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

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Sizes of Quotient Rings of DVRs with Finite Residue Field

If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are ...
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Indecomposable commutative rings [on hold]

Let $R$ be a commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?
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Generalization of Singular locus and non-free locus to an algebra

Let $R$ be a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ and $ \Lambda $ be a noetherian $ R $-algebra. Recall that: (1) The singular locus of $R$, denoted by $\mathsf{Sing} ...
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commutativity taking the complement and taking fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...
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56 views

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

A simplicial complex $\Delta$ on the vertex set $\{x_1,\dots,x_n\}$ is shellable if the facets of $\Delta$ can be ordered, say $F_ 1 , . . . , F _s$, such that for all $1 \leq i < j ...
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1answer
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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 ...
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20 views

Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$

What is the Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$ where $A$ is a Noetherian, commutative ring and $\mathfrak{a} = \langle f_1, \ldots, f_s \rangle$ and each $f_i$s is monic in $A$? Is ...
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2answers
59 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
55 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 ...
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31 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} ...
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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, ...
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1answer
71 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 $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$. ...
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41 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 $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
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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
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$R\subset S$ 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 $R$. ...
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1answer
34 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!
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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 ...
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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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8 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 ...
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1answer
44 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 ...
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1answer
23 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 ...
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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: ...
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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. ...
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56 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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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 ...
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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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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
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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
64 views

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

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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
74 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 ...
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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 ...
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1answer
128 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 $R$, ...
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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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39 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 ...
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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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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 ...
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2answers
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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 ...
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67 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
44 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, ...
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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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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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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 ...