Invertible lattices

Esther Beneish University of Wisconsin-Parkside Ming-chang Kang National Taiwan University

Number Theory mathscidoc:1608.24001

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.
Integral representation, algebraic tori, permutation lattice
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@inproceedings{estherinvertible,
  title={Invertible lattices},
  author={Esther Beneish, and Ming-chang Kang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818172009063893249},
}
Esther Beneish, and Ming-chang Kang. Invertible lattices. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818172009063893249.
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