Descent theory for semiorthogonal decompositions
type: Item
instance of: scholarly article
title: Descent theory for semiorthogonal decompositions
zbMATH Open document ID: 1300.14019
author: A. D. Elagin
published in: Sbornik: Mathematics
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.2881
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 13D09
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6084149
zbMATH Keywords: derived category
zbMATH Keywords: semiorthogonal decomposition
zbMATH Keywords: descent theory
zbMATH Keywords: algebraic variety
MaRDI profile type: Publication
arXiv ID: 1206.2881
DOI: 10.1070/SM2012V203N05ABEH004238
review text: In the paper under review, the author describes a method to construct a semiorthogonal decomposition of the equivariant derived category of a quasi-projective \(k\)-scheme \(X\) under the action of an algebraic group \(G\), starting from a semiorthogonal decomposition on the standard derived category.NEWLINENEWLINESuppose that \(G\) acts on \(X\) via an action \(a:G\times X\longrightarrow X\), and let \(p_{2}:G\times X\longrightarrow X\) be the projection. An equivariant sheaf on \(X\) is a pair \((F,\theta)\) where \(F\) is a sheaf on \(X\) and \(\theta:p_{2}^{*}F\longrightarrow F\) is a morphism verifying some cocycle condition depending on \(a\). We write \(D(X)\) (resp. \(D^{G}(X)\)) for the derived category of quasi-coherent sheaves (resp. equivariant quasi-coherent sheaves), \(D^{b}(X)\) (resp. \(D^{b}(\text{coh}^{G}(X))\)) for the bounded derived category of coherent sheaves (resp. equivariant coherent sheaves) and \(D^{\text{perf}}(X)\) (resp. \(D^{\text{perf},G}(X)\)) for the category of perfect complexes in \(D(X)\) (resp. in \(D^{G}(X)\)).NEWLINENEWLINEIn section 2, the author provides a description of \(D^{G}(X)\) using the language of cosimplicial categories and comonads. A comonad on a category \(\mathcal{C}\) is a triple \(\mathrm{T}=(T,\epsilon,\delta)\) given by a functor \(T:\mathcal{C}\longrightarrow\mathcal{C}\) and two natural transformation \(\epsilon:T\longrightarrow id_{\mathcal{C}}\) and \(\delta:T\longrightarrow T^{2}\) satisfying some commutativity condition. A comodule over a comonad \(\mathrm{T}\) on \(\mathcal{C}\) is a pair \((F,h)\) where \(F\) is an object of \(\mathcal{C}\), and \(h:F\longrightarrow TF\) is a morphism verifying some commutativity condition. The comodules over the comonad \(\mathrm{T}\) form a category \(\mathcal{C}_{\mathrm{T}}\). If \(\mathcal{C}\) is abelian and \(\mathrm{T}\) is a comonad on it, then \(\mathcal{C}_{\mathrm{T}}\) is abelian. In general, if \(\mathcal{C}\) is triangulated, we do not have \(\mathcal…
reviewed by: Arvid Perego
description: scientific article; zbMATH DE number 6084149