Countably Generated Matrix Algebras

TL;DR

This paper introduces a new way to complete associative algebras in the context of modules, linking it to deformation theory and formal algebra constructions, with applications to noncommutative deformation functors.

Contribution

It defines a novel completion process for associative algebras in module sets, connecting it to GMMP-algebras and deformation theory, providing a unique prorepresenting hull.

Findings

Defines completion of associative algebras in modules

Links completion to GMMP-algebras and deformation functors

Establishes uniqueness of the formal algebra hull

Abstract

We define the completion of an associative algebra $A$ in a set $M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module $A/\mathfrak a$ is $\hat A^M\simeq \hat A^{\mathfrak a}.$ This works by defining $\hat A^M$ as a formal algebra determined up to a computation in a category called GMMP-algebras. From deformation theory we get that the computation results in a formal algebra which is the prorepresenting hull of the noncommutative deformation functor, and this hull is unique up to isomorphism.