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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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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 $\mathbb{F}_{p}$,…
The Thin Whistler
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For a scheme $X$, does the functor $R \mapsto \operatorname{Hom}(\operatorname{Spec} R,X)$ for local rings $R$ characterize $X$?

Let $\bf{Sch, Sets, Ring}$ be the categories of schemes, sets, and commutative rings, respectively. By Yoneda's lemma, a scheme $X$ is characterized by the contravariant functor $$\operatorname{Hom}(*, X): \bf{Sch}^{op}\to Sets$$ Now thinking …
Yos
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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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What exactly is sheafification?

I have recently learned about the very BASICS of sheaves, but I was wondering is there an easier definition for sheafification? I could not find anywhere an easier definition for sheafification. I kind of compare it to local rings, where in local…
combinatorialist46Carey2
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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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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.
Strongart
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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…
Diego
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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…
user456828
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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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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 true? "$P$ is torsion-free if and only if $P$ is…
mathlover314
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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 assume $R$ is normal? My thoughts: If $R$ is a…
feder
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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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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…
DobryZiom
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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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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…
E.R
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