Periodic phenomena in the classical Adams spectral sequence

Abstract:

We investigate certain periodic phenomena in the classical Adams sepctral sequence which are related to the polynomial generators ${\nu _n}$ in ${\pi _{\ast }}(\operatorname {BP} )$. We define the notion of a class $a$ in ${\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2)$ being ${\nu _n}$-periodic or ${\nu _n}$-torsion and prove that classes that are ${\nu _n}$-torsion are also ${\nu _k}$-torsion for all $k$ such that $0 \leqslant k \leqslant n$. This allows us to define a chromatic filtration of ${\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2)$ paralleling the chromatic filtration of the Novikov spectral sequence ${E_2}$-term given in [13].