The Modular Homology of Inclusion Maps and Group Actions

✍ Scribed by Valery Mnukhin; Johannes Siemons

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 470 KB
• Volume: 74
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.0051

✦ Synopsis

Let 0 be a finite set of n elements, R a ring of characteristic p>0 and denote by M k the R-module with k-element subsets of 0 as basis. The set inclusion map : M k Ä M k&1 is the homomorphism which associates to a k-element subset 2 the sum (2)=1 1 +1 2 + } } } +1 k of all its (k&1)-element subsets 1 i . In this paper we study the chain

arising from . We introduce the notion of p-exactness for a sequence and show that any interval of (*) not including M nÂ2 or M n+1Â2 respectively, is p-exact for any prime p>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on 0 of order not divisible by p. If an interval of (*) , or an equivalent sequence arising from a permutation group on 0, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application are p-rank formulae for orbit inclusion matrices.