Let R be a left and right Noetherian ring and n, k be any non-negative integers. R is said to satisfy the Auslander-type condition Gn（k） if the right fiat dimension of the （i ＋ 1）-th term in a minimal injective resolution of RR is at most i ＋ k for any 0 ≤ i ≤ n - 1. In this paper, we prove that R is Gn（k） if and only if so is a lower triangular matrix ring of any degree t over R.
We introduce and study the relative lett derive functor Torn（￡,￡1） category, which unifies several related left derived functors. Then we give some criteria for computing the -resolution dimensions of modules in terms of the properties of Torn（￡,￡1） . We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.
