Homotopy classification and the generalized Swan homomorphism

摘要

In his Fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism (Z/|G|)* -> (K-0) over tilde (Z[G]) for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [181, 1211). We extend the definition so that, for suitable modules J over reasonably general rings Lambda, it takes the form S : Aut(Der)(J) -> (K-0) over tilde(Lambda); here Der is the quotient of the category of Lambda-homomorphisms obtained by setting 'projective = 0'. We then employ it to give an exact classification of homotopy classes of extensions 0 -> J -> F-n -> ... -> F-0 -> M -> 0 where each F-r is finitely generated Free.