Semi-addivitity and acyclicity

TL;DR

This paper introduces an ordinal-valued invariant for finitely generated modules over Noetherian rings, demonstrating its semi-additivity and applying it to prove general acyclicity theorems.

Contribution

It generalizes the concept of length to an ordinal-valued invariant and establishes its semi-additivity, leading to new acyclicity results.

Findings

Invariant is semi-additive on short exact sequences

Proves general acyclicity theorems

Extends length concept to ordinal values

Abstract

We generalize the notion of length to an ordinal-valued invariant defined on the class of finitely generated modules over a Noetherian ring. A key property of this invariant is its semi-additivity on short exact sequences. As an application, we prove some general acyclicity theorems.