Computation of the ANSS E2-Term
File(s): none available
Status: planning
Abstract
Problem: find a practical way to compute the Ext of the BP Hopf-algebroid.
Project details
Recall Miller's algebraic Novikov spectral sequence
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.
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 = BP ⊗ E(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.