Knörrer Periodicity and Bott Periodicity

Authors

Michael K. Brown, University of Nebraska-LincolnFollow

Date of this Version

5-2015

Document Type

Article

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Mark E. Walker. Lincoln, Nebraska: May, 2015

Copyright (c) 2015 Michael K. Brown

Abstract

The main goal of this dissertation is to explain a precise sense in which Knörrer periodicity in commutative algebra is a manifestation of Bott periodicity in topological K-theory. In Chapter 2, we motivate this project with a proof of the existence of an 8-periodic version of Knörrer periodicity for hypersurfaces defined over the real numbers. The 2- and 8-periodic versions of Knörrer periodicity for complex and real hypersurfaces, respectively, mirror the 2- and 8-periodic versions of Bott periodicity in KU- and KO-theory. In Chapter 3, we introduce the main tool we need to demonstrate the compatibility between Knörrer periodicity and Bott periodicity: a homomorphism from the Grothendieck group of the homotopy category of matrix factorizations associated to a complex (real) polynomial f into the topological K-theory of its Milnor fiber (positive or negative Milnor fiber). A version of this map first appeared in the setting of complex isolated hypersurface singularities in the paper “An Index Theorem for Modules on a Hypersurface Singularity", by Buchweitz and van Straten. We show that, when f is non-degenerate quadratic (over the real or complex numbers), this map recovers the Atiyah-Bott-Shapiro construction in topology. In Chapter 4, we prove that when f is a complex simple plane curve singularity, this homomorphism is injective.