On Tate duality and a projective scalar property for symmetric algebras

TL;DR

This paper extends the characterization of Kn"orr lattices and certain arithmetic properties from finite group algebras to a broader class of symmetric algebras over complete discrete valuation rings, using explicit Tate duality descriptions.

Contribution

It identifies a new class of symmetric algebras over complete discrete valuation rings where known lattice characterizations and arithmetic properties apply, generalizing previous results.

Findings

Extension of Kn"orr lattice characterization to new algebra class

Explicit description of Tate duality for symmetric algebras

Arithmetic properties of finite group representations extend to this class

Abstract

We identify a class of symmetric algebras over a complete discrete valuation ring $\mathcal O$ of characteristic zero to which the characterisation of Kn\"orr lattices in terms of stable endomorphism rings in the case of finite group algebras, can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on an explicit description of Tate duality for lattices over symmetric $\mathcal O$-algebras whose extension to the quotient field of $\mathcal O$ is separable.