Homotopy groups of homotopy fixed point spectra associated to E_n

Abstract

ETHAN S DEVINATZWe compute the mod.p/ homotopy groups of the continuous homotopy fixed point spectrum E hH 2 2 for p > 2, where E n is the Landweber exact spectrum whose coefficient ring is the ring of functions on the Lubin-Tate moduli space of lifts of the height n Honda formal group law over ‫ކ‬ p n , and H n is the subgroup W ‫ކ‬ p n Ì Gal.‫ކ‬ p n ‫ކ=‬ p / of the extended Morava stabilizer group G n . We examine some consequences of this related to Brown-Comenetz duality and to finiteness properties of homotopy groups of K.n/ -local spectra. We also indicate a plan for computing .E hH n n^V .n 2//, where V .n 2/ is an E n -local Toda complex. 55Q10, 55T25

Towards the finiteness of π*LK(n)S0

“…Unfortunately, π * (E hH 2 2 ∧ M) is not of finite type. In fact, subsequent (unpublished) calculations of mine seemed to indicate that π * (E hG 2 ∧ M) is almost never of finite type, yet almost always "almost" of finite type in the sense of [5], the meaning of which we now recall.…”

“…Also write ν : π * (E hG n ∧ X) → π * (E hG n ∧ X) for the map on homotopy groups induced by E hG n ∧ ν. Then π * (E hG n ∧ X) is an F p [ν]-module and, as in [5,Proposition 4.5], π * (E hG n ∧ X) is good. Since v n−1 self-maps are essentially unique (cf.…”

“…More generally, if π * L K(n) X is of finite type for some finite X in the E(n) * -local category, then π * L K(n) Y is of finite type for any finite Y in the E(n) * -local category such that (See [2] for a discussion of the E(n) * -local thick subcategory theorem.) If X is a K(n − 1) * -acyclic finite spectrum in the E(n) * -local category, then it is standard that π * L K(n) X is of finite type [5,Proposition 4.1]. The next case to consider is therefore when X is K(n − 2) * -acyclic.…”

“…(See [5] for a discussion of this.) For subgroups G whose continuous cohomology with trivial coefficients F p is easily understood, one might hope to compute π * (E hG n ∧ V (n − 2)) and see that it is of finite type.…”

“…In [5], we carried out this computation in the case where p 3 and G is the "diagonal" subgroup H 2 of G 2 . Unfortunately, π * (E hH 2 2 ∧ M) is not of finite type.…”