An Injectivity Result for Hermitian Forms over Local Orders

Let Λ be a ring endowed with an involution a 7→ ã. We say that two units a and b of Λ fixed under the involution are congruent if there exists an element u ∈ Λ× such that a = ubũ. We denote by H(Λ) the set of congruence classes. In this paper we consider the case where Λ is an order with involution in a semisimple algebraA over a local field and study the question whether the natural map H(Λ)→ H(A) induced by inclusion is injective. We give sufficient conditions on the order Λ for this map to be injective and give applications to hermitian forms over group rings. Introduction and motivation Let R be a ring endowed with an involution ̃ : R → R (that is, an antiautomorphism of order 2). For a left R-module M we denote by M∗ the dual module HomR(M,R) with the left R-module structure given by (aφ)(m) = φ(m)ã, for all a ∈ R, φ ∈ HomR(M,R), m ∈ M . Let ∈ R be a fixed central element satisfying ̃ = 1, for example = ±1. A (unimodular) -hermitian form over R is a pair (M, h) consisting of a reflexive R-module M and an isomorphism of R-modules h : M → M∗ satisfying h∗ = h. The notion of isometry of -hermitian forms is defined in the obvious way. It is a natural question to ask for a classification of -hermitian forms over R. An obvious necessary condition for two forms (M1, h1) and (M1 , h2) to be isometric is that their underlying R-modules M1 and M2 be isomorphic. This leads us to fix an R-module M and consider the set of all -hermitian forms on M . Assuming that this set is not empty, we fix once and for all an -hermitian form h0 : M → M∗ and we equip the endomorphism ring Λ = EndR(M) with the involution given by f̃ = h−1 0 f h0. (1) A straightforward calculation shows that all the -hermitian forms on M are of the form h = h0a, with a ∈ Λ× satisfying ã = a, and that two such forms h = h0a and g = h0b are isometric if and only if there exists u ∈ Λ× such that uaũ = b. Note that this is a particular case of the so-called transfer to the endomorphism ring in hermitian categories (see [15, Chapter 7, Section 4] or [13]). The above construction motivates the introduction of the following equivalence relation for any ring Λ equipped with an involution .̃ a ∼ b ⇐⇒ there exists u ∈ Λ× such that uaũ = b The second–named author was supported by Louisiana Education Quality Support Fund grant No LEQSF(RF1995-97)-RD-A-40.