Reconstruction of a superscheme from its derived category
A superspace \(X\) is a ringed space \((X,\mathcal{O}_X)\) such that \(\mathcal{O}_X\) is a sheaf of supercommutative superrings and every stalk is a local superring. 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 \(\mathcal{O}_{X,0}\).NEWLINENEWLINERecall that, 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 computed it in some examples. In the present paper the authors define the notion of a \(\mathbb{Z}/2\mathbb{Z}\)-graded tensor triangulated category \(\mathcal{T}\) and show that Balmer's definitions and results work in this setting, that is, there exists a spectrum of \(\mathcal{T}\). They then define perfect complexes on superschemes and show that if \(X\) is a superscheme with an ample family of line bundles, then the category of perfect complexes \(\mathcal{T}=D^{\text{perf}}(X)\) on the superscheme \(X\) is the category of compact objects in the bounded derived category of coherent sheaves \(D^b(X)\) on \(X\) and that the spectrum of \(D^{\text{perf}}(X)\) is isomorphic to \(X\) as a superscheme.
scientific article; zbMATH DE number 6187619