Abstract
As a step towards understanding the mathrm {tmf}-based Adams spectral sequence, we compute the K(1)-local homotopy of mathrm {tmf}wedge mathrm {tmf}, using a small presentation of L_{K(1)}mathrm {tmf} due to Hopkins. We also describe the K(1)-local mathrm {tmf}-based Adams spectral sequence.
Highlights
This paper calculates the K (1)-local homotopy of tmf ∧ tmf
An initial difficulty with this spectral sequence is the fact that bo∗bo does not satisfy Adams’ flatness assumption, resulting in the E2-term not having a description in terms of Ext
This paper is almost entirely set inside the K (1)-local category. This leads to some unusual choices about notation, for the sake of which we encourage even the expert reader to take a look at Sect. 1.2 below
Summary
This paper calculates the K (1)-local homotopy of tmf ∧ tmf. The motivation behind this traces back to Mahowald’s work on bo-resolutions. It is worth noting that, for the sake of calculating Adams spectral sequences, one is interested in the coalgebra of bo∗bo as much as its algebra – and the original, non-θ -algebraic calculation π∗ L K (1)(bo ∧ bo) = K O∗ ⊗ Mapscts(Z×p /μ, Zp) is better suited for this purpose It is this realization, and a search for an analogue for tmf, that eventually led to the proof of Theorem B. While the following is essentially a restatement of the original calculation, it is of independent enough interest to deserve explicit mention: Theorem C At the primes 2 and 3, the ring of ordinary 2-variable p-adic modular forms is generated as a θ -algebra by j−1, j−1, and a single other generator
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