Weighted homological regularities

TL;DR

This paper introduces weighted homological invariants for graded algebras, unifying and extending classical regularity concepts, and provides identities that connect these invariants in novel ways.

Contribution

It develops weighted versions of key homological invariants for graded algebras, enabling finite measures where classical invariants may be infinite, and unifies various homological identities.

Findings

Weighted invariants generalize classical regularities.

Some infinite invariants become finite under weighting.

New homological identities are established.

Abstract

Let $A$ be a noetherian connected graded algebra. We introduce and study homological invariants that are weighted sums of the homological and internal degrees of cochain complexes of graded $A$-modules, providing weighted versions of Castelnuovo--Mumford regularity, Tor-regularity, Artin--Schelter regularity, and concavity. In some cases an invariant (such as Tor-regularity) that is infinite can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. We prove a few weighted homological identities some of which unify different classical homological identities and produce interesting new ones.