Weak homological equivalence, canonical factorisability and chinburg's invariants

Abstract

Let N/K be a Galois extension of number fields with Galois group T. T.Chinburg has constructed invariants of the extension N/K lying in the locally free classgroup Cl(ZӶ). In the first chapter we generalise this construction by defining weak homological equivalences and their projective invariants over any Noetherian ring A.In the case where A is an order in a semisimple algebra, we obtain for each A-latticeM an effectively computable subgroup Δ(M) of the kernel group D(A). Specialising tothe case A = ZT we relate Δ subgroups to generalised Swan subgroups and we describe a representative of the coset of the Swan subgroup T(ZӶ) containing Chinburg's invariant Ω(N/K, 1) in terms of the projective invariant of a homomorphism. In the second chapter we generalise A. Frohlich's canonical factorisability from abelian to arbitrary finite groups. We obtain a canonical factorisation function - related to the ring of integers O(_N) - which determines a unique coset of Cl(ZӶ) / D(ZӶ) equal to the coset generated by Chinburg's invariant Ω(N/K, 2). Thus we establish "modulo D(ZӶ)" a conjecture of Chinburg