The spectrum of a triangulated category and its applications to algebraic geometry

Abstract：

Tensor triangular geometry is an epoch-making theory initiated by Balmer. For a tensor triangulated category, Balmer defined the ringed space called the Balmer spectrum of T, whose underlying set consists of prime thick ideals of T. This theory has been successfully applied to the study of tensor triangulated categories appearing in various areas of mathematics. However, this theory cannot be applied to triangulated categories without tensor structures, and it is badly behaved concerning triangulated functors that do not preserve tensor structures. In this talk, we define the ringed space for a given triangulated category without using tensor structure and develop a tensor-free analog of Balmer’s tensor triangular geometry. Also, we apply the theory to study perfect derived categories of algebraic varieties, Fourier-Mukai partners, and reconstruction problems.