K-Theory in categorical geometry

Doctor of PhilosophyDepartment of MathematicsZongzhu LinIn the endeavor to study noncommutative algebraic geometry, Alex Rosenberg defined in [13] the spectrum of an Abelian category. This spectrum generalizes the prime spectrum of a commutative ring in the sense that the spectrum of the Abelian category R − mod is homeomorphic to the prime spectrum of R. This spectrum can be seen as the beginning of “categorical geometry”, and was used in [15] to study noncommutative algebriac geometry. In this thesis, we are concerned with geometries extending beyond traditional algebraic geometry coming from the algebraic structure of rings. We consider monoids in a monoidal category as the appropriate generalization of rings–rings being monoids in the monoidal category of Abelian groups. Drawing inspiration from the definition of the spectrum of an Abelian category in [13], and the exploration of it in [15], we define the spectrum of a monoidal category, which we will call the monoidal spectrum. We prove a descent condition which is the mathematical formalization of the statment “R − mod is the category of quasi-coherent sheaves on the monoidal spectrum of R − mod”. In addition, we prove a functoriality condidition for the spectrum, and show that for a commutative Noetherian ring, the monoidal spectrum of R − mod is homeomorphic to the prime spectrum of the ring R. In [1], Paul Balmer defined the prime tensor ideal spectrum of a tensor triangulated category; this can be thought of as the beginning of “tensor triangulated categorical geometry”. This definition is very transparent and digestible, and is the inspiration for the definition in this thesis of the prime tensor ideal spectrum of an monoidal Abelian category. It is shown that for a polynomial identity ring R such that the catgory R − mod is monoidal Abelian, the prime tensor ideal spectrum is homeomorphic to the prime ideal spectrum