Comparison of Morava E-theories

Abstract

We show that the nth Morava E-cohomology group of a finite spectrum with action of the nth Morava stabilizer group can be recovered from the (n + 1)st Morava E-cohomology group with action of the (n + 1)st Morava stabilizer group.

HKR characters, $p$-divisible groups and the generalized Chern character

“…In this section we review the generalized Chern character constructed in [32]. We recall the construction of a commutative ring spectrum B n and two ring spectrum maps ch : E n+1 → B n and i : E n → B n .…”

“…In [3], Ando, Morava and Sadofsky have considered a generalization of the Chern character which is a multiplicative natural transformation from a height (n + 1) cohomology theory to a height n cohomology theory. In [32], we have refined this construction. We have constructed a commutative ring spectrum B n and a ring spectrum map i : E n → B n , where B n is even-periodic and Landweber exact of height n. The ring spectrum map i induces a faithfully flat ring homomorphism E 0 n → B 0 n on the degree 0 homotopy rings, and hence we can consider that B n is an extension of E n .…”

“…Then there is a natural B 0 n -algebra homomorphism B 0 n (BG) → HKR(G; B n ) which induces an isomorphism after tensoring with Q. The generalized Chern character constructed in [32] has the following form:…”

On E∞ -structure of the generalized Chern character

“…Hence the classical Chern character map is a multiplicative map from a height 1 theory to a height 0 theory. In [11] we constructed a generalization of the Chern character map…”

“…Therefore we can consider that G acts on E n and E n+1 through the projections. In [11] we showed that the group G acts on B in the stable homotopy category. Furthermore, we have shown that ch and i are equivariant with respect to the action of G in the stable homotopy category.…”