Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie

A new construction of cyclic homology

Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by…

Non-commutative Cartier operator and Hodge-to-de Rham degeneration

We introduce a version of the Cartier isomorphism for de Rham cohomology valid for associative, not necessarily commutative algebras over a field of positive characteristic. Using this, we imitate the well-known argument of P. Deligne and L. Iluusie and prove, in some cases, a conjecture of M. Kontsevich which claims that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates.

Deligne's conjecture on extensions of 1-motives

We introduce the notion of extension of 1-motives. Using the dictionary between strictly commutative Picard stacks and complexes of abelian sheaves concentrated in degrees -1 and 0, we check that an extension of 1-motives induces an extension of the corresponding strictly commutative Picard stacks. We compute the Hodge, the de Rham and the l-adic realizations of an extention of 1-motives. Using these results we can prove Deligne's conjecture on extensions of 1-motives.

通过范数构造的非交换相对德拉姆-维特复形

Noncommutative relative de Rham--Witt complex via the norm

该研究引入相对霍赫希尔德-维特同调，统一了德拉姆-维特复形的相对与非交换两种推广，证明其HKR定理并建立与相对德拉姆-维特复形的关联，还补全了Kaledin相关研究的缺失部分。

Réalisations de Hodge des motifs de Voevodsky

Over a subfield of the field of complex numbers, the Hodge realization of a geometrical motive is defined and represented as the cohomology of a mixed Hodge DG-complex in the sense of Deligne. Both filtrations are represented by truncation functors, on a Bondarko weight complex for the weight filtration and on the De Rham motivic complex for the Hodge one. The Deligne-Beilinson realization is also constructed. This preprint partially replaces the former "Réalisation des complexes motiviques de Voevodsky".

关于A-无穷代数、A-无穷范畴与非交换几何的札记 I

Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I

本文基于分次向量空间范畴的形式概形概念，提出研究A-infinity代数与范畴的几何方法，厘清相关结构问题，推广多个经典猜想与概念，并给出对应二维拓扑场论的结构等价性结论。