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Can someone give me a reference (note I am looking for a reference and not a proof) for the following:

If a complex $C$ has a dg-algebra structure, then the hypercohomology $H^0R\pi_*C$ has an algebra structure, and if $M$ is a dg-module for $C$, then $H^0R\pi_*M$ is a module under $H^0R\pi_*C$. (Here I am thinking of $C$ as a complex of sheaves on some scheme $S$ with $\pi : S \rightarrow T$, but this should just be a fact of homological algebra.)

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Since you want references, not a proof, maybe you can look at the paper by Hinich that Leonid already mentioned, but also at the original book by Godement, "Topologie Algébrique et Théorie des Faisceaux". There he treats also the problem with multiplicative structures in sheaf cohomology (and you can adapt it to the derived functor of direct images). As for "Thom-Whitney functors", mentioned by Minhyong, you should look at the original paper by V. Navarro-Aznar, "Sur la thérorie de Hodge-Deligne", Inv. Math. 90 (1987), 11-76. But be aware that if you are not working with commutative dg algebras you don't need in fact the essential tool of this paper (the "Thom-Whitney simple/total functor"): the usual total functor of double complexes (together with the Alexander-Whitney map) will work as well and looks easier. Or you can also ask me for a preprint about that subject that I'm going to finish one of these days. Sorry for this self-advertising. :-) (One last hint: you don't need to restrict yourself to the H^0 cohomology: you have a dg multiplicative structure already in R\pi_*C : the multiplicative structure in the H^0 is inherited from that one.)

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Sorry, when I said "about that subject", I didn't mean your subject, but just about deriving direct images, with or without products. And I forgot also to warn you that Navarro's paper only works in characteristic zero, but his techniques with the usual total complex of a double complex, instead of the Thom-Witney simple/total functor, work over any ring of coefficients. – a.r. Dec 3 '09 at 19:35

Sheaves of DG-algebras were studied by Hinich here, and he also gives some other references, but I cannot say whether it contains what you are asking about.

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