Algorithms for determination of t-module structures on some extension groups

TL;DR

This paper develops a comprehensive algorithm to determine the t-module structure on certain extension groups, generalizing previous methods and introducing new concepts like τ-composition series for specific t-modules.

Contribution

It extends prior work by providing a complete algorithm for computing t-module structures on Ext^1 groups for modules with rank conditions, under new invertibility assumptions.

Findings

Algorithm applicable when the matrix at the highest τ-power in Φ_t is invertible.

Introduction of τ-composition series for the additive category of t-modules.

Generalization of previous methods to broader classes of t-modules.

Abstract

In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(\Phi,\Psi )$ where $\Phi $ and $\Psi$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on $\Ext^1(\Phi,\Psi )$ for \tm modules $\Phi $ and $\Psi$ such that $\rk \Phi > \rk \Psi.$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $\tau$ in $\Phi_t$ is invertible. We also introduce a notion of ${\tau}$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $\Phi $ and $\Psi$ our algorithm is also executable.