Mapping algebras and the Adams spectral sequence

For a suitable ring spectrum, such as $\mathbf{E}=\mathbf{H}\mathbb{F}_p$, the $E_2$-term of the $\mathbf{E}$-based Adams spectral sequence for a spectrum $\mathbf{Y}$ 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. 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 $\operatorname{Fun}(\mathbf{Y},\mathbf{M})$ for various $\mathbf{E}$-modules $\mathbf{M}$, equipped with the action of $\operatorname{Fun}(\mathbf{M},\mathbf{M}^\prime)$ on them.

Keywords

spectral sequence, truncation, differentials, cosimplicial resolution, mapping algebra

2010 Mathematics Subject Classification

Primary 55T15. Secondary 55P42, 55U35.

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Received 4 February 2020

Received revised 1 July 2020

Accepted 2 July 2020

Published 15 October 2020