On Lattices over Valuation Rings of Arbitrary Rank

TL;DR

This paper generalizes key results about p-adic lattices to lattices over valuation rings of arbitrary rank, establishing conditions for uniqueness and classification of indecomposable lattices.

Contribution

It extends classical lattice theory results to more general valuation rings, including cases where 2 is not invertible, with new classification invariants.

Findings

Uniqueness of Jordan decomposition when 2 is invertible.

Witt Cancellation Theorem holds under certain conditions.

Classification of indecomposable rank 2 lattices without invertibility of 2.

Abstract

We show how several results about p-adic lattices generalize easily to lattices over valuation ring of arbitrary rank having only the Henselian property for quadratic polynomial. If 2 is invertible we obtain the uniqueness of the Jordan decomposition and the Witt Cancelation Theorem. We show that the isomorphism classes of indecomposable rank 2 lattices over such a ring in which 2 is not invertible are characterized by two invariants, provided that the lattices contain a primitive norm divisible by 2 of maximal valuation.