nLab pseudocompact ring

Definition

A pseudocompact ring is a complete Hausdorff topological ring, $R$, which admits a base at $0$ of two-sided open ideals $I$ for each of which the quotient ring $R/I$ is Artinian. Equivalently, it is a regular pro-object in the category of Artinian rings.

(For the definition of regular pro-object, see the page 327 of KG71.)

More generally let $R$ be a commutative pseudocompact ring. A complete Hausdorff topological ring $\Lambda$ will be called a pseudocompact algebra over $R$ if

(i) $\Lambda$ is an $R$ algebra in the usual sense, and

(ii) $\Lambda$ admits a system of open neighbourhoods of 0 consisting of two-sided ideals $I$ such that $\Lambda/I$ has finite length as an $R$ module.

Motivations and applications

1. Grothendieck developed the theory of formal groups over pseudocompact rings.

2. The completed group algebras of profinite groups are naturally pseudocompact algebras, usually considered over finite fields or the profinite completion of the ring of integers.

Related concepts

• linear topological ring

References

The homological algebra of such rings and the corresponding modules are discussed in

• A. Brumer, Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966) 442-470 [doi:10.1016/0021-8693(66)90034-2, pdf, MR202790]

The connection between pseudocompact rings and pro-objects is discussed in section 4.2 of:

• J. F. Kennison, Dion Gildenhuys: Equational completion, model-induced triples and pro-objects, Journal of Pure and Applied Algebra 1 4 (1971) 317-346 [doi:10.1016/0022-4049(71)90001-6, MR306289]

category: algebra