Syzygies and Diagonal Resolutions for Dihedral Groups

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ωr(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” Hr(G, N) = Hom𝒟er(Ωr(Z), N). We describe the Ωr(Z) explicitly for the dihedral groups D4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.