nLab degeneration of Hodge to de Rham spectral sequence

1. 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 $A_\infty$-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.

2. References

• Degeneration of the Hodge-de-Rham spectral sequence (pdf)
• D. Kaledin, Non-commutative Hodge-to-de Rham degeneration via the method of

  Deligne-Illusie, Pure Appl. Math. Quat. 4 (2008), 785–875.

• D. Kaledin, Spectral sequences for cyclic homology, in Algebra, Geometry and Physics in the 21st Century (Kontsevich Festschrift), Birkhäuser, Progress in Math. 324 (2017), 99–129.
• D. Kaledin, A. Konovalov, K. Magidson, Spectral algebras and non-commutative Hodge-to-de Rham degeneration, arxiv/1906.09518