  Let $L$ denote a finite lattice with at least two points and let $A$ denote
the incidence algebra of $L$. 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.
