Linear Algebra over Z_p[[u]] and related rings

TL;DR

This paper develops efficient methods for linear algebra operations over rings of formal power series with coefficients in a complete discrete valuation ring, addressing challenges due to non-principal modules and introducing approximation techniques.

Contribution

It introduces algorithms for sum and intersection of modules over S=R[[u]], using quasi-isomorphisms to handle non-principal modules, with applications in Iwasawa and p-adic Hodge theories.

Findings

Efficient computation of module operations over S=R[[u]]

Approximation techniques mitigate non-principality issues

Applications in Iwasawa theory and p-adic Hodge theory

Abstract

Let R be a complete discrete valuation ring, S=R[[u]] and n a positive integer. The aim of this paper is to explain how to compute efficiently usual operations such as sum and intersection of sub-S-modules of S^d. As S is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting of these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the p-adic and u-adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with S-modules has applications in Iwasawa theory and p-adic Hodge theory.