Lattices and cohomological Mackey functors for finite cyclic p-groups

For a finite cyclic p-group G and a discrete valuation domain R of characteristic 0 with maximal ideal pR the R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups (cf. Theorem A). As a consequence any R[G]-lattice can be presented by R[G]-permutation modules (cf. Theorem C). The proof of these results is based on a detailed analysis of the category of cohomological G-Mackey functors with values in the category of R-modules. It is shown that this category has global dimension 3 (cf. Theorem E). A crucial step in the proof of Theorem E is the fact that a gentle R-order category (with parameter p) has global dimension less than or equal to 2 (cf. Theorem D).