  Let $R$ be a left noetherian ring, $S$ a right noetherian ring and $_RU$ a
generalized tilting module with $S={\rm End}(_RU)$. The injective dimensions of
$_RU$ and $U_S$ are identical provided both of them are finite. Under the
assumption that the injective dimensions of $_RU$ and $U_S$ are finite, we
describe when the subcategory $\{{\rm Ext}_S^n(N, U)|N$ is a finitely generated
right $S$-module$\}$ is closed under submodules. 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.

Rings and Algebras;Representation Theory