nLab descent spectral sequence

Idea

Given a morphism of discrete commutative rings, $\phi:A\to B$, one is often interested in computing the Amitsur cohomology? with respect to $\phi$. 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 $B$-module, contains useful information regarding descent along $\phi$. More generally, when we pass to a morphism of, say, ring spectra $\phi:A\to B$, 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 $\phi:A\to B$ converges to the homotopy groups of $B$.

Examples

• The (unstable) Adams spectral sequence is an example for descent along $Spec(E) \to Spec(R)$. (…) (Hess 10, section 5.3)

References

For higher monadic descent: section 5.3 of

• Kathryn Hess, A general framework for homotopic descent and codescent, arXiv/1001.1556

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