The relation of quadratic k-theory to hermitian k:-Theory

Abstract

It is known that the Hermitian algebraic K-theory of a ring with antistructure is the fixed point theory of a certain involution on the algebraic K-theory of the ground ring. It is shown here that the (unitary) algebraic K-theory of nonsingular split quadratic forms over a ring with antistructure (R,α,u) is the same as the algebraic K-theory of nonsingular Hermitian forms over a related ring with antistructure (S⟪v). Specifically, S = R = R[d]/(d2) is the ring of dual numbers of R, the conjugation β is the antiautomorphism of S given by β{r + r’d) = α(r) - α(r’)d, and the unit of symmetry v= u(l + d). This makes it possible to regard split quadratic algebraic K-theory of R as a fixed point theory of an involution on algebraic K-theory of R . On the level of Ko (Grothendieck) groups, quadratic and split quadratic algebraic K-theory are the same. This is applied to show that the Kervaire-Arf invariant of skew-quadratic forms over Z (here (α,u) = (id,-l)) becomes identified with a generalized Hasse-Witt...