On Modular Homology in the Boolean Algebra

✍ Scribed by Valery Mnukhin; Johannes Siemons

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 165 KB
• Volume: 179
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0009

✦ Synopsis

Let ⍀ be a set, R a ring of characteristic p ) 0, and denote by M the k R-module with k-element subsets of ⍀ as basis. The set inclusion map

is the homomorphism which associates to a k-element subset ⌬ the sum

q⌫ of all its k y 1 -element subsets ⌫ . In this paper we

arising from Ѩ. We introduce the notion of p-exactness for a sequence. If ⍀ is Ž . infinite we show that ) is p-exact for all prime characteristics p ) 0. This result can be extended to various submodules and quotient modules, and we give general constructions arising from permutation groups with a finitary section. Two particular applications are the following: The orbit module sequence of such a permutation group on ⍀ is p-exact for every prime p, and we give a formula for the p-rank of the orbit inclusion matrix if the group has finitely many orbits on k-element subsets.