http://mathoverflow.net/questions/22032/what-is-descent-theory
http://www.ncatlab.org/nlab/show/cohomological+descent
Menini and Dragos on Descent theory and Amitsur cohomology of triples, with a nice general introduction on descent theory in the algebraic setting.
arXiv:1103.3135 Cohomological descent theory for a morphism of stacks and for equivariant derived categories from arXiv Front: math.AG by Alexey Elagin In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.
arXiv:1001.1556 A general framework for homotopic descent and codescent from arXiv Front: math.KT by Kathryn Hess In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as -category-theoretic, as our framework is constructed in the universe of simplicially enriched categories, which are a model for -categories
We provide general criteria, reminiscent of Mandell’s theorem on -algebra models of -complete spaces, under which homotopic (co)descent is satisfied. Furthermore, we construct general descent and codescent spectral sequences, which we interpret in terms of derived (co)completion and homotopic (co)descent
We show that a number of very well-known spectral sequences, such as the unstable and stable Adams spectral sequences, the Adams-Novikov spectral sequence and the descent spectral sequence of a map, are examples of general (co)descent spectral sequences. There is also a close relationship between the Lichtenbaum-Quillen conjecture and homotopic descent along the Dwyer-Friedlander map from algebraic K-theory to étale K-theory. Moreover, there are intriguing analogies between derived cocompletion (respectively, completion) and homotopy left (respectively, right) Kan extensions and their associated assembly (respectively, coassembly) maps.
arXiv:0911.3554 Flat descent for Artin n-stacks from arXiv Front: math.CT by B. Toen We prove two flat descent statements for Artin n-stacks. We first show that an n-stack for the etale topology which is an Artin n-stack in the sense of HAGII, is also an n-stack for the fppf topology. Moreover, an n-stack for the fppf topology which possess a fppf n-atlas is an Artin n-stack (i.e. possesses a smooth n-atlas). We deduce from these results some comparison statements between fppf and etale (non-ablelian) cohomolgies. This paper is written in the setting of derived algebraic geometry and its results are also valid for derived Artin n-stacks.
arXiv:0808.3684 (Co)Simplicial Descent Categories from arXiv Front: math.AT by Beatriz Rodriguez Gonzalez Let D be a category and E a class of morphisms in D. In this paper we study the question of how to transfer homotopic structure from the category of simplicial objects in D, sD, to D through a `good' functor s : sD --> D, which we call simple functor. For instance, the homotopy colimit in a Quillen simplicial model category is a good simple functor. As a remarkable example outside the setting of Quillen models we include Deligne's simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also show that sD is in a natural way a Brown category of cofibrant objects. We use these facts to produce cofiber sequences on the localized category of D by E, which give rise to a natural Verdier triangulated structure in the stable case.
http://mathoverflow.net/questions/72140/schemes-do-not-form-a-stack-in-the-etale-topology
http://mathoverflow.net/questions/6979/what-is-etale-descent
http://mathoverflow.net/questions/37970/do-coequalizers-in-ringspc-automatically-lead-to-descent
There is some descent stuff in Waterhouse, with group schemes in mind. In alg gps and gp schemes folder.
http://www.ncatlab.org/nlab/show/descent+in+noncommutative+algebraic+geometry
nlab: monadic descent
nLab page on Descent II