No abstract

We present a new way of forming a grothendieck group with respect to exact sequences. A ‘defect’ is attached to each non-split sequence and the relation that would normally be derived from a collection of exact sequences is only effective if the (signed) sum of the corresponding defects is zero. The theory of the localization exact sequence and, in particular, of the relative group in this sequence is developed. The (‘locally free’) class group of a module category with exactness defect is defined and an idèlic formula for this is given. The role of torsion and of torsion-free modules is investigated. One aim of the work is to enhance the locally trivial, ‘class group’, invariants obtainable for a module while keeping to a minimum the local obstructions to the definition of such invariants.

No abstract

This paper generalises Chinburg's construction [4, 5] of invariants in the class group of an integral group ring from two‐fold extensions of (Galois) modules. The two main results are the expression of invariants of endomorphisms of a non‐projective lattice over an order (which lie in the kernel group) in terms of reduced norms of local automorphisms, and the description of a coset of the Swan subgroup of the class group, which contains Chinburg's invariant Ω(N/K, 1) of a finite Galois extension N/K of number fields, in terms of invariants of homomorphisms.