Computation of the ANSS E2-Term

File(s): none available
Status: planning


Problem: find a practical way to compute the Ext of the BP Hopf-algebroid.

Project details

Recall Miller's algebraic Novikov spectral sequence
ExtAodd(Fp) ⇒ ExtBP*BP(BP*)
where Aodd denotes the "oddified" associated graded of the Steenrod algebra. One way to set up this spectral sequence is based on the spectrum
EBP = BPE(β0, β1,...)
Here EBP should be understood as a wedge of suspended copies of BP. EBP is an associated graded of the Eilenberg-MacLane spectrum HFp and there is a related differential on EBP with (βi) = vi.

The cooperations EBP*EBP form a differential Hopf algebroid whose -homology is the Hopf algebra dual to Aodd. Miller's spectral sequence can then be realized by computing a lift K of an Aodd-resolution into the world of EBP*EBP-modules.

I vaguely think that the following might be true:

Idea 1: Such a lift K can be computed recursively as K/I^n for n=1,2,...

Here K/In should require computations in EBP*EBP/In+1 and leave K/In-1 alone.

Idea 2: You only need K/I to draw a chart of the algebraic Novikov spectral sequence (including extensions).

Combined these ideas might lead to a little miracle: a computation of the Ext of BP*BP which only requires working in EBP*EBP/I2.