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degeneration of Hodge to de Rham spectral sequence

Contents

Idea

In algebraic geometry, there is a Hodge to de Rham spectral sequence and the statement of the sufficient conditions when it degenerates. Maxim Kontsevich has conjectured an extension of this to noncommutative algebraic geometry based on A A_\infty-categories. A somewhat weaker case, in the framework of dg-categories has been proved by Dmitri Kaledin. Although the conjecture is in characteristic zero, Kaledin has used a method in positive characteristic, combining the cyclic homology with ideas from the one of the proofs of the classical Hodge-dR ss degeneration with positive characteristic methods due Pierre Deligne and Luc Illusie, involving Frobenius automorphism and so-called Cartier operator. This is one of the most nontrivial facts in noncommutative geometry.

References

  • Degeneration of the Hodge-de-Rham spectral sequence (pdf)

Revised on September 12, 2012 12:01:24 by Urs Schreiber (82.169.65.155)