Spectrum of some triangulated categories
type: Item
instance of: scholarly article
title: Spectrum of some triangulated categories
zbMATH Open document ID: 1258.14020
author: Umesh V. Dubey
author: Vivek M. Mallick
published in: Journal of Algebra
publication date: 2013-02-21T00:00:00Z
review text: For any tensor triangulated category, \textit{P. Balmer} [J.\ Reine Angew.\ Math.\ 588, 149--168 (2005; Zbl 1080.18007)] defined a locally ringed space, called the spectrum and, in particular, proved that the spectrum of the triangulated category of perfect complexes \(\mathcal{D}^{\text{per}}(X)\) (which coincides with the bounded derived category if the variety is smooth) on any topologically Noetherian scheme \(X\) is in fact \(X\). There exist other examples where the spectrum is known, but in general it is fairly difficult to determine. The paper under review computes the spectrum in the following two cases. Firstly, let \(X\) be a quasi-projective smooth variety over a field on which a finite group \(G\) acts. We either assume that the characteristic \(p\) of the field is \(0\) or that the order of \(G\) is coprime to \(p\) if \(p\) is positive. Recall that in this setting an equivariant sheaf is a sheaf \(F\) on \(X\) equipped with isomorphisms \(F\cong g^*F\) for any \(g\in G\) satisfying a cocycle condition. The authors prove that the spectrum of the bounded derived category of coherent \(G\)-equivariant sheaves and the spectrum of \(\mathcal{D}^{\text{per}}(X/G)\) are isomorphic as locally ringed spaces. The isomorphism is induced by \(\pi^*\), where \(\pi: X \to X/G\) is the quotient map. The proof uses some results from representation theory. Secondly, consider a superspace \(X\), that is, \((X,\mathcal{O}_X)\) is a ringed space such that every stalk is a local ring and the sections of \(\mathcal{O}_X\) over any open subset form a supercommutative ring. A superspace is a superscheme if the even part \((X,\mathcal{O}_{X,0})\) is a scheme and \(\mathcal{O}_{X,1}\) is a coherent sheaf over \(\mathcal{O}_{X,0}\). The authors prove that if \(X\) is a split quasi-compact, quasi-separated superscheme, then \(X_0\), the \(0\)-th part of the superscheme, and the spectrum of \(\mathcal{D}^{per}(X)\) are isomorphic as locally ringed spaces. For the proof the author…
reviewed by: Pawel Sosna
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6138039
zbMATH Keywords: tensor triangular geometry
zbMATH Keywords: spectrum
zbMATH Keywords: equivariant sheaves
zbMATH Keywords: superschemes
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.04.030
OpenAlex ID: W2140807452
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DOI: 10.1016/J.JALGEBRA.2012.04.030
description: scientific article; zbMATH DE number 6138039