nLab Artinian local ring

Contents

Context

Algebra

higher algebra

universal algebra

Contents

Definition

A Artinian local ring or local Artinian ring or local Artin ring or Artin local ring or Weil ring is a local ring which satisfies one of the equivalent conditions

These are also called Artinian local algebras, local Artinian algebras, Artin local algebras, or local Artin algebras or Weil algebras, since Artinian local rings are commutative algebras over the residue field $\mathbb{K}$.

In synthetic differential geometry, the term Weil algebra is used for real Artinian local algebras, see infinitesimally thickened point for more details.

Properties

Every Artinian local $\mathbb{K}$-algebra $A$ has a maximal ideal $\mathfrak{m}_A$, whose residue field $A / \mathfrak{m}_A$ is $\mathbb{K}$ itself. As a $\mathbb{K}$ vector space one has a splitting $A=\mathbb{K}\oplus \mathfrak{m}_A$. Moreover, the descending chain condition implies that $(\mathfrak{m}_A)^n=0$ for some $n\gg 0$, a consequence of Nakayama lemma. This implies that the maximal ideal is a nilradical.

Given a field $K$ and a Artinian local $K$-algebra $A$, let $I$ be the nilradical of $A$. There is a function $v:I \to \mathbb{N}$ which takes a nilpotent element $r \in I$ to the least natural number $n$ such that $r^{v(r) + 1} = 0$, such that given nilpotent elements $r \in I$ and $s \in I$ and non-zero scalars $a \in K$ and $b \in K$,

• $v(a r + b s) = v(r) + v(s)$
• $v(r s) = \min(v(r), v(s))$

Spectra

Passing from commutative rings to their spectra (in the sense of algebraic geometry), Artinian local algebras correspond to infinitesimal pointed spaces. As such, they appear as bases of deformations in infinitesimal deformation theory. For instance $Spec(\mathbb{K}[\epsilon]/(\epsilon^2))$ is the base space for 1-dimensional first order deformations. Similarly, $Spec(\mathbb{K}[\epsilon]/(\epsilon^{n+1}))$ is the base space for 1-dimensional $n$-th order deformations.

An Artinian local algebra has a unique prime ideal, which means that its spectrum consists of a single point, i.e., $Spec(A)$ is trivial as a topological space. It is however non-trivial as a ringed space, since its ring of functions is $A$. By this reason spectra of Artinian local algebras are occasionally called fat points in the literature.

Examples

• A classical example is the ring of dual numbers $\mathbb{K}[\epsilon]/(\epsilon^2)$ over a field $\mathbb{K}$.

• Every prime power local ring is a local Artinian ring.

Local Artinian $\infty$-algebras are discussed in