Generalized tilting modules with finite injective dimension

Let R be a left noetherian ring, S a right noetherian ring and a generalized tilting module with . The injective dimensions of and US are identical provided both of them are finite. Under the assumption that the injective dimensions of and US are finite, we describe when the subcategory is submodule-closed. As a consequence, we obtain a negative answer to a question posed by Auslander in 1969. Finally, some partial answers to Wakamatsu Tilting Conjecture are given.