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descent spectral sequence

Contents

Context

Locality and descent

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Given a morphism of discrete commutative rings, ϕ:AB\phi:A\to B, one is often interested in computing the Amitsur cohomology? with respect to ϕ\phi. This is the dualization of Cech cohomology for a morphism of spaces, except it takes place before applying a “Spec” functor to the rings. Often, the Amitsur cohomology of a morphism of rings, especially with coefficients in some BB-module, contains useful information regarding descent along ϕ\phi. More generally, when we pass to a morphism of, say, ring spectra ϕ:AB\phi:A\to B, we obtain a cosimplicial? ring spectrum, and in this homotopical context, the generalization of cohomology groups are the homotopy groups of its totalization. Indeed, if we restrict to Eilenberg-Mac Lane spectra whose homotopy is concentrated in degree zero, we regain the usual notion of discrete Amitsur cohomology. Thus, the descent spectral sequence is the spectral sequence associated to cosimplicial Amitsur complex of a morphism of algebra objects in some category. In nice cases, the descent spectral sequence for a morphism ϕ:AB\phi:A\to B converges to the homotopy groups of BB.

Examples

References

For higher monadic descent: section 5.3 of

  • Menini, Claudia; Ştefan, Dragoş, Descent theory and Amitsur cohomology of triples, J. Algebra 266 (2003), no. 1, 261–304.

Last revised on January 16, 2015 at 05:16:33. See the history of this page for a list of all contributions to it.