Chamber basis of the Orlik-Solomon algebra and Aomoto complex

A deformation of the Orlik-Solomon algebra

A deformation of the Orlik-Solomon algebra of a matroid M is defined as a quotient of the free associative algebra over a commutative ring R with 1. It is shown that the given generators form a Groebner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as R-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Groebner basis and hence the algebra is Koszul

Aomoto复形的$q$-形变

$q$-deformation of Aomoto complex

本文研究Aomoto复形的$q$-形变，通过将上边缘矩阵元素替换为$q$-模拟构造，并证明在实结构导出的特殊基下其恢复为上链复形且具有特化普适性。

The spectral sequence of an equivariant chain complex and homology with local coefficients

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 boun…