Questions tagged [local-rings]
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.
408
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left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, ...
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Local ring which is module finite algebra over a complete local ring [closed]
Let $(S, \mathfrak n)\to (R,\mathfrak m)$ be a local homomorphism of Commutative Noetherian local rings such that via this map, $R$ becomes a module finite $S$-algebra. If $S$ is $\mathfrak n$-...
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Ideal of definition of local ring
Definition. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$. An ideal $I$ is called an ideal of definition of $R$ if $\mathfrak{m}^n \subset I \subset \mathfrak{m}$ for some $n\...
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When is the intersection of every non-zero ideal with socle again non-zero?
Let $(R,\mathfrak m)$ be a Noetherian local ring such that $(0:_R \mathfrak m)\neq 0$. Then, is it true that for every non-zero ideal $I$ of $R$, one also has $I\cap (0:_R \mathfrak m)\neq 0$ ?
I know ...
- 69
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Freeness of $I/I^2$ implies that of $\dfrac{I}{I^2+xR}$ over $R/I$?
Let $(R,\mathfrak m)$ be a Noetherian local and $I$ be an ideal of $R$ such that $\sqrt{I}=\mathfrak m$. Let $x\in I\setminus I \mathfrak m$ be such that $x$ is a non-zerodivisor on $R$. If $I/I^2$ is ...
- 437
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Residue field of DVR
Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and residue field $K = A/\mathfrak{m}$ and suppose that $A$ is also a $k$-algebra. Under what conditions can I say that $K/k$ is a finite ...
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The set of non-zero-divisors is dense in the $\mathfrak m$-adic topology for a Noetherian local ring of positive depth
Let $(R,\mathfrak m)$ be a Noetherian local ring with at least one non-zero-divisors in $\mathfrak m$. Let $a\in \mathfrak m$ and $p\in \mathbb N$. Then we can find $a'\in \mathfrak m^p$ such that $a+...
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Derived tensor product of finitely generated module of finite projective dimension and bounded above chain complex of finitely generated free modules
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring. Let $0\neq G$ be a finitely generated $R$-module of finite projective dimension. Let $M$ be a bounded above chain complex of finitely ...
- 131
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Ideals of zero intersection are in powers of the maximal ideal
let $A$ be a noetherian local ring, with maximal ideal $\frak m$.
I suppose moreover that $A$ is complete for the $\frak m$-adic topology, if that helps.
Let $(I_n)_{n \ge 1}$ be a decreasing sequence ...
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Eventual vanishing of $\operatorname{Ext}^i_R(M,N)$ Vs. eventual vanishing of $\operatorname{Ext}^i_R(M,R)$, where $N$ has projective dimension $2$
Let $(R,\mathfrak m)$ Noetherian local reduced ring. Let $N$ be a finitely generated $R$-module of projective dimension $2$ ,i.e. there exists positive integers $a,b,c$ and an exact sequence $0\to ...
- 131
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Making $I/Rx$ maximal Cohen-Macaulay $R$-module when $I$ itself is maximal Cohen-Macaulay
Let $R$ be a local Cohen-Macaulay ring of positive dimension such that $R$ is not an integral domain. Let $I$ be an ideal of $R$ containing a non-zero-divisor of $R$ such that $I$ is a maximal Cohen-...
- 131
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Dimension of $(A[[X]])_\mathfrak{m} \geq k+1$, regular ring, chain of prime ideals
Assume $A$ is a regular ring and $m$ a maximal ideal of $A$. We define $R= A[[X]]$.Then $\mathfrak{M}=mR + XR$ is a maximal ideal of $R$. Lets assume $h(mA_m) = k$.
I want to show, that
\begin{align*}
...
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Localization in power series
Let $A$ be a comm. ring with unity.
Say $\mathfrak{M}$ is a maximal ideal of $A[[X]]$.
Is following statement generally true?
\begin{align*}
(A[[X]])_\mathfrak{M} \cong (A_{\mathfrak{M} \cap A}[[X]])_\...
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If a Noetherian local ring contains a field, then so does its $\mathfrak m$-adic completion? [closed]
Let $(R, \mathfrak m)$ be a Noetherian local ring containing a field. Then, does the $\mathfrak m$-adic completion $\widehat R$ of $R$ also contain a field?
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Every element in a local ring is either invertible or nilpotent
I would like to ask about the correctness of the proof of 2.4. Proposition. I think it is not true. This proposition is stated as follows:
Let $R$ be a local ring. Then, every element in $R$ is ...
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Local Noetherian ring with several properties is a PIR
Let $R$ be a local Noetherian ring such that every nonzero prime ideal is maximal and every $P$-primary ideal is a power of $P$. I want to show that $R$ is a principal ideal ring.
Let $I \...
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Show that local rings on a curve are PID
I'm reading the notes on Elliptic Curves from this MIT course, more specifically this part where the local ring of a curve $C$ at point $P$ is defined, as the set of rational functions $f$ on $C$ such ...
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Vanishing of $\text{Ext}^i_R(M,N)$ vs. $\text{Ext}^i_R(M,R)$, for large $i$, where $N$ has projective dimension $1$
Let $M,N$ be finitely generated modules over a commutative Noetherian local ring $(R, \mathfrak m)$. Assume that $\text{Ext}^i_R(M,N)=0$ for all large integers $i\gg 0$, and also that $N$ has ...
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Relation between $\operatorname{depth}(R)$ and $\operatorname{depth}(R/xR)$.
Let $(S,\mathfrak m)$ be a local regular ring, $I\subset \mathfrak m$ an ideal and $x\in \mathfrak m\setminus I$.
If $x$ is regular on $R=S/I$, then $\operatorname{depth}(R/xR)=\operatorname{depth}(R)-...
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Finitely generated modules over local domains
Let $R$ be an integral domain. Let $P$ be a finitely generated $R$-module.
The problem:
If we additionally assume that $R$ is a local ring (that is, $R$ is a local domain), is the following statement ...
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Commutative rings are McCoy [duplicate]
I am having difficulty to prove that commutative rings are McCoy. It means if $R$ is any commutative ring and $R[x]$ is its polynomial ring, whenever two polynomials $f(x),g(x)$ $\in R[x]$ annihilate ...
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A characterization of Noetherian regular ring
Proposition.
Let $A$ be a Noetherian ring. The folliwing fact are equivalent:
$A_P$ is a local regular ring for all $P$ prime ideals of $A$
$A_{\mathfrak{M}}$ is a local regular ring for all $\...
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A finite algebra over a local ring is semi-local
Here, rings are commutative. The question comes from Commutative Algebra by Matsumura.
$A$ is a local ring with maximal ideal $\mathfrak{m}$. $B$ is a finite $A$-algebra, that is, there exists $b_1, \...
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Example of a $1$-dimensional reduced local ring $R$ admitting a special kind of module finite birational extension
What is an example of a reduced local ring $(R,\mathfrak m)$ of dimension $1$ and a module finite ring extension $R\subseteq S$ such that $S\subseteq Q(R)$ and $S^{**}\cong S^*\ncong S$ ?
(Here, $Q(R)$...
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Elements in a local ring written as a sum.
Set $A$ as a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ generated by $m_1,\ldots,m_n$.
I believe, every element $a \in A$ can be written as a finite sum:
\begin{align*}
a= \...
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Highest possible place of non-zero Koszul homology
Let $(R,\mathfrak m)$ be a Commutative Noetherian local ring. Let $K^R$ be the Koszul complex on a minimal generating set of $\mathfrak m$. Then, is it true that $H_i(K^R)=0$ for all $i>\mu(\...
- 29
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Weakening the assumptions for showing that a local ring homomorphism is surjective
I was looking at Homomorphism of local rings which is about proving a particular theorem on local rings. More precisely, we are given a local homomorphism $f\colon(A,\mathfrak m_A)\to(B,\mathfrak m_B)$...
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Homologies of Koszul complex of the maximal ideal of $R$ and $R/xR$
For a Noetherian local ring $(R,\mathfrak m)$, let $K^R$ denote the Koszul complex on a minimal generating set of $\mathfrak m$ (it is well-known that given any two minimal generating set, the Koszul ...
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Invertible matrix over a local ring
In this paper by Kaplansky on his theorem of projective modules over local rings, he states
Hence the matrix $(c_{ij})$ is non-singular; for it has units down the main diagonal and non-units ...
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Prove or disprove that if $A$ is a local ring, then $A[X]/\langle X^n\rangle$ is local ring.
I can prove that it is true when $A$ is a field. When $A$ is a field any $\langle X\rangle$ is a maximal ideal of $A[X]$. Any maximal ideal of $A[X]/\langle X^n\rangle$ is of the form $M/\langle X^n\...
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About local ring defined in a different way [duplicate]
Let $R$ be a commutative ring with $1$.
a. Show that for all $a,b\in R$, $ab$ is invertible iff $a,b$ are.
b. Show that if the set of non-invertible elements in $R$ is closed under addition then for ...
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Existence of $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal
Let $(R,\mathfrak m)$ be a Noetherian local domain of dimension at least $2$.
Then, must there exist $x\in \mathfrak m \setminus \mathfrak m^2$ such that $xR$ is a prime ideal of $R$? What if we also ...
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On maximal Cohen-Macaulay property of a special kind of ideal
Let $R$ be a local Cohen-Macaulay domain of dimension at least $2$. Let $M$ be a maximal Cohen-Macaulay $R$-module such that localization of $M$ at every height $1$ prime ideal of $R$ is free. ...
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On the number of generators of power of ideal in a local ring
let $(R,m)$ be a local ring and let $I,J$ be an ideals of $R$, let $\mu(I)$ be the minimal number of generators of $I$. I've proved that $\mu(IJ)\leq\mu(I)$ but somehow this claim does not feel right ...
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If $(R:_{Q(R) } S)$ is non-zero, then does $(R:_{Q(R) } S)$ contain a non-zero-divisor?
Let $R$ be a reduced Noetherian local ring of dimension $1$ with total ring of fractions $Q(R)$. Consider a ring extension $R \subseteq S \subseteq Q(R)$.
If $(R:_{Q(R) } S)$ is non-zero, then does $(...
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Property about submodules of noetherian modules over noetherian local rings
$\def\m{\mathfrak{m}}
$I am trying to show the following commutative algebra result:
Proposition. Let $(R,\m)$ be a noetherian local ring, $M$ be a noetherian module and $M_1,M_2\subset M$ be ...
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About the hypotheses of the Nakayama Lemma
I know the statement of Nakayama's Lemma
Let $R$ be a local ring with maximal ideal $M$ and $I$ be a finite generated ideal of $I$ such that $MI=I$ then $I=(0)$
In this case I know the demonstration,...
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Monic polynomials with coefficients in a local ring
Proposition. Let $A$ be a local ring with maximal ideal $P$, and $f=x^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\in A[x]$ such that $a_i\in P$ for all $i=0,\dots, n-1$, with $a_0\notin P^2$. Consider the $A$-...
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Local rings k[[T]] and k{{T}}
I have a question of algebraic geometry:
A local ring is a ring having a unique maximal. Prove that rings k[[T]] and k{{T}} are local. What are their maximal ideals ?
I see how to manage the fact ...
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Element in ring of integers of $\overline{\mathbb{Q}}_p$
Is it true that every element $x \in \mathcal{O}_{\overline{\mathbb{Q}}_p}$ can be written as $x=u \cdot p^{m/p^s}$ where $u$ is a unit, $m \in \mathbb{N}$ and $s \in \mathbb{N}$?
Note that $p$ is a ...
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$I.M+N=M \implies M=N$
Some time ago I used some result saying that if $M$ is a finitely generated $R$ module, where $R$ is local noetherian (I don't know if that's necessary), then for $N$ a submodule of $M$ and $I$ a ...
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If $M$ is a separable module over a complete local ring, $M/mM\cong K[[x]]/(\bar{f})$
I am looking at some notes on structure of complete local rings by Hochster.
Let $A$ be a Noetherian, complete local ring with maximal ideal $m$.
I am looking at the ring of formal power series $R=A[[...
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When is $R$ a direct summand of Frobenius pushforward?
Let $(R,\mathfrak m)$ be a Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module structure ...
- 437
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81
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linear map between finitely generated free modules (over local ring) which become injective after tensoring with the residue field
Let $F,G$ be finitely generated free modules over a Noetherian local ring $(R,\mathfrak m)$. Let $f: F\to G$ be an $R$-linear map such that the induced map $\bar f: F/\mathfrak m F \to G/\mathfrak m G$...
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59
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When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?
For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...
- 131
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Local nature of completion in an example of Hartshorne
Hartshorne gives the example of completing the local ring of the affine variety defined by $y^2 - x^2 - x^3$ in $\mathbb{A}^2$ at $(0,0)$ on page 35. The takeaway seems to be that when we complete the ...
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Is the set $\bigcup_{x\in R}\text{Ass}(R/xI)$ finite for a Noetherian local domain $R$?
Let $(R,\mathfrak m)$ be a Noetherian local domain of dimension at least $2$. Let $I$ be an ideal of $R$, let $Q(R)$ be the fraction field of $R$.
Then, is it true that the set $\bigcup_{x\in(R:_{Q(R)}...
- 131
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244
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Stuck showing that Milnor's topological intersection multiplicity agrees with the algebraic definition
I am trying to solve problem 3 from Appendix B from Milnor's “Singular Points of Complex Hypersurfaces” (p. 115). This shows that Milnor's topological definition of intersection multiplicity agrees ...
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Why is a semilocal domain $A$ a UFD, if its localisations at maximal ideals $A_{\mathfrak m}$ are UFDs?
This is Exercise 20.2 in Matsumura's Commutative Ring Theory:
Let $A$ be an integral domain. We say that $A$ is locally UFD if $A_{\mathfrak m}$ is a UFD for every maximal ideal $\mathfrak m$. If $A$ ...
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Is the following module flat over $A$?
Let $B \to A$ be a surjection where $B$ and $A$ are Artin local rings which are $k$-algebras and both having residue field $k$. Let $M$ be the kernel of the surjection and $M^2 = 0$. This induces an $...
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