers
75 views

Non-domain of Krull dimension zero

Let $F$ be a field, and $V$ be an $F$-vector space. Make $R=F⊕V$ a ring by putting $xy=0$ for $x,y\in V$. Is it true that the Krull dimension of $R$ is equal to zero? If this is so, $R$ would be an ...
3
votes
1answer
76 views

If $A$ is a finitely generated $R$-module, is $\operatorname{Hom}_R(A,R)$ finitely generated? [duplicate]

Let $R$ be an utterly arbitrary commutative, unital ring. Let $A$ be a finitely generated $R$-module. Is $\operatorname{Hom}_R(A,R)$ finitely generated as an $R$-module? Intuitively and based on ...
3
votes
1answer
49 views

Stalks of the sheaf of total quotient rings

Let $X$ be a scheme, for each $U$ open in $X$, let $S(U)$ be the set consisting of elements of $O_X(U)$ whose image in $O_{X,p}$ is a non-zerodivisor for every $p\in U$. In particular, if $U = ...
3
votes
2answers
2k views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...
0
votes
1answer
16 views

Height of associated prime ideal is zero

Let $P\in\operatorname{Ass}(0)$ in a Noetherian ring $R$, and assume the local ring $R_P$ is a domain. I want to prove that the height of $P$ is zero. I know that in a Noetherian ring, each ideal ...
1
vote
2answers
36 views

Closure of subset of affine plane

Inspired by this question, I wonder if one can prove the following Let $ k $ be an algebraically closed field. Is the closure of $ \{(x,y):x^{2}+y^{2}=1,x\ne 0\} $ in the affine plane over $ k $ ...