# 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.

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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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### 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 ...
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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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### 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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### 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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### 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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### 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 \$...