Off-campus UNL users: To download campus access dissertations, please use the following link to log into our proxy server with your NU ID and password. When you are done browsing please remember to return to this page and log out.
Non-UNL users: Please talk to your librarian about requesting this dissertation through interlibrary loan.
Knörrer periodicity and bott periodicity
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.
Subject Area
Mathematics
Recommended Citation
Brown, Michael K, "Knörrer periodicity and bott periodicity" (2015). ETD collection for University of Nebraska-Lincoln. AAI3689620.
https://digitalcommons.unl.edu/dissertations/AAI3689620