ON A DUALITY THEOREM OF WAKAMATSU

Let R be a left coherent ring, S a right coherent ring and RU a generalized tilting module, with S=End(RU) satisfying the condition that each finitely presented left R-module X with ExtRi(X,U)=0 for any i≥1 is U-torsionless. If M is a finitely presented left R-module such that ExtRi(M,U)=0 for any i≥0 with $i \neq n$ (where n is a nonnegative integer), then $\mathrm {Ext}_S^n( \mathrm {Ext}_R^n(M, U), U)\cong M$ and ExtSi(ExtRn(M,U),U)=0 for any i≥0 with $i\neq n$. A duality is thus induced between the category of finitely presented holonomic left R-modules and the category of finitely presented holonomic right S-modules.

PROPERLY STRATIFIED ALGEBRAS AND TILTING

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.