## Is there triangulated category version of Barr-Beck’s theorem?

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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?

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## 2 Answers

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

1. pullback is conservative (it respects isomorphisms), and
2. pullback respects certain limits (namely totalizations of cosimplicial objects, which are split after pullback)

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!

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There is a Beck's theorem for Karoubian triangulated categories, proposed by Konstevich and Rosenberg in July 2004, which is proved by using the Verdier's abelianization functor and graded monads; see page 36 of A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57, pdf.

In fact they got simulatenously (July 15, 2004) both versions: A-infinity and triangulated. The reference to the triangulated is above: while for A-infinity there is no write up, but Kontsevich gave a talk (I think Nov 2004, van den Bergh birthday conference) where he formulated and used the result; one of nice applications was to glue certain ordinary commutative schemes to get certain formal schemes. I remember very well the weeks preceding the result when we discussed possible shape of the result seeked at IHES. Later at the conference in Split, Kontsevich gave a talk where he gave some usages in noncommutative algebraic geometry.

I disagree with the statement: "Beck's theorem for comonad is equivalent to Grothendieck flat descent theory on scheme". Namely Grothendieck gave both the flat descent theory for quasicoherent sheaves (SGA I.8.1) which is a special case of Beck's theorem (though it has some symmetries which general noncommutative case does not have), but also the (stronger) flat descent theorem for affine schemes (SGA I.8.2) and for morphisms (cf. SGA I.8.5), which unlike the descent for quasicoherent sheaves, does not generalize to the noncommutative algebras and consequently to categorical setup either.

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Thanks Zoran! The link to Rosenberg's pdf didn't work for me. I'm very curious to see how the triangulated theorem is used - I understand how you'd use the $A_\infty$ version (which is presumably subsumed now by Lurie) but I'm wondering if there are some settings where the triangulated version is useful eg for gluing. – David Ben-Zvi Mar 9 2010 at 13:01
@David and Skoda: Now I saw Rosenberg used this triangulated version of Beck's theorem for induction theorem in triangulated category. Because in triangulated version, we do not require the exactness of functors,so the induction theorem works very smooth in this settings. For induction theorem, I mean cohomological induction. Rosenberg then used t-structures to turn this triangulated picture to abelian picture. I will elaborate this in an additional answers – Shizhuo Zhang Mar 18 2010 at 8:00
David, I corrected the link, for some reason MPI changed the format of the URL in the meantime and www at the beginning is now mandatory. – Zoran Škoda Mar 23 2010 at 16:11
Yes, in the usual geometric situations with qcoh sheaves on open neighborhoods the descent for derived categories does not work as we know from elementary counterexamples. But there are lots of examples with Cohn localization which is non-flat, and these are examples in which if we would restrict to noncommutative subvarieties which are close to the commutative the descent would suddenly become flat, and everything would work at the abelian level already. You should push Sasha & Maxim to finish their "secret" preprint started in 1999, whose delay was initially due to lack of similar tools. – Zoran Škoda Mar 23 2010 at 16:12

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Just before Prop. 3.4, Rosenberg says on p.36, that given a monad F on a triangulated category, the category of F-modules is triangulated in such a way that the forgetful functor is exact. This would imply in particular that modules over ring spectra up to homotopy form a triangulated category. Etc, etc. Do you believe this? Specifically, I don't see how to put a module structure on the cone of a morphism of modules, unless some assumption is made about the monad (like being separable, in which case it works). – Paul Balmer Mar 23 2011 at 22:10
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