Hermitian forms and higher algebraic 𝐾-theory

Abstract

Introduction. This note promulgates certain invariants for Hermitian forms over a ring with antistructure which are defined utilizing the algebraic ^-theory of the ring in a Z2-equivariant fashion. Low degree cases of these invariants extend the classical rank, discriminant, and Hasse-Witt invariants. The principal result described here is that these invariants (using all degrees) yield the stable classification of forms over many fields and algebras for which low degree cases alone do not usually suffice (because of signature invariants that may be defined). The manner in which this is accomplished is quite amusing, in that it sheds light on why the reductions of the signatures modulo certain powers of 2 are in fact determined by the low degree invariants. More general results can be obtained, e.g. for higher unitary algebraic A'-theory. However, these are suppressed here in favor of an outline of the equivariant algebraic Ktheory upon which this work so crucially relies. This will be dealt with at length in [Gi2], and arithmetic aspects will be pursued further elsewhere. 1. Hermitian forms. If R is a ring and MR is the category of right R-modules, then PR will denote the full, exact subcategory of M# determined by the finitely generated projective modules, and KnR, n E Z, will denote the (GerstenWagoner) algebraic K-theory groups of R (cf. [Ge] ). Recall that an antistructure (a, u) on R consists of an antiautomorphism r »—* r of R and a unit uGR such that uu = 1 and r* = unT, rER. An Hermitian form over (R, a, w), or an (a, u)-form over R, is a pair (P, A) with P E ?R and X: P x P —+ R such that \(x, y 4zf) = X(x, y) + X(x, z)r, \(y, x) = \(x, y)u for xf y, z E P, r E R, and (Ad X): P —> Da? is an isomorphism where (Ad XX*)00 = X(JC, y), x, y E R. Here, Da: VR —* VR is the a-conjugate duality cofunctor Dal = Hom/2(?, R) where HomR(P, R) has right R-module structure (f° r)(x) = r /(x), x E P, r E R, ƒ E Z)aP. There are evident notions of isometry and orthogonal direct sum of (a, w)-forms over R (cf. [Wa] ), resulting in a permutative category U(R, a, u) and (Gersten-Wagoner type) unitary algebraic K-theory groups KUn(R, a, w), n E Z, of (R, a, w). Of primary interest here is KU0(R, a, u) which is just the Grothendieck group of U(R, a, u).