Formal moduli of modules over local $k$-algebras

Abstract

We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local $k$-algebra $R$, not necessarily commutative, where $k$ is a field. The class of topological modules we consider include all those of finite rank over $k$ and some of infinite rank as well, namely those with a Schauder basis in the sense of $\S 1$. This generalizes the results of [Sh], where the result was obtained in a different way in case the ring $R$ is the completion of the local ring of a plane curve singularity and the module is ${k^n}$. Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called "growth functions" to handle explicit epsilonics involving the convergence of formal power series in noncommuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of [Sch] which were used in [Sh] and is more conceptual.