Ordinal length and the canonical topology

TL;DR

This paper introduces an ordinal-valued invariant for finite-dimensional Noetherian modules, linking their submodule lattice to homological properties, and defines a canonical topology where all morphisms are continuous.

Contribution

It extends the classical length function to an ordinal-valued invariant and establishes a canonical topology on modules ensuring morphism continuity.

Findings

Ordinal length can be computed via the fundamental cycle.

The canonical topology makes all module morphisms continuous.

Links lattice structure to homological invariants.

Abstract

We extend the classical length function to an ordinal-valued invariant on the class of all finite-dimensional Noetherian modules. We show how to calculate this combinatorial invariant by means of the fundamental cycle of the module, thus linking the lattice of submodules to homological properties of the module. Using this, we define on a module its canonical topology, in which every morphism is continuous.