The splitting theorem of Bass, Heller and Swan [8] and Bass [7] for the torsion group of the Laurent polynomial extension A[z, z–1] of a ring A

will now be generalized to the torsion group of the finite Laurent extension A[z, z–1] of a filtered additive category A

The proof makes use the Mayer-Vietoris presentations of §8 and the nilpotent objects of §9 to obtain a split exact sequence

and the analogue with K1iso replace by K1

as well as a version for the Whitehead torsion and reduced class groups

Given an object L in A and j ∈ Z define objects in G1(A)

The projections onto ζjL+ and ζjL– define morphisms in G1(A)

definition 10.1 The split surjection

sends the torsion of an isomorphism in A[z, z–1]

with inverse

to the end invariant

and the reduced nilpotent classes

of the objects (P±, v±) in Nil(P0(A)) given by

There are two distinct ways of splitting K1iso (A[z, z–1]), as the algebraically significant direct sum system

and the geometrically significant direct sum system

Similarly for K1 instead of K1iso. The algebraically significant splitting j! of i! is induced by the functor

definition 10.2 (i) The algebraically significant injection

is the splitting of B ⊕ N+ ⊕ N– with components