On the Hodge to de Rham spectral sequence in positive characteristic, after A. Petrov
On the Hodge to de Rham spectral sequence in positive characteristic, after A. Petrov
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Luc Illusie, University of Paris
Let $k$ be a perfect field of characteristic $p>0$. A. PetrovÌýhas recently constructed a projective, smooth scheme $X/W(k)$, ofÌýrelative dimension $p+1$, such that the Hodge to de Rham spectralÌýsequence of $X_0/k$, where $X_0 = X \otimes_{W(k}k$, does not degenerateÌýat $E_1$, thus solving a question of Deligne-Illusie left open sinceÌý1987. I will explain the main ideas of the proof, which uses inputs fromÌýBhatt-Lurie's theory of diffracted Hodge complexes and combinesÌý techniques of homotopical algebra and representation theory of algebraicÌýgroups.