Whitehead Groups of Localizations and the Endomorphism Class Group

We compute the Whitehead groups of the associative rings in a class which includes (twisted) formal power series rings and the augmentation localizations of group rings and polynomial rings. For any associative ring A, we obtain an invariant of a pair (P,\alpha) where P is a finitely generated projective A-module and \alpha:P \to P is an endomorphism. This invariant determines (P,\alpha) up to extensions, yielding a computation of the (reduced) endomorphism class group of A. We also refine the analysis by Pajitnov and Ranicki of the Whitehead group of the Novikov ring.