The E2-term of the K(n)-local E-Adams spectral sequence

Abstract

Abstract. Let E = En be Morava E-theory of height n. In [DH04] Devinatz and Hopkins introduced the K(n)-local En-Adams spectral sequence and showed that, under certain conditions, the E 2 -term of this spectral sequence can be identified with continuous group cohomology. We work with the category of L-complete E ∨ * E-comodules, and show that in a number of cases the E 2 -term of the above spectral sequence can be computed by a relative Ext group in this category. We give suitable conditions for when we can identify this Ext group with continuous group cohomology.

The chromatic splitting conjecture at n = p = 2

“…Proof We first prove that the E 2 -term is isomorphic to H * (G C , (E C ) * X). This can be deduced directly from Barthel and Heard [1,Theorem 4.3]. Nonetheless, we sketch the proof here.…”

Topological resolutions in K(2)‐local homotopy theory at the prime 2

“…The final statement follows from Lemma 4.1.Now let q : E hS1 2 → E hG24 be the augmentation in the topological duality resolution of Proposition 3.19. This is also the projection from the top to the bottom of the duality resolution tower (3.21).…”

“…The towers (3.10) and (3.11) determine each other. This is because there is a diagram with rows and columns cofibration sequences (3.12) Using the tower over E hS 1 2 of (3.11) we get a number of spectral sequences; for example, if Y is any spectrum, we get a spectral sequence for the function spectrum F (Y, E hS 1 2 )…”

“…Let p : E hS 1 2 → E hG24 × E hG24 be the augmentation in the topological centralizer resolution of Theorem 3.7. This is the same map as from the top to the bottom of the centralizer resolution tower (3.10).…”

“…We have E hG 1 2 = (E hS 1 2 ) hGal where Gal = Gal(F 4 /F 2 ) and S 1 2 = S 2 ∩ G 1 2 . For many computational applications, the difference between E hG 1 2 and E hS 1 2 is innocuous. See Lemma 1.37.…”