Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie
type: Item
instance of: scholarly article
title: Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie
zbMATH Open document ID: 1189.14013
published in: Pure and Applied Mathematics Quarterly
publication date: 2009-02-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0611623
review text: Given a smooth projective variety \(X\) over a field of characteristic 0, according to the Hodge theory, the Hodge-to-de Rham spectral sequence \(H^p (X, \Omega^q_X) \implies H^{p+q}_{DR} (X)\) degenerates at the first term, so that there is a noncanonical decomposition of the form \(H^{p+q}_{DR} (X) = \bigoplus H^p (X, \Omega^q_X)\). The terms in the Hodge-to-de Rham spectral sequence can be generalized to the case of a site \(X\) equipped with a sheaf of noncommutative associative algebras \(\mathcal A\) by replacing the differential forms with the Hochschile cohomology sheave \(HH_\bullet (\mathcal A)\) and the de Rham complex with the cyclic homology complex \(HC_\bullet (\mathcal A)\). Thus the Hodge-to-de Rham spectral sequence is well-defined for noncommutative algebras, and it would be natural to ask when it degenerates. A recent conjecture of Kontsevich and Soibelman provides a condition under which the Hodge-to-de Rham spectral sequence degenerates in the noncommutative case. In this paper, the author proves the Kontsevich and Soibelman conjecture under some additional assumptions.
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 5503843
zbMATH Keywords: Hodge theory
zbMATH Keywords: de Rham cohomology
zbMATH Keywords: cyclic cohomology
zbMATH Keywords: Hodge-to-de Rham spectral sequences
author: Dmitry Kaledin
MaRDI profile type: Publication
Wikidata QID: Q57258733
arXiv ID: math/0611623
OpenAlex ID: W2963947855
DOI: 10.4310/PAMQ.2008.V4.N3.A8
description: scientific article