The modular homology of inclusion maps and group actions
Given a finite set \(\Omega\) with power set \(2^\Omega\) and \(R\) an associative ring with 1, \(R2^\Omega\) is the free unitary left \(R\)-module with basis \(2^\Omega\) which becomes an associative ring with identity \(1\emptyset\) if the product \(f\cup g=(\sum f_\Delta\Delta)\cup(\sum g_\Gamma\Gamma)=\sum f_\Delta g_\Gamma(\Delta\cup\Gamma)\), \(\Delta,\Gamma\in 2^\Omega\). Then, if \((\Gamma)=\Gamma_1-+\Gamma_k\) is the formal sum of the distinct \((k-1)\)-subsets of the \(k\)-set \(\Gamma\), it follows that \(\partial(f\cup g)=\partial(f)\cup g+f\cup\partial(g)\) and \(\partial:M_k\to M_{k-1}\), where \(M_k\) is the submodule of \(R2^\Omega\) generated by the \(k\)-sets of \(\Omega\). The inclusion (boundary) mapping is thus a derivation, which leads to special properties useful in determining details of homology, including the characteristic \(p>0\) case. Given that many results have been obtained elsewhere, explicitly or in different guises, as recognized by the authors, they have nevertheless been able to take advantage of several nice devices (e.g., corollary 2.3 and lemma 2.4) to prove exactness of \(0\leftarrow M_0\leftarrow M_1\leftarrow\cdots\leftarrow M_m\) for all \(m\leq|\Omega|\) if \(R\) has characteristic 2 and a related notion of \(p\)-exactness (\(M_k\leftarrow M_{k+i}\leftarrow M_{k+p}\) is exact for \(0\leq k\leq m-p\) and \(i\), \(1\leq i\prec p\)) yielding \(0\leftarrow M_0\leftarrow M_1\leftarrow\cdots\leftarrow M_k\) \(p\)-exact if \(R\) has characteristic (prime) \(p\succ 0\) and \(2k\leq|\Omega|\). Having done this group actions on \(\Omega\) become linear maps on \(R2^\Omega\) and the machinery derived can be employed efficiently in obtaining a variety of expressions and forms including an inclusion-exclusion principle and version of Betti-numbers. The techniques employed are successful enough to be used in several similar papers with somewhat varying emphasis and look to be adequate to the tackling of further questions, some mentioned in a concluding remark.
scientific article; zbMATH DE number 883815