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The algebraic Hodge theorem was proved in a beautiful 1987 paper by Deligneand Illusie, using positive characteristic methods. We argue that the centralalgebraic object of their proof can be understood geometrically as a linebundle on a derived scheme. In this interpretation, the Deligne-Illusie resultcan be seen as a proof that this line bundle is trivial under certainassumptions.  We give a criterion for the triviality of this line bundle in a more generalcontext. The proof uses techniques from derived algebraic geometry,specifically arguments which show the formality of certain derivedintersections. Applying our criterion we recover Deligne and Illusie's originalresult. We also apply these techniques to the result of Barannikov-Kontsevich,Sabbah, and Ogus-Vologodsky concerning the formality of the twisted de Rhamcomplex.

Derived intersections and the Hodge theorem