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.
Publié le : 1999-06-15
Classification:
11E39,
11E08,
11E70,
19G38
@article{1255985221,
author = {Fainsilber, Laura and Morales, Jorge},
title = {An injectivity result for Hermitian forms over local orders},
journal = {Illinois J. Math.},
volume = {43},
number = {3},
year = {1999},
pages = { 391-402},
language = {en},
url = {http://dml.mathdoc.fr/item/1255985221}
}
Fainsilber, Laura; Morales, Jorge. An injectivity result for Hermitian forms over local orders. Illinois J. Math., Tome 43 (1999) no. 3, pp. 391-402. http://gdmltest.u-ga.fr/item/1255985221/