ag.algebraic-geometrytriangulated-categories
It is well known that Beck's theorem for Comonad is equivalent to Grothendieck flat descent theory on scheme.
There are several version of derived noncommutative geometry. I wonder whether someone developed the triangulated version of Beck's theorem. And What does it mean,if exists?
The question actually fits into the more general setting that is described in David Ben-Zvi's answer to this MO question (which is about categorification of the Chern character).
Shklyarov is the one who developped RR theorem for noncommutatve derived schemes (by this one should understand smooth proper DG algebras): http://arxiv.org/abs/0710.1937.
Shklyarov's result has been improved recently by Petit, Lunts, and also Polishchuk-Vaintrob in the context of matrix factorizations.
Let me explain what the statement is. Let $A$ be a proper and homologically smooth dg algebra $A$ (proper means that $\sum_n dim(H^n(A))<+\infty$, and homologicaly smooth means that $A$ is perfect in $D(A\otimes A^{op})$). Let $M$ be a perfect $A$-module. There is a trace map $ch:Hom_{D^{perf}(A)}(M,M)\to HH_0(A)$ (see e.g. this paper of Caldararu-Willerton for a very nice description of the kind of traces I am speaking about), which you can consider as being the Chern character.
Now for an $A$-module $M$ and an $A^{op}$-module $N$ we can consider the $k$-module $N\otimes_AM$ (all my tensor products are derived). Then for $f:M\to M$ and $g:N\to N$ we can consider $ch(g\otimes f)=str(g\otimes f)\in HH_0(k)=k$. Finally the formula is $$ ch(g)\cup ch(f)=ch(g\otimes f) $$ where $\cup:HH_*(A^{op})\otimes HH_*(A)\to HH_*(k)=k$ is the so-called (categorical) Mukai pairing. This is actually more a Lefshetz type formula.
The Todd class is actually hidden in the Mukai pairing (the point is that for associative algebras there is no analogon neither for the Todd class, nor for the usual pairing given by integration).
To my knowledge the first one who proved a RR Theorem for D-modules is Laumon (Sur la categorie derivee des D-modules filtres, Algebraic Geometry, M. Raynaud and T. Shioda eds, Lecture Notes in Math. Springer-Verlag 1016 pp. 151–237, 1983). Then Schapira and Scheinders also considered it (Index theorem for elliptic pairs II. Euler class and relative index theorem, Asterisque 224 Soc. Math. France, 1994) and made a very important conjecture which has been proved by Bressler-Nest-Tsygan using methods of deformation quantization ( http://arxiv.org/abs/math/9904121 and http://arxiv.org/abs/math/0002115) developped by Fedosov.
There is also a paper of Engeli and Felder that gives a Lefschetz type formula. Their approach has been later clarified by Ramadoss (he has many paper on this subject that you can find on arXiv).
The subject really moved to deformation quantization. You can learn a lot about all this (with also more details on who one should credit for what) in Section 4, 5 and 6 of this book by Kashiwara and Schapira.
If you have a formal noncommutative deformation $A_\hbar$ of the structure sheaf $\mathcal O_X$ (maybe as a twisted resheaf, or algebroid stack - this is what Kashiwara and Schapira call a DQ algebroid), then you can play the same game as with a smooth proper dg algebras : define a trace with values in Hochschild homology, and state a HRR type Theorem (better, Lefshetz formula) about the compatibility of the cup product on Hochschild homology with the composition of kernels. The reason for that relies on some finiteness and duality properties for cohomologicall complete $A_\hbar$-modules. The main difficulty is then to prove such a result.
To conclude this paragraph, let me observe that (the Rees algebra of) $\mathcal D_X$ can be viewed as a deformation of $\mathcal O_{T^*X}$.
Last but not least, the relation between the derived non-commutative geometry and deformation quantization stuff is adressed in a very recent preprint of Petit: his strategy to prove HRR for DQ algebroids is to use the result for smooth proper DG algebras. Namely, he proves that some derived category of cohomologically complete modules over a DQ-algebroid on a projective variety has a compact generator.
This subject is currently very active.
I apologize for that this answer reduces to a (non-exhaustive) list of references. If you have a more specific question I can tell you where to go in these references in order to (hopefully) find an answer.
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.
Best Answer
There isn't a descent theory for derived categories per se - one can't glue objects in the derived category of a cover together to define an object in the base. (Trying to apply the usual Barr-Beck to the underlying plain category doesn't help.)
But I think the right answer to your question is to use an enriched version of triangulated categories (differential graded or $A_\infty$ or stable $\infty$-categories), for which there is a beautiful Barr-Beck and descent theory, due to Jacob Lurie. (This is discussed at length in the n-lab I believe, and came up recently on the n-category cafe (where I wrote basically the same comment here..) This is proved in DAG II: Noncommutative algebra. In the comonadic form it goes like this. Given an adjunction between $\infty$-categories (let's call the functors pullback and pushforward, to mimic descent), if we have
then the $\infty$-category downstairs is equivalent to comodules over the comonad (pullback of pushforward). (There's an opposite monadic form as well) This can be verified in the usual settings where you expect descent to hold. In other words if you think of derived categories as being refined to $\infty$-categories (which have the derived category as their homotopy category), then everything you might want to hold does.
So while derived categories don't form a sheaf (stack), their refinements do: you can recover a complex (up to quasiisomorphism) from a collection of complexes on a cover, identification on overlaps, coherences on double overlaps, coherences of coherences on triple overlaps etc. More formally: define a sheaf as a presheaf $F$ which has the property that for an open cover $U\to X$, defining a Cech simplicial object $U_\bullet=\{U\times_X U\times_X U\cdots\times_X U\}$, then $F(X)$ is the totalization of the cosimplicial object $F(U_\bullet)$. Then enhanced derived categories form sheaves (in appropriate topologies) as you would expect. This is of course essential to having a good theory of noncommutative algebraic geometry!