Wakamatsu Tilting Modules, $U$-Dominant Dimension and $k$-Gorenstein Modules

Abstract

Let $\Lambda$ and $\Gamma$ be left and right noetherian rings and ${}_{\Lambda }U$ a Wakamatsu tilting module with $\Gamma =\mathrm{End}{(}_{\Lambda }T)$. We introduce a new definition of $U$-dominant dimensions and show that the $U$-dominant dimensions of ${}_{\Lambda }U$ and $U_{\Gamma }$ are identical. We characterize $k$-Gorenstein modules in terms of homological dimensions and the property of double homological functors preserving monomorphisms. We also study a generalization of $k$-Gorenstein modules, and characterize it in terms of some similar properties of $k$-Gorenstein modules.

24 pages. Lect. Notes in Pure and Applied Math. 249, 2006. Some misprints in Theorems 5.2 and 5.4 are corrected (1. In the statement 5.2(2) and the statement 5.4(5): "torsionless" is replaced by "$U$-torsionless". 2. In p.18, line -12: "with $P$ projective" is replaced by "with $P$ in add${}_{\Lambda }U$".)