The power operation structure on Morava E – theory of height 2 at the prime 3

Suppose E is a commutative S–algebra, in the sense of Elmendorf, Kriz, Mandell and May [6], and A is a commutative E–algebra. We want to capture the properties and underlying structure of the homotopy groups π∗A = A∗ of A, by studying operations associated to the cohomology theory that E represents. An important family of cohomology operations, called power operations, is constructed via the extended powers. Specifically, consider the m’th extended power functor PE (−) := (−)E/Σm : ModE → ModE on the category of E–modules, which sends an E–module to its m-fold smash product over E modulo the action by the symmetric group on m letters. The PE (−)’s assemble together to give the free commutative E–algebra functor PE(−) := ∨