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 ν n {\nu _n} in π ∗ ( BP ) {\pi _{\ast }}(\operatorname {BP} ) . We define the notion of a class a a in Ext A ( Z / 2 , Z / 2 ) {\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2) being ν n {\nu _n} -periodic or ν n {\nu _n} -torsion and prove that classes that are ν n {\nu _n} -torsion are also ν k {\nu _k} -torsion for all k k such that 0 ⩽ k ⩽ n 0 \leqslant k \leqslant n . This allows us to define a chromatic filtration of Ext A ( Z / 2 , Z / 2 ) {\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2) paralleling the chromatic filtration of the Novikov spectral sequence E 2 {E_2} -term given in [13].