Common hyperbolic bases for chains of alternating or quadratic lattices

TL;DR

This paper provides a simple bilinear proof that chains of p-elementary lattices over p-adic integers or similar rings share common hyperbolic bases, aiding the study of Bruhat-Tits buildings.

Contribution

It introduces a purely bilinear proof of the existence of common hyperbolic bases for chains of p-elementary lattices, simplifying previous approaches.

Findings

Chains of p-elementary lattices have common hyperbolic bases.

The proof is purely bilinear and simpler than previous methods.

This result is useful for studying Bruhat-Tits buildings.

Abstract

We give a short and purely bilinear proof of the fact that two chains of $p$-elementary lattices with quadratic form or alternating bilinear form over the $p$-adic integers ore more generally over a complete discrete valuation ring have common hyperbolic bases. This fact, which is useful for the study of Bruhat-Tits buildings, has been proven before with different methods by Abramenko and Nebe and by Frisch.