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Questions tagged [local-rings]

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Relation between lifts of simple roots and lifts of idempotents (Henselian property)

Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
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1answer
131 views

commutative ring satisfying descending chain condition on radical ideals

Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
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1answer
120 views

Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?

If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...
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1answer
208 views

Minimal resolution of local cohomology module

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$ Question Can we say anything about Betti numbers ...
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Bezout theorem for germs of holomorphic functions

UPDATE. It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample. Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
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2answers
349 views

Reduced ring with all non-prime ideals finitely generated

Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ? Without reduced assumption, it is not true even ...
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1answer
320 views

Rings with all non-prime ideals finitely generated

Motivated by this question, I would like to ask: If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case? Note that ...
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1answer
632 views

local ring all whose non-maximal ideals are finitely generated

Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ? It is easy to see ...
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1answer
138 views

Valuation ring whose maximal ideal and every ideal of finite height are principal

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
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111 views

When is a zero dimensional local ring a chain ring?

A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that a zero dimensional local ring is a chain ...
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2answers
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Is a filtered colimit of complete module complete?

This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the ...
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1answer
94 views

Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$

Let $(R, \mathfrak m)$ be a non-field local domain with fraction field $Q(R)$ . Let $k_{R}:=R/m$. We know that there is a Valuation ring $(V,\mathfrak m_V)$ such that $R \subseteq V \subsetneq Q(R)$ ...
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a generalization of group (monoid with order-by-order invertible elements)

Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
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3answers
510 views

$K[[X_1,…]]$ is a UFD (Nishimura's Theorem)

Let us define the infinitely-many-variable formal power series ring $$ K[[X_1,\ldots]] \colon= \underset{m \geq 1}{\varprojlim}\,K[[X_1,\ldots,X_m]]. $$ $K[[X_1,\ldots]]$ is known to be a UFD by a ...
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156 views

What kind of module is this?

Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...
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0answers
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Integral closure of lexsegment ideal

Let $R=k[x_1,\ldots,x_d]$ where $k$ is a field and $I$ be a lexsegment ideal of $R$ and $l(I)=d$ (where $l(I)$ is analytic spread of $I$). Is $I$ integrally closed? If I is generated by elements ...
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1answer
226 views

indecomposable module over a local ring

I ask this in mathematics for some days.it doesn't have an answer up to now. https://math.stackexchange.com/questions/2565828/indecomposable-module-over-a-local-ring As we all know, for an arbitrary ...
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2answers
459 views

The projective covers of Artinian module

The injective hull for a module always exists, however over certain rings modules may not have projective covers. I have a question. If $A$ is an Artinian module on a Noetherian local ring $R$ then $...
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112 views

Local weak factorization

This is a follow-up to question Locally toric resolutions of compactifications, answered by Jason Starr. In a series of papers (see https://arxiv.org/abs/math/9904076), Jaroslaw Wlodarczyk proves ...
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2answers
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Canonical module of a semigroup ring

Let $S$ be a numerical semigroup and $k[S]$ is the associated semigroup ring. I would like to compute canonical module $\omega$ of $k[S].$ I want to show that $\omega=k[t^{-n}:n\in\mathbb Z\setminus ...
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1answer
127 views

Length of a module and Frobenius map

Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map. How to compute $l(R/m^{[p^e]})?.$ I ...
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Analytic spread of an ideal after reduction

Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$ My question is what is the relation between $l(I)$ anf $l(I+(x)/(x))?$
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Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?

Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
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1answer
134 views

Localization of a maximal Cohen-Macaulay module

Let $(R,m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $M$ an module with depth$M=d.$ Is there any example of $M$ such that $(1)$ $M_p$ is not free for some $p\in Ass(R)$ and $(2)$...
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1answer
85 views

Adic filtration and integral closure

Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$. Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. ...
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0answers
166 views

Meaning of the statement “$a\in I$ is a general element of $I$”

Suppose $I$ is an ideal in a Noetherian local ring $(R,m)$. In some papers I have seen the following statement: "$a\in I$ is a general element of $I$". What is the definition of general element ...
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1answer
59 views

A question on infinite local rings which are not division ring

Is it true that if $(R,m)$ is a (not necessary commutative) local ring then $R$ and $m$ have the same cardinal ? (Exclude TWO trivial cases: when $R$ is finite and when $R$ is a division ring) On ...
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171 views

Analytic spread of an ideal

How to calculate analytic spread of the ideal $I=\left<xyw^2,xyz^2,xw^2+yz^2\right>$ in $\mathbb Q[x,y,z,w]?$ I think it is 3.
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Analytic spread of localization of an ideal

Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$. Q) Are there ...
5
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1answer
287 views

how to pass from algebraic power series to the analytic ones

Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \...
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107 views

Asymptotic stability of prime divisors

Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively. My question is What ...
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1answer
97 views

Behaviour of length function under faithfully flat extension

Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$. Can we relate $l_R(I/J)$ and $l_S(IS/JS)$? PS: Here $l(-)$ ...
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126 views

Intersections of Noetherian regular local rings of finite Krull dimensions

Let us consider Noetherian regular local rings $R_i$ of finite Krull-dimensions for each $i \geq 1$ such that \begin{equation*} R_1 \supset R_2 \supset \cdots \end{equation*} Suppose each embedding $\...
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2answers
217 views

Almost complete intersection ideal and $d$-sequence

In a Noetherian local ring $R$, an ideal $I$ is called an almost complete intersection ideal if $\mu(I)=\text{ht}(I)+1$. Q) Is it true that $I$ is generated by a $d$-sequence?
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1answer
346 views

Example of a locally complete intersection ideal

Let $(R,\mathfrak m)$ be a Noetherian local ring. Definition: $I$ is called locally complete intersection ideal if $I_p$ is a complete intersection for all $p\in V(I)$. I want an example of an ideal ...
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Uniform Artin-Rees bound for annihilators in Noetherian local rings

Let $(A,\mathfrak{m})$ be a Noetherian local ring. If $I$ is an ideal of $A$, then by (a weak version of) the Artin-Rees lemma, there exists $j \geq 0$ such that for all $i \geq j$, $$\mathfrak{m}^i \...
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1answer
158 views

Properties of d-sequence

Let $x_1,\ldots,x_n$ is a sequence in a Noetherian local ring $R$. We say $x_1,\ldots,x_n$ is a $d$-sequence if 1) $x_i\notin (x_1,\ldots,\hat{x_i},\ldots,x_n),$ 2) for all $k\geq i+1$ and all $i\...
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0answers
166 views

Modern dictionary for “old” homological terms

I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors. Let $X = SpecA$ be a spectrum of a local ring $(A,...
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1answer
47 views

When is $rad(L)[x_1,\ldots]$ radical in $Ker(\varphi_\ast)$?

Suppose we have a local ring $L$ (not necessarily commutative) such that $L/rad(L)$ is a division algebra (here $rad(L)$ is the Jacobson radical of $L$). We clearly have the canonical surjection $\...
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1answer
127 views

Is there a commonly used name and notation for $\beta(n)=\dim_{A/\mathbf{m}}\mathbf{m}^{n}/\mathbf{m}^{n+1}$, where $(A,\mathbf{m})$ is a local ring?

I've recently found myself doing some work on local rings, and I found the following quantity keeps popping up- Let $A$ be a local commutative unital ring, with maximal ideal $\newcommand{\mfr}{\...
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1answer
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uniqueness of uniformizers

Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local). If $p$ is a height one prime, then the localization $R_p$ is a dvr, hence the maximal ideal $...
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Question about Ext$^1$ in local commutative algebras

Given a local commutative (commutative only if needed...) selfinjective (non-semisimple) finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{...
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Ext^1 for a local finite dimensional selfinjective algebra

Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here: http://arxiv.org/pdf/1609.00588.pdf. There it is ...
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3answers
502 views

Krull-dimension of local domain

Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated. ...
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1answer
324 views

Can K[[T_1,…,T_∞]] be embedded into K[[X,Y]]?

In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the ...
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1answer
82 views

Is there a characterization of r(M) by local cohomology instead of Ext

For a Noetherian local ring $(R,m)$, and a finite $R$-module $M$ with $\operatorname{depth} M=t,$ type of $M$ is defined to be $r(M):=dim_{R/m}Ext^t \ (R/m, M).$ Is there a characterization of $...
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0answers
133 views

Popescu-Neron Desingularization for K[[T_1,…,T_∞]]

Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$. Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
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1answer
255 views

Let $f:R\to S$ be a local finite monomorphism .If $M$ is an Artinian $S$-module, is it an Artinian $R$-module?

$(R,m)$ and $(S,n)$ are local rings (commutative Noetherian with 1). Let $f:R\to S$ be a local homomorphism/monomorphism ($f(m)\subseteq n$), such that the natural induced homomorphism $R/m\to S/n$ ...
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1answer
102 views

initial and terminal objects in local rings [closed]

in a class gave me the concept of initial and terminal object. I started to look for this objects in different categories. I already proved that $\mathbb{Z}$ is initial and the zero-ring is terminal ...
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0answers
222 views

Structure theorem for non-Noetherian local rings

Is there a structure theorem (like Cohen 's structure theorem) for non-Noetherian local rings? I am adding what I am looking for as someone asked in the comment. If $R$ is a local domain (not ...