Stabilization of the Witt Group

In this Note, using an idea due to Thomason [8], we define a “homology theory” on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find Balmers’s higher Witt groups. For more general rings, this homology is isomorphic to the KT-theory of Hornbostel [3], inspired by the work of Williams [9]. For real or complex C*-algebras, we recover up to 2 torsion topological K-theory. 1. Let A be a ring with an antiinvolution a € a and let ε be an element of the center of A such that ε ε = 1. We assume also that 2 is invertible in the ring. There are now well known definitions of the higher hermitian K-group (denoted by εLn(A), as in [5]) and the higher Witt group εWn(A) : this is the cokernel of the map induced by the hyperbolic functor Kn(A) zzc εLn(A) where the Kn(A) denote the Quillen K-group (which is defined for all values of n [ Z). One of the fundamental results of higher Witt theory is the periodicity isomorphism (where Z’ = Z[1/2], cf.[4]) εWn(A)* Z’ – εWn-2(A) * Z’ It is induced by the cup-product with a genuine element u2 [ -1L-2(Z’). By analogy with algebraic topology, we shall call u2 the Bott element in Witt theory. This element is explicitly described in the following way. We consider the 2 x 2 matrix (with the involution defined by z = z-1 and t = t-1 and where we put λ = λ = 1/2). M =    b(t+t -1 -2) a + (1-a)t -(1-d+dt -1 ) -c    where    a b c d    = u p0 u-1 with p0 =    1 0 0 0    and u =    λz + λ λλ(z 1) z 1 λz + λ    This 2 x 2 matrix represents an element of -1L0(Z’[t, t-1, z, z-1]) whose image in -1L-2(Z’) – Z ⊕ Z/2 is a free generator (cf. [5] for the details). 2. The higher Witt groups εWn(A) do not have all the nice formal properties one should expect. For instance, a cartesian square of rings with antiinvolutions (where the vertical maps are surjective) A zzc A1