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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Formal moduli of modules over local $ k$-algebras


Authors: Allan Adler and Pradeep Shukla
Journal: Trans. Amer. Math. Soc. 337 (1993), 143-158
MSC: Primary 16S80; Secondary 11S20, 13D10, 14B10, 14B12, 14D20, 15A33, 16W60
DOI: https://doi.org/10.1090/S0002-9947-1993-1091702-6
MathSciNet review: 1091702
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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 $ \S1$. 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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1091702-6
Keywords: Formal moduli space, artin local ring, prorepresentable functor, Schauder basis, formal power series in noncommuting variables, free monoid, deformations of Galois representations, free topological $ A$-module, matrix of a continuous endomorphism, growth function, order of a matrix, admissible homomorphism, formal power series in the narrow sense, formal power series in the wide sense, universal deformation ring
Article copyright: © Copyright 1993 American Mathematical Society