Coxeter transformations of the derived categories of coherent sheaves
type: Item
instance of: scholarly article
title: Coxeter transformations of the derived categories of coherent sheaves
zbMATH Open document ID: 1307.14020
author: Xinhong Chen
author: Ming Lu
published in: Journal of Algebra
publication date: 2014-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.2368
reviewed by: Andreas Hochenegger
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 6329084
zbMATH Keywords: Coxeter transformation
zbMATH Keywords: Jordan normal form
zbMATH Keywords: rational surface
zbMATH Keywords: toric variety
MaRDI profile type: Publication
OpenAlex ID: W2963075737
arXiv ID: 1211.2368
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DOI: 10.1016/J.JALGEBRA.2013.10.003
review text: The authors study the Coxeter transformation of the derived category \(\mathcal{D}^b(X)\) of a smooth, projective \(n\)-dimensional variety \(X\). This transformation is given by \(S_X[-1] = - \otimes \omega_X[n-1]\). In the article, a short survey is given about the representation-theoretic origins of these transformations: They were introduced by Coxeter to study compact Lie groups. Passing representation theory of finite-dimensional algebras, where they are known as Auslander-Reiten translations, the found their way to algebraic geometry by tilting theory.NEWLINENEWLINEHere, the corresponding linear map on the Grothendieck group \(K_0(X)_{\mathbb Q}\) is the main object of study, in the case that its rank \(m\) is finite. The main result is that its characteristic polynomial is \((x+(-1)^n)^m\). Therefore, one obtains a necessary condition that \(\mathcal{D}^b(X)\) is equivalent to the derived category of a finite-dimensional algebra.NEWLINENEWLINEAnother central topic is the Jordan canonical form of the Coxeter transformation. The authors can describe it expicitly in the case of toric varieties in terms of the combinatorics of the associated fans, whenever the (anti)canonical divisor is a Lefschetz element (e.g.\ if ample). The key ingredients to the proof are the hard Lefschetz theorem and that the cohomology \(H^\bullet(X,\mathbb Z)\) is concentrated in even degrees. As an example, the Jordan blocks for all toric del Pezzo surfaces and toric Fano three-folds are given.NEWLINENEWLINEThese ideas are also applied to rational surfaces \(X\), where \(K_X\) is a Lefschetz element if and only if the rank of the Picard group is not 10. This induces also a dichotomy of the structure of the Jordan blocks.NEWLINENEWLINEFinally, as an application, a formula is given for the Jordan normal form of tensor products of invertible matrices.
description: scientific article; zbMATH DE number 6329084