Invertible Lattices

TL;DR

This paper generalizes a theorem on invertible lattices over Dedekind domains, providing a short proof and extending previous results about flabby and coflabby lattices related to finite groups.

Contribution

It extends Endo and Miyata's theorem from integers to Dedekind domains under specific conditions, and offers a simplified proof and partial generalization of related results.

Findings

All flabby and coflabby lattices are invertible if Sylow subgroups are cyclic.

Provides a shorter proof of a known theorem.

Partially generalizes a result by Torrecillas and Weigel.

Abstract

Theorem. Let $\pi$ be a finite group of order $n$, $R$ be a Dedekind domain satisfying that (i) $\fn{char}R=0$, (ii) every prime divisor of $n$ is not invertible in $R$, and (iii) $p$ is unramified in $R$ for any prime divisor $p$ of $n$. Then all the flabby (resp.\ coflabby) $R\pi$-lattices are invertible if and only if all the Sylow subgroups of $\pi$ are cyclic. The above theorem was proved by Endo and Miyata when $R=\bm{Z}$ \cite[Theorem 1.5]{EM}. As applications of this theorem, we give a short proof and a partial generalization of a result of Torrecillas and Weigel \cite[Theorem A]{TW}, which was proved using cohomological Mackey functors.