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.

A ring $R$ is a local ring if it has any one of the following equivalent properties:

(1) $R$ has a unique maximal left ideal. (2) $R$ has a unique maximal right ideal. (3) $1 \neq 0$ and the sum of any two non-units in $R$ is a non-unit. (4) $1\neq 0$ and if $x$ is any element of $R$, then $x$ or $1- x$ is a unit. (5) If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies $1\neq 0 $).

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical.

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Is there a non-regular depth 1 noetherian local ring with this property?

Let $(R, \mathfrak{m})$ be a non-regular depth 1 noetherian local ring. Then if $x$ is any regular element of $R$ the module $R/(x)$ will have depth 0, and so it has $\mathfrak{m}$ as an associated prime. This means that there is an element $y\in R$…
Brendan Murphy
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Does the category of local rings with residue field $F$ have an initial object?

Let $F$ be a field. Does the category $C_F$ of local rings with residue field isomorphic to $F$ have an initial object? This is, for instance, true if $F=\mathbb{F}_{p}$ for some prime $p$: If $R$ is a local ring with residue field $\mathbb{F}_{p}$,…
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For local ring $R$, does funcotor $\operatorname{Hom( Spec}R, X)$ characterize scheme $X$?

Let $\bf{Sch, Sets, Ring}$ be a category of schemes, sets, commutative rings. By Yoneda's lemma, scheme $X$ is characterized by contravariant functor $$\operatorname{Hom}(*, X): \bf{Sch}^{op}\to Sets$$ Now thinking glueing of schemes by affine…
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Compute the Hilbert-Samuel function

Let $f\in R=k[x,y,z]_{(x,y,z)}$ be a homogeneous polynomial of degree $d$, monic in $x$. Show that $(y,z)$ is an ideal of finite colength on $M=R/(f)$. Compute the corresponding Hilbert-Samuel function. Maybe someone can do it as an example.
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$GL_2(R)$ action on binary cubic forms

Suppose that $R$ is a local ring. Then $GL_2(R)$ acts on the space of binary cubic forms $$p(x,y)=ax^3+bx^2y+cxy^2+dy^3, \quad a,b,c,d\in R,$$ by $$g\cdot p(x,y)=p((x,y)g).$$ My question is how to show that the action is faithful. I'm reading the…
Eclipse Sun
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Krull dimension of Noetherian local rings is finite

Does anyone know an "elementary" proof of the fact that a Noetherian local ring has finite Krull dimension? The one I know is from Atiyah&Macdonald's book Introduction to Commutative Algebra, where they use Hilbert functions (which is not an…
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Local ring with finite maximal ideal is finite

Let $(R, m)$ be a commutative local ring which is not a field such that $m$ is finite. Then is it true that $R$ is finite ? I can see that $R$ has finitely many ideals and all proper ideals are finite; so in particular $R$ is Artinian. Moreover…
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Why are local rings called local?

I gather that rings of germs of functions at a point $p$ on a manifold/variety/etc. are local with the maximal ideal containing exactly the germs of functions which vanish at $p$. So in some sense, these rings, which happen to be local, describe the…
Vercassivelaunos
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Why does a finite module over a Noetherian local ring supported only at the maximal ideal have the residue field as a submodule and a quotient?

I am reading the book “Fourier-Mulkai transforms in algebraic geometry” by Daniel Huybrechts. In the proof of Lemma 4.5, in page 92, it is written that if $M$ is a finite module over a local noetherian ring $(A,m)$ with…
Pouya Layeghi
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Let $A$ be an integrally closed noetherian domain, and $(R, \mathfrak{m})$ local with $A \subseteq R \subseteq K(A)$. Is $R$ a localization of $A$?

Let $K(A)$ denote the fraction field of $A$. For context, I'm trying to prove $A = \bigcap_{\text{ht}(\mathfrak{p}) = 1} A_{\mathfrak{p}}$ for an integrally closed domain $A$, from Atiyah-Macdonald's corollary 5.22 which states that $A$ is the…
Daniel
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Finite dimensional local rings with infinitely many minimal prime ideals

Is there a finite dimensional local ring with infinitely many minimal prime ideals? Equivalent formulation: Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of minimal prime sub-ideals of $\mathfrak p$ is…
Pierre-Yves Gaillard
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Precise general statement of flat morphisms being "equidimensional"

I am given to understand that the main property of flat morphisms is that they give some precise notion of "continuously varying family of fibres". I realise there is a fair bit of literature on this, but I really wanted to nail down what exactly is…
Luke
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A characterization for semilocal rings

A commutative ring with 1 is called semi-local if it has finitely many maximal ideals and is called local if it has only one maximal ideal. There are some algebraic charactrizations for local rings. For example a ring $R$ is local ring if and only…
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The localization of a ring at a maximal ideal

I am working on the following problem: If $R$ is a local ring whose maximal ideal is denoted $\mathfrak{p}$ then show that $R \cong R_\mathfrak{p}$. $R_\mathfrak{p} := \{\frac{r}{u} : r\in R, u\in R\setminus\mathfrak{p}\}/\sim$ where $\frac{r}{u}…
2dalimit2015
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derivations of the ring of germs of $C^{\infty}$ functions

Let $\mathcal{O}_{\mathbb{R},0}$ be the ring of germs of $C^{\infty}$ funcitons on the real line. A derivation of $\mathcal{O}_{\mathbb{R},0}$ is a $\mathbb{R}$-linear map $\partial:\mathcal{O}_{\mathbb{R},0}\to\mathcal{O}_{\mathbb{R},0}$ that…
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