Canonical deformations of de Rham complexes
type: Item
instance of: scholarly article
title: Canonical deformations of de Rham complexes
zbMATH Open document ID: 0859.16022
author: Amnon Yekutieli
published in: Advances in Mathematics
publication date: 1996-03-31T00:00:00Z
review text: Let \(\Omega^\bullet_{A/k}\) be the de Rham complex of the commutative polynomial ring \(A=k[x_1,\dots,x_n]\) relative to a field \(k\). The author investigates a special deformation of \(\Omega^\bullet_{A/k}\) in the category of differential graded algebras that appeared in the paper of \textit{S. Woronowicz} [Commun. Math. Phys. 122, No. 1, 125-170 (1989; Zbl 0751.58042)]. In fact, such a deformation \(\Omega^\bullet_q\) can be considered as an analog of de Rham complex of the non-commutative multi-parameter quantum deformation \(A_q\) of \(A\) in the sense of \textit{M. Artin} et al. [Commun. Pure Appl. Math. 44, No. 8/9, 879-895 (1991; Zbl 0753.17015)]. Using the formalism of duality developed by the author [J. Algebra 153, No. 1, 41-84 (1992; Zbl 0790.18005)], he describes an integration over \(A_q\) by means of a canonical isomorphism \(\int:\text{Ext}^n_{A_q}(k,\Omega^n_q)\to k\) induced by the natural extension of the standard contraction of differential forms along the Euler vector field of the de Rham complex \(\Omega^\bullet_{A/k}\) to its deformation \(\Omega^\bullet_q\). More generally, similar results hold in the case where \(A\) is the skew homogeneous coordinate ring.
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 13D25
zbMATH DE Number: 848988
zbMATH Keywords: dualizing complexes
zbMATH Keywords: equivariant sheaves
zbMATH Keywords: de Rham complexes
zbMATH Keywords: categories of differential graded algebras
zbMATH Keywords: non-commutative multi-parameter quantum deformations
zbMATH Keywords: integration
zbMATH Keywords: differential forms
zbMATH Keywords: homogeneous coordinate rings
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: aima.1995.1056
OpenAlex ID: W2045921486
DOI: 10.1006/AIMA.1995.1056
description: scientific article; zbMATH DE number 848988