The Critical and Cocritical Degrees of a Totally Acyclic Complex over a Complete Intersection

TL;DR

This paper extends the concept of critical degree to totally acyclic complexes over complete intersection rings, introducing the critical and cocritical degrees and a new measure called the critical diameter.

Contribution

It generalizes the notion of critical degree from modules to complexes, providing dual analogues and a new invariant for totally acyclic complexes over complete intersections.

Findings

Defined critical and cocritical degrees for complexes

Introduced the critical diameter as a new measure

Extended the concept of critical degree to a broader class of complexes

Abstract

It is widely known that the minimal free resolution of a module over a complete intersection ring has nice patterns eventually arising in its Betti sequence. In 1997, Avramov, Gasharov, and Peeva defined the notion of critical degree for finitely generated modules, proving that this degree is finite whenever the module has finite CI-dimension. This paper extends the notion of critical degree via complete resolutions, thus defining the critical and cocritical degrees of an object in the category of totally acyclic complexes over a complete intersection ring of the form $R=Q/(f_1,\dots,f_c)$. In particular, we provide the appropriate dual analogue to critical degree which enables us to introduce a new measure for complexes, called the critical diameter.