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Artin ring

Definition

A ring $R$ is said to be an Artinian ring if it satisfies the descending chain condition on ideals.

Depending whether this condition is satisfied by left ideals, right ideals or two-sided ideals, one speaks of left Artinian, right Artinian, or two-sided Artinian rings, respectively. Clearly, in the context of commutative algebra all these notions coincide.

A local Artin algebra with residue field $𝕂$ is a finitely generated $𝕂$-algebra $A$, which is a commutative Artin ring (with unit) with a unique maximal ideal ${𝔪}_A$, such that the residue field $A/{𝔪}_A$ is $𝕂$. As a $𝕂$ vector space one has a splitting $A=𝕂\oplus {𝔪}_A$. Moreover, the descending chain condition implies that $({𝔪}_A{)}^n=0$ for some $n>>0$. This is a consequence of Nakayama lemma?.

Examples

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

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

Note that, since a local Artin algebra has a unique prime ideal, its spectrum consists of a single point, i.e., $\mathrm{Spec}(A)$ 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 Artin algebras are occasionally called fat points in the literature.

References

Local Artin $\mathrm{\infty }$-algebras are discussed in

• Jacob Lurie, Formal moduli problems (pdf)