The BP<n> cohomology of elementary Abelian groups

Neil P. Strickland

20J06; 55N20; 14L05

math.AT/0011120

Department of Pure Mathematics
University of Sheffield
Hicks Building
Hounsfield Road
Sheffield S3 7RH
UK

N.P.Strickland@sheffield.ac.uk

In this paper we study E^*BV_k, where E=BP<m,n> is a cohomology theory
with coefficient ring F_p[v_m,...,v_n] (if m>0) or Z_(p)[v_1,...,v_n]
(if m=0).  We use ideas from the theory of multiple level structures,
developed in earlier work of the author with John Greenlees.  Our
results apply when k is less than or equal to w=n+1-m.  If k<w we find
that E^*BV_k has no v_m-torsion.  When k=w, we show that the
v_m-torsion is annihilated by the ideal I_{n+1}=(v_m,...,v_n), and
that it is a free module on one generator over the ring
F_p[[x_0,...,x_{w-1}]].  We give three very different formulae for
this generator; it is not at all obvious that these give the same
element, and we only have a rather indirect proof of this.