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Artin ring

Context

Algebra

Formal geometry

Contents

Definition

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

Examples

A classical example is the ring of dual numbers 𝕂[ϵ]/(ϵ 2)\mathbb{K}[\epsilon]/(\epsilon^2) over a field 𝕂\mathbb{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 Spec(𝕂[ϵ]/(ϵ 2))Spec(\mathbb{K}[\epsilon]/(\epsilon^2)) is the base space for 1-dimensional first order deformations. Similarly, Spec(𝕂[ϵ]/(ϵ n+1))Spec(\mathbb{K}[\epsilon]/(\epsilon^{n+1})) is the base space for 1-dimensional nn-th order deformations.

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

References

Local Artin \infty-algebras are discussed in

Revised on April 7, 2014 08:02:42 by Urs Schreiber (88.128.80.110)