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I am now taking a course focusing on triangulated geometry. The professor has formulated the Beck's theorem for Karoubian triangulated category. The proof is very simple. Just using the universal homological functor(equivalent to Verdier abelianization)back to abelian settings(in particular, Frobenius abelian category), then using the Beck's theorem for abelian category to get the proof.

When he finished the proof, he made a remark that the cohomological descent theory can be taken as a consequence of triangulated version of Beck's theorem.

As we know, the Beck's theorem for abelian category is equivalent to Grothendieck flat descent theory(Beck's theorem may be more general). I have two questions:

  1. Is there any reference(other than SGA 4)in English explaining the relationship of usual descent theory and cohomolgical descent theory? What I am looking for is not a very thick book but a lecture notes with some examples.

  2. I know Jacob Lurie developed the derived version of Beck's theorem for his infinity category(correct me if I make mistake). But I have never read his paper very carefully. I wonder whether he explained the relationship of Beck's theorem and "cohomogical descent in his settings"(if there exists such terminology).

All the other comments are welcome. Thank you

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up vote 17 down vote accepted

The relationship between cohomological descent and Lurie's Barr-Beck is exactly the same as the relationship between ordinary descent and ordinary Barr-Back. To put things somewhat blithely, let's say you have some category of geometric objects $\mathsf{C}$ (e.g. varieties) and some contravariant functor $\mathsf{Sh}$ from $\mathsf{C}$ to some category of categories (e.g. to $X$ gets associated its derived quasi-coherent sheaves, or as in SGA its bounded constructible complexes of $\ell$-adic sheaves). Now let's say you have a map $p:Y \rightarrow X$ in $\mathsf{C}$, and you want to know if it's good for descent or not. All you do is apply Barr-Beck to the pullback map $p^\ast:\mathsf{Sh}(X)\rightarrow \mathsf{Sh}(Y)$. For this there are two steps: check the conditions, then interpret the conclusion. The first step is very simple -- you need something like $p^\ast$ conservative, which usually happens when $p$ is suitably surjective, and some more technical condition which I think is usually good if $p$ isn't like infinite-dimensional or something, maybe. For the second step, you need to relate the endofunctor $p^\ast p_\ast$ (here $p_*$ is right adjoint to $p^\ast$... you should assume this exists) to something more geometric; this is possible whenever you have a base-change result for the fiber square gotten from the two maps $p:Y \rightarrow X$ and $p:Y \rightarrow X$ (which are the same map). For instance in the $\ell$-adic setting you're OK if $p$ is either proper or smooth (or flat, actually, I think). Anyway, when you have this base-change result (maybe for p as well as for its iterated fiber products), you can (presumably) successfully identify the algebras over the monad $p^\ast p_\ast$ (should I say co- everywhere?) with the limit of $\mathsf{Sh}$ over the usual simplicial object associated to $p$, and so Barr-Beck tells you that $\mathsf{Sh}(Y)$ identifies with this too, and that's descent. The big difference between this homotopical version and the classical one is that you need the whole simplicial object and not just its first few terms, to have the space to patch your higher gluing hopotopies together.

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Oh, this is exactly I am looking for! – Shizhuo Zhang Feb 10 '10 at 14:10

Stacks-GIT develops this theory in (what was at one point) chapter 16 in the chapter on hypercoverings. It is explained in 16.7, although I haven't read it, so I can't vouch for its quality.

Specifically, descent by cohomology is covered in chapter 27, but it relies very much on the machinery developed in chapter 16.

Another useful thing you might want to look at (in French):

Cours 5 develops homotopic descent, which will be very useful for studying Lurie and Toen-Vezzosi.

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For 1 -- Brian Conrad's notes here are great and in english. Lemma 6.8 and the discussion preceding it explain the relation to descent theory.

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