Syzygies and Diagonal Resolutions for Dihedral Groups

Abstract

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ω_r(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” H^r(G, N) = Hom_𝒟er(Ω_r(Z), N). We describe the Ω_r(Z) explicitly for the dihedral groups D_4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.

Key Words:

• Diagonal resolution
• Dihedral group
• Free resolution
• Syzygy

2010 Mathematics Subject Classification:

• Primary: 16E05
• 20C10
• Secondary: 18E30