Lattices over orders: finitely presented functors and preprojective partitions

Abstract:

Suppose $R$ is a commutative noetherian equidimensional Gorenstein ring and $\Lambda$ an $R$-algebra which is finitely generated as an $R$-module. A $\Lambda$-module $M$ is a lattice if ${M_{\underline {\underline p} }}$ is ${\Lambda _{\underline {\underline p} }}$-projective and ${\text {Ho}}{{\text {m}}_R}{(M,R)_{\underline {\underline p} }}$ is $\Lambda _{\underline {\underline p} }^{{\text {op}}}$-projective for all nonmaximal prime ideals $\underline {\underline p}$ in $R$. We assume that $\Lambda$ is an $R$-order in the sense that $\Lambda$ is a lattice when viewed as a $\Lambda$-module. The first main result is to show that simple contravariant functors from lattices to abelian groups are finitely presented. This is then applied to showing that if $R$ is also local and complete, then the category of lattices has a preprojective partition. This generalizes previous results of the authors in the cases $R$ is artinian or a discrete valuation ring.