Minimal free resolutions of lattices over finite groups
Let G be a finite group and let M be a \({\mathbb{Z}}G\)-lattice. The first author [Proc. Lond. Math. Soc., III. Ser. 48, 91-107 (1984; Zbl 0507.20004)] has proved that two minimal projective resolutions of M are in the same genus (as complexes). The main object of this article is the analogous question with respect to minimal free resolutions of M. It is shown that in general two minimal free resolutions have different rank sequences. However, the following result holds. Theorem. If \({\mathbb{Z}}G\) allows cancellation (this is e.g. the case, if the Eichler condition is present), then all minimal free resolutions of a \({\mathbb{Z}}G\)- lattice M belong to one genus class. The final part of the paper investigates the difference between minimal projective and minimal free resolutions. A short description (in the authors' own words) is that this difference is concentrated in a finite interval. Clearly the results are based on cancellation properties, generation properties (Swan modules) and the usual techniques of homological algebra. However, the authors present some interesting refinements of these methods, which are also certainly of interest with respect to other questions of integral representations of finite groups.
scientific article; zbMATH DE number 4116786