## Illinois Journal of Mathematics

### An injectivity result for Hermitian forms over local orders

#### Abstract

Let $\Lambda$ be a ring endowed with an involution $a \mapsto \tilde{a}$. We say that two units $a$ and $b$ of $\Lambda$ fixed under the involution are congruent if there exists an element $u \in \Lambda^{\times}$ such that $a = ub\tilde{u}$. We denote by $\mathcal{H}(\Lambda)$ the set of congruence classes. In this paper we consider the case where $\Lambda$ is an order with involution in a semisimple algebra $A$ over a local field and study the question of whether the natural map $\mathcal{H}(\Lambda) \rightarrow \mathcal{H}(\Lambda)$ induced by inclusion is injective. We give sufficient conditions on the order $\Lambda$ for this map to be injective and give applications to hermitian forms over group rings.

#### Article information

Source
Illinois J. Math., Volume 43, Issue 2 (1999), 391-402.

Dates
First available in Project Euclid: 19 October 2009

https://projecteuclid.org/euclid.ijm/1255985221

Digital Object Identifier
doi:10.1215/ijm/1255985221

Mathematical Reviews number (MathSciNet)
MR1703194

Zentralblatt MATH identifier
0939.11020

#### Citation

Fainsilber, Laura; Morales, Jorge. An injectivity result for Hermitian forms over local orders. Illinois J. Math. 43 (1999), no. 2, 391--402. doi:10.1215/ijm/1255985221. https://projecteuclid.org/euclid.ijm/1255985221