The triangular spectrum of matrix factorizations is the singular locus

Abstract

The singularity category of a ring/scheme is a triangulated category defined to capture the singularities of the ring/scheme. In the case of a hypersurface R / f R/f , it is given by the homotopy category of matrix factorizations [ M F ( R , f ) ] [MF(R,f)] . In this paper, we apply Balmer’s theory of tensor triangular geometry to matrix factorizations by taking into consideration their tensor product. We show that the underlying topological space of the triangular spectrum of [ M F ( R , f ) ] [MF(R,f)] is the singular locus of the hypersurface by using a support theory developed by M. Walker.