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,α), where P is a finitely generated projective A-module and α :P→P is an endomorphism. This invariant determines (P,α) up to extensions, yielding a computation of the (reduced) endomorphism class group . We also refine the analysis by Pajitnov and Ranicki of the Whitehead group of the Novikov ring, a computation which Pajitnov has used in work on circle-valued Morse theory.