Pure minimal injective resolutions and perfect modules for lattices

TL;DR

This paper explores the conditions under which distributive lattices have pure minimal injective coresolutions, linking lattice theory with homological algebra and classifying perfect modules in this context.

Contribution

It provides a classification of perfect antichain modules and characterizes when incidence algebras of distributive lattices admit pure minimal injective coresolutions.

Findings

Distributive lattices with pure minimal injective coresolutions are characterized.

A classification of perfect antichain modules is established.

Existence of Auslander-Gorenstein rings without pure injective coresolutions is demonstrated.

Abstract

In a recent article, Iyama and Marczinzik showed that a lattice is distributive if and only if the incidence algebra is Auslander regular, giving a new connection between homological algebra and lattice theory. In this article we study when a distributive lattice has a pure minimal injective coresolution, a notion first introduced and studied in a work of Ajitabh, Smith and Zhang. We will see that this problem naturally leads to studying when certain antichain modules are perfect modules. We give a classification of perfect antichain modules under the assumption that their canonical antichain resolution is minimal and use this to give a completion classification in lattice theoretic terms of incidence algebras of distributive lattices with pure minimal injective coresolution. We use our results to answer a question raised by Ajitabh, Smith and Zhang by showing that there exist Auslander-Gorenstein polynomial identity rings without a pure injective coresolution.