A splitting theorem for certain cohomology theories associated to BP* (?)

Abstract

Let P(n)*(−) be Brown-Peterson cohomology modulo In and put B(n)*(−)=P(n)*(−)[1/vn]. In this note we construct a canonical multiplicative and idempotent operation Ωn in a suitable completion\(\bar B\) (n)*(−) of B(n)*(−) which has the property that its image is canonically isomorphic to the n-th Morava K-theory K(n)*(−). In particular, the ring theory K(n)*(−) is contained as a direct summand in the theory\(\bar B\) (n)*(−). A similar result is not true before completing. pleting. Because the completion map B (n)*(−) →\(\bar B\) (n)*(−) is injective, the above splitting theorem contains also information about B(n)*(−). The proof of the theorem depends on a result about the behaviour of formal groups of finite height over complete graded Fp.