The spectral sequence of an equivariant chain complex and homology with local coefficients
The spectral sequence of an equivariant chain complex and homology with local coefficients
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等变链复合体的谱序列及其与局部系数的同源性
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发表时间：
2007
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通讯作者：
Alexander I. Suciu
中科院分区：
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作者：
S. Papadima;Alexander I. Suciu
我们研究了连通CW-复形X的泛覆盖的(扭曲)等变链复形上与增广理想的幂滤有关的谱序列。在这个过程中，我们根据H*(X，k)的余代数结构和扭转系数上的kπ1(X)-模结构来识别d1微分。特别地，这以对偶形式恢复了Reznikov关于非球面复数的环p-覆盖的mod p上同调的结果。这个方法提供了关于X的所有Galois覆盖的同调的信息，它还得到了关于X的上同调群的秩的可计算的上界，其中系数为素数幂阶数，排名为一局部系统。当X允许极小胞胞分解时，我们将泛阿贝尔覆盖的等变余链复形的线性化与由H*(X，k)的杯积结构产生的Aomoto复形联系起来，从而推广了Cohen和Orlik的一个结果。
We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d 1 differential in terms of the coalgebra structure of H * (X, k) and the kπ 1 (X)-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod p cohomology of cyclic p-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper bounds on the ranks of the cohomology groups of X, with coefficients in a prime-power order, rank one local system. When X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of H* (X, k), thereby generalizing a result of Cohen and Orlik.