Adically Finite Chain Complexes

TL;DR

This paper explores the properties of adic finiteness in chain complexes over noetherian rings, extending known isomorphisms and homological dimensions, and examines how adic finiteness transfers along ring homomorphisms.

Contribution

It extends isomorphism properties and characterizations of homological dimensions from homologically finite to adically finite complexes, and studies transfer of adic finiteness.

Findings

Extended isomorphism properties to adically finite complexes

Characterized homological dimensions for adic finiteness

Analyzed transfer of adic finiteness along ring homomorphisms

Abstract

We investigate the similarities between adic finiteness and homological finiteness for chain complexes over a commutative noetherian ring. In particular, we extend the isomorphism properties of certain natural morphisms from homologically finite complexes to adically finite complexes. We do the same for characterizations of certain homological dimensions. In addition, we study transfer of adic finiteness along ring homomorphisms, all with a view toward subsequent applications.