Spectrum of some triangulated categories
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 authors, in particular, use a localization result due to Neeman to establish the isomorphism of the structure sheaves.
scientific article; zbMATH DE number 6138039
scientific article