[For the entire collection see Zbl 0677.00008.]
A finitely generated left or right module M over a left and right Noetherian ring A is said to satisfy the Auslander condition if for every integer v and every submodule N of \(Ext^ v_ A(M,A)\), it follows that \(Ext^ i_ A(N,A)=0\) when \(i<v\). An Auslander-Gorenstein ring is a left and right noetherian ring A with finite injective dimension such that every finitely generated A-module satisfies the Auslander condition. A ring A with finite global dimension such that every finitely generated left or right A-module satisfies the Auslander condition is said to be Auslander regular.
Various properties of these rings (sometimes using different terminology) have been studied by Fossum, Bass, Reiten, Roos, and others. A large part of the present paper is expository, offering some detailed proofs, especially in connection with the bidualizing complex for modules over Auslander-Gorenstein rings, filtered rings, and pure modules. In particular he provides a proof of this result of Gabber: If M is a pure module over an Auslander-Gorenstein ring A and T is an A-module containing M such that every finitely generated submodule of T is pure, then T contains a unique largest tame and pure extension of M. This theorem is used to describe the characteristic ideal and the characteristic cycle of certain quotients of pure modules over filtered rings.