Geometry of tensor triangulated categories [HBNI Th 47]

Abstract

Given a quasi-projective scheme X with an action of a finite group G, consider the tensor triangulated category DG(X). The present study relates the spectrum of this category, as defined by P. Balmer, with the spectrum of the category of all perfect complexes over the scheme X=G. Similarly, consider the category of perfect complexes Dper(X) over a split super-scheme X. It gives isomorphism of the spectrum of Dper(X) with the spectrum of Dper(X0). Here X0 denotes the even part of the super-scheme X ; it is a scheme in the usual sense. The computation of these two spectrums gives examples of two distinct categories with isomorphic Balmer spectrums. The result also shows the limitations of the geometric notion spectrum beyond the category of schemes. This Report suggests some possible generalisations of Balmer's notion of spectrum.