Syzygy Computations and the D(2)-Problem for the Metacyclic Group G(p,3)

Abstract

Let $p=3d+1$ be prime, let $G(p,3)=C_p\rtimes C_3$, and put $\Lambda=\mathbb{Z}[G(p,3)]$. We study tensor products of $\Lambda$-lattices representing syzygies and generalised syzygies of the trivial module. For $i=1,\,2,\,3$, we prove that $K(3)\otimes R(i)\cong R(i)\oplus\Lambda^{2d}$ and $K(3)\otimes K(i)\cong K(i)\oplus\Lambda^{2(p-d)}$. Thus, $K(3)$ acts as an identity on the corresponding stable classes under tensor product. We then construct explicit lattices $L$ and $Y$, representing dual stable classes associated with $K(1)$ and $K(2)$, and determine decompositions of several tensor products involving $R(1)$, $L$, and $Y$. As an application, we show that the stable isomorphism $L\sim Y^{\ast}$ is sufficient for $G(p,3)$ to have Wall's $\mathcal{D}(2)$-property and, under the same hypothesis, construct an associated six-periodic free resolution. Finally, we show that Johnson's ideal class injectivity condition implies $L\sim Y^{\ast}$, and formulate the remaining stable-isomorphism problem as a concrete integral-representation calculation.