Root invariants in the Adams spectral sequence

Abstract

Let E E be a ring spectrum for which the E E -Adams spectral sequence converges. We define a variant of Mahowald’s root invariant called the ‘filtered root invariant’ which takes values in the E 1 E_1 term of the E E -Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E E -Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2 2 . We use the filtered root invariants to compute some low-dimensional root invariants of v 1 v_1 -periodic elements at the prime 3 3 . We also compute the root invariants of some infinite v 1 v_1 -periodic families of elements at the prime 3 3 .