Factor Equivalence of Rings of Integers and Chinburg's Invariant in the Defect Class Group

Let Γ be a finite group. We introduce the factorizability defect , fd, defined on exact sequences in mod(ZΓ). Let ℒ be the subcategory of mod (ZΓ) of modules with finite projective dimension away from a finite set S of integer primes. We examine the defect class group ℒℓ(ℒ fd ) (the subgroup of locally trivial elements in the defect Grothendieck group, 0 (ℒ fd )) and show that it is isomorphic to the direct sum class group ofℒ. Let N/K be a Galois extension of algebraic number fields with group Γ which is tamely ramified outside S . We show that [ N ] — [ K Γ] lies in ℒℓ(ℒ fd ) and equals the image of Ω( N/K ,2), Chinburg's second invariant. We also show that if M and M ′ lie in ℒ and [ M ] — [ M ′] ∈ ℒℓ(ℒ fd ) then M is factor equivalent to M ′ in a very strong sense. In particular N is factor equivalent in this way to a free ZΓ‐module.