The Adams Spectral Sequence for Topological Modular Forms

On a conjecture of Mahowald on the cohomology of finite sub-Hopf algebras of the Steenrod algebra

“…This spectral sequence first appeared in [5] and was referred to as the "Koszul spectral sequence" in [7]. It has recently been rechristened as the Davis-Mahowald spectral sequence by Bruner, Rognes and their coworkers in [3] and [11].…”

The Adams spectral sequence for 3-local $\mathrm{tmf}$

“…So we just need to check the second condition of May's Convergence Theorem, that there are no crossing differentials. Note that v 2 1 ∈ E 6,2,10 1 and that E m,2,10 2 = 0 for all m. So condition (2) is satisfied here. Likewise, note that (v 1 α 1 ) 2 ∈ E 8,4,18 1 and that E m,3,18 1 = 0 for all m. Thus there are no nonzero differentials to worry about.…”

“…[1,9]), but on the E 2 -term, there are the nonzero classes v 2 2 α 2 and v 2 c 6 b 4 α 1 which are not killed by previously established d 2 -differentials. The only way for v 2 2 α 2 to be killed is by a d 2 -differential given by the claimed differential. The hidden v 0 -extension established in Proposition 4.6 gives us the second differential.…”

“…Thus, the combined pattern only receives differentials. It receives its differentials from the free pattern Ext A(1) * (F 3 ) • {v 2 2 b 4 }. The differentials (5.14), (5.15), and (5.17) propagate to give the following differentials:…”

Classifying and extending $Q_0$-local $\mathcal{A}(1)$-modules

“…Here, we describe the construction of the Davis-Mahowald spectral sequence in the case where n = 1 and construct N • := (R • 1 ) * directly. The use of this spectral sequence for the computation of h −1 0 Ext •,• A(1) (-, F 2 ) was suggested to the author by John Rognes, and the following development of the spectral sequence relies heavily on upcoming work of Rognes and Bruner [5].…”