Distributive lattices and Auslander regular algebras

Abstract

Let L denote a finite lattice with at least two points and let A denote the incidence K-algebra of L over a field K. We prove that L is distributive if and only if A is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, A has an explicit minimal injective coresolution, whose i-th term is given by the elements of L covered by precisely i elements. We give a combinatorial formula of the Bass numbers of A. We apply our results to show that the order dimension of a distributive lattice L coincides with the global dimension of the incidence algebra of L. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.