Mapping algebras and the Adams spectral sequence

Abstract

The $E_2$-term of the Adams spectral sequence for $\mathbf{Y}$ may be described in terms of its cohomology $E^{\ast }\mathbf{Y}$, together with the action of the primary operations $E^{\ast }\mathbf{E}$ on it, for ring spectra such as $\mathbf{E}=\mathbf{H}{\mathbb{F}}_p$. We show how the higher terms of the spectral sequence can be similarly described in terms of the higher order truncated $\mathbf{E}$-mapping algebra for $\mathbf{Y}$ $-$ that is truncations of the function spectra $\mathrm{Fun}(\mathbf{Y},\mathbf{M})$ for various $\mathbf{E}$-modules $\mathbf{M}$, equipped with the action of $\mathrm{Fun}(\mathbf{M},{\mathbf{M}}^{\prime })$ on them.

To appear in "Homology, Homotopy and Applications"