Next: 3. Finding a regular Up: An algorithm for constructing Previous: 1. Introduction

# 2. The space of transformation matrices

Throughout this paper we assume R to be a finitely generated, local (or graded) k-algebra without zerodivisors and any (any graded) module M as given as cokernel of a map of free modules. The presentation matrix of this map is assumed to represent a minimal system of generators of its image and is denoted by A(M) or simply A.

The case of a graded module is handled analogously. Therefore, we restrict ourself to the local settings.

Let two modules M and M' be given by their representation matrices A and A'. Assume the modules to be isomorphic. There are two quadratic matrices U and V such that UAV=A'. By the above properties of the representations both matrices have to be invertible, and .

Now, let's look from the other side: Starting with two representation matrices A and A' a necessary condition for an isomorphism of the represented modules M and M' is an identical size of both matrices. Denote this size by . The sufficient condition for an isomorphism is again the existence of matrices and with UAV=A'. That means, we have to resolve the equation

 (1)

and to look for a pair (X0,Y0) of regular matrices among all solutions of (1).

First, we determine the module of all possible transformations . Let us consider the matrix A as map

just by multiplying with A from the right. Analogously, we set

to be the map induced by the multiplication with A' from the left. The map

given by is a well defined module homomorphism and its kernel is the module Tr(A,A') of possible transformations. Tr(A,A') can be computed in a single syzygy computation.

Next: 3. Finding a regular Up: An algorithm for constructing Previous: 1. Introduction
| ZCA Home | Reports |