nLab
pseudocompact ring

A pseudocompact ring is a complete Hausdorff topological ring, RR, which admits a base at 00 of two-sided open ideals II for each of which R/IR/I is an Artinian ring. Equivalently, it is a regular pro-object in Artinian rings.

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

(i) Λ\Lambda is an RR algebra in the usual sense, and

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

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, MR202790, doi pdf

category: algebra

Revised on April 7, 2014 00:14:40 by Tim Porter (2.26.40.190)