AI 中文总结
研究亚循环群\(G(p,3)\)的合冲计算，通过\(-\otimes_\mathbb{Z}-\)讨论合冲相互作用，明确其与Wall的\(\mathcal{D}(2)\)问题的关系。
AI 中文摘要
设\(p = 3d + 1\)为素数，其中\((\mathbb{Z}/p\mathbb{Z})^\times=\langle 2,\,3\rangle\)，令\(\Lambda=\mathbb{Z}[C_p\rtimes C_3]\)为阶为\(3p\)的亚循环群\(G(p,\,3)\)的整群环。合冲\(\Omega_r(\mathbb{Z})\)是平凡模的自由\(\Lambda\)-分解中中间模的稳定类。我们明确讨论了\(-\otimes_\mathbb{Z}-\)下合冲的相互作用，并将其与Wall的\(\mathcal{D}(2)\)问题相关联。
英文摘要
Let $p=3d+1$ be prime, let $G(p,3)=C_p\rtimes C_3$, and put $Λ=\mathbb{Z}[G(p,3)]$. We study tensor products of $Λ$-lattices representing syzygies and generalised syzygies of the trivial module. For $i=1,\,2,\,3$, we prove that $K(3)\otimes R(i)\cong R(i)\oplusΛ^{2d}$ and $K(3)\otimes K(i)\cong K(i)\oplusΛ^{2(p-d)}$. 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.
CommentsWe have discovered an improvement upon the original argument which now works for all primes of the form 3d+1. This is a substantial improvement upon the original
