Pure minimal injective resolutions and perfect modules for lattices

TL;DR

This work analyzes when pure minimal injective coresolutions exist for incidence algebras of distributive lattices and related finite‑dimensional algebras, tying homological purity to lattice structure. It proves that for Auslander regular algebras this purity is equivalent to being right diagonal with all indecomposable injectives perfect, and specializes to $A=K L$ where this occurs exactly for $L$ the order‑ideals lattice of an upward‑linear poset (with the two‑sided variant characterizing divisor lattices). The authors classify distributive lattices with perfect injective modules via Boolean antichains, yielding precise combinatorial criteria and a full classification in terms of upward‑linear posets, plus a computational appendix using QPA for verification. They also construct Auslander–Gorenstein polynomial identity incidence algebras that nevertheless lack a pure minimal injective coresolution, answering a question of Ajitabh–Smith–Zhang in the negative.

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.

Figures (1)

• Figure 1: From left to right, $P$ and $\mathcal{O}(P)$

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.6
• Theorem 2.1: Birkhoff's representation theorem
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• proof