# 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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### Valuation in the local ring of a curve

Let $k$ be an algebraically closed field and $A = k[X,Y]/(Y^2-X^3)$. Let $M$ be the ideal of $A$ generated by $X$, $Y$, and $A_M$ be the localization of $A$ at $M$. Is $A_M$ a discrete valuation ring?...
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### Is $K[[X^2,X^3]]$ a local ring where $K$ is field? [closed]

I came a cross with this question "Is $K[[X^2,X^3]]$ a local ring where $K$ is field?" This is a ring of formal power series in which the terms are generated by $X^2$ and $X^3$ i.e. This ...
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Let $A=\oplus_{i \in N \cup \{0\}} A_i$ be a positively graded ring of dimension $d$ with $A_0=k$ and $k$ is a field. If $B$ is a Noetherian graded subring of $A$. Can we say dimension of $B$ cannot ...
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### Module over local ring which is free after cutting down by a non-zero-divisor

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M\neq \mathfrak m M$. Let $x\in \mathfrak m$ be a non-zero-divisor on both $R$ and $M$. If $M/xM$ is a ...
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### Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?

Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
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### How to understand a situation where one can use Nakayama's lemma even when the situation is not Tailor-made.

In commutative algebra we have the following version of Nakayama's Lemma(also calle NAK lemma): NAK Lemma: Let $R$ be a local ring and $\mathbf m$ be the unique maximal ideal of $R$.Let $M$ be a ...
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### How to express elements of a DVR as power series of the uniformizer?

Let $R$ be a DVR with a fixed uniformizer $t$. My main considering example is the local ring of an algebraic curve over an algebraically closed field $k$, so let's assume $R$ is a $k$-algebra and has ...
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### Is a flat extension of DVRs always finite?

Is it true that if $S/\mathbb{Z}_p$ is a flat extension of discrete valuation rings then $S$ is finite over $\mathbb{Z}_p$?
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The question is from a Persian text book: Let $R$ be a commutative local ring with $1$, and $\mathfrak m$ its maximal ideal. If $M$ is a non-zero module over $R$ prove that $\text{Hom}_R(M,R/\mathfrak ... • 124 3 votes 1 answer 85 views ### Serre conditions and local isomorphisms I encountered the following in the introduction of the paper "Duality for Koszul Homology over Gorenstein Rings": I assume it's easy, but why is this fact true? I played with modding out by ... • 1,441 -2 votes 1 answer 98 views ### How can I show that if$x$is a nonunit then$1-x$is a unit? [duplicate] Let$R$be a commutative local ring. I want to show that then$x\in R$is not a unit implies$1-x$is a unit. My idea was the following: Since$R$is a local ring we can chose$\mathfrak{m}$to be ... • 299 1 vote 1 answer 67 views ### On pushforward functor and freeness of modules Let$R$be a commutative Noetherian ring. Given an$R$-module$M$and a ring homomorphism$\phi:R\to R$, let$\phi^* M$be the$R$-module whose underlying abelian group is the same as$M$but the$R$-... • 433 1 vote 1 answer 63 views ### Equivalent definitions of a local algebra Let$A$be a unital, associative commutative algebra over a field$k$. In particular,$A$is a ring and we call$A$local if it is local as a ring. Namely, call$A$local if$A$has a unique maximal ... • 1,769 0 votes 0 answers 91 views ### Betti numbers of associated graded ring Let$(R,\mathfrak{m})$be a (not necessarily commutative) local ring with commutative residue field$k$. Define it's betti numbers as $$\beta_i(R):=\dim_k\mathrm{Tor}^i_R(k,k).$$ The associated graded ... • 1,220 3 votes 0 answers 109 views ### Is the ring of formal power series a localization of some non local ring at prime ideals? Consider$F = K[[x]],$the formal series over field$K.$We know that$K$is a local ring with maximal ideal$(x).$Does there exist a non local ring$R$and prime ideal$p$such that$R_p = K[[x]]?$... • 751 1 vote 1 answer 51 views ### Do we have$\bigcap_{1\leq i\leq r}(\mathfrak{m}^n+\mathfrak{p}_i)=\mathfrak{m}^n$in a local ring, where$\mathfrak{p_i}$are the minimal primes?$\newcommand{\cO}{\mathcal{O}}\newcommand{\fm}{\mathfrak{m}}\newcommand{\fp}{\mathfrak{p}}$Let$(\cO,\fm)$be a Noetherian reduced local ring, essentially of finite type over an algebraically closed ... • 2,714 0 votes 0 answers 29 views ### On Elkik's Theorem of Henselian pair. As far as I know, the celebrated Elkik's result often quoted boil down as follows$\colon$Let$A$be a heneselian local ring with its maximal ideal${\frak m}$and choose an arbitrary ideal$I \subset ...
Let $(K,v)$ be an algebraically closed complete valuation field. Let $(R,v|_R)$ be a valuation subring of $K$. Denote by $F^{\operatorname{sep}}$ the separable closure of $\operatorname{Frac}R$ in $K$....
Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of ...