On the structure of modules indexed by small categories

TL;DR

This paper studies modules over small categories, introducing a quasi-tame cover that approximates modules with stable local structures, and explores conditions for when this approximation is exact.

Contribution

It defines a quasi-tame cover for C-modules with stable local structures and characterizes when this cover precisely recovers the original module.

Findings

QTC(M) is a finite sum of quasi-blocks approximating M.

The associated graded local structure of QTC(M) is isomorphic to that of M.

The surjection QTC(M) -> M is an isomorphism iff the excess of M vanishes.

Abstract

Given a small category C, a C-module M is a functor from C to the category of finite-dimensional vector spaces over a field k. Associated to M is its local structure, given as a functor from C to the category of bi-closed multi-flags over k. When the local structure of M is stable (a condition satisfied whenever both the category C and the field k are finite), it determines a quasi-tame cover QTC(M) (a finite direct sum of quasi-blocks), indexed by the same category, for which the associated graded local structure is canonically isomorphic to that of M. QTC(M) represents the closest approximation to M by a quasi-tame module, and recovers M precisely when M itself is quasi-tame. In the case M has stable local structure and is equipped with an inner product compatible with that structure, there exists a C-module surjection QTC(M) -> M inducing the above-mentioned isomorphism on associated graded local structures. This map is an isomorphism iff the excess of M vanishes (where the excess numerically measures the failure of the local structure of M to be in general position).