DR-decomposablity of a smooth variety is an important property which leads to the first algebraic proof of Kodaira vanishing and degeneration of Hodge-de Rham spectral sequence. In the case of the semi-stable variety, which appears at the boundary of the moduli of manifolds, the DR-decomposablity is established by K.Kato in pure characteristic. The DR-decomposablity in mixed characteristic is posted by L.Illusie as a problem in a 1996 survey. In this talk, I will report my recent work (joint with Mao Sheng) on this problem. It turns out that Illusie’s problem generally has a negative answer.

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