AI 中文总结
本文研究层级群上模复形的结构，证明无环投射或平坦复形的可缩性或纯无环性，并应用于完全分解、周期上同调及Gorenstein维数。
AI 中文摘要
设$\overline{\mathfrak{F}}$为从有限群类$\mathfrak{F}$出发，通过迭代应用Kropholler的运算${\scriptstyle{\bf LH}}$和Talelli的运算$\Phi$所得到的群类。我们描述了以交换环为系数的$\overline{\mathfrak{F}}$-群的群代数上投射模的无环复形的余核的结构。若$G$是一个$\overline{\mathfrak{F}}$-群，我们证明任意投射（相应地，平坦）$\mathbb{Q}G$-模的无环复形是可缩的（相应地，纯无环的）。若$G$是无挠的，同样的结论对于投射或平坦$\mathbb{Z}G$-模的无环复形也成立。类似的结果对于内射模的无环复形也有效。我们给出了一些应用，涉及具有完全分解的模、在若干步后具有周期上同调的群，以及群代数上模的Gorenstein同调维数与普通同调维数之间的关系。
英文摘要
Let $\overline{\mathfrak{F}}$ be the group class obtained from the class $\mathfrak{F}$ of finite groups by iterated applications of Kropholler's operation ${\scriptstyle{\bf LH}}$ and Talelli's operation $Φ$. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of $\overline{\mathfrak{F}}$-groups with coefficients in a commutative ring. If $G$ is an $\overline{\mathfrak{F}}$-group, we show that any acyclic complex of projective (respectively, flat) $\mathbb{Q}G$-modules is contractible (respectively, pure acyclic). If $G$ is torsion-free, the same conclusions hold for acyclic complexes of projective or flat $\mathbb{Z}G$-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.
