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

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.