Free Resolutions, K-Theory and dg-Categories

University Of Nebraska-Lincoln, Lincoln NE

Investigators

Abstract

This research project concerns topics related to commutative and homological algebra and looks at formal systems of equations. While the rules for manipulating such equations are the same as the ones learned in high-school algebra, the general setting in which they are studied yields powerful tools that put commutative algebra at the heart of much of pure mathematics with applications to many other areas of study, such as number theory (the study of properties of the integers) and algebraic geometry (the study of geometric properties of solutions to systems of polynomial equations). Homological algebra is the branch of algebra related to the field of algebraic topology (the study of topological spaces, i.e."shapes"). A central object of study in homological algebra is that of a complex of modules, which can be thought of an abstraction of the notion of a topological space ("shape"). One specific goal of this project is to settle some long-standing conjectures concerning such complexes and the analogous topological spaces. The award will also support graduate students working on affiliated topics. In more detail, this research project aims to settle various conjectures concerning bounded complexes of finite rank free modules over commutative rings, in particular concerning the possible ranks of the modules appearing in such complexes that have finite length homology. These topics relate to conjectures of Halperin and Carlsson concerning free actions of tori and elementary abelian p-groups on topological spaces. The success of the proposed research on these topics will thus advance our understanding of homological algebra over commutative rings, of algebraic topology, and of the interaction of the two. This research also will pursue several conjectures about smooth and proper dg-categories, focusing on the example of the dg-category of matrix factorizations associated to a hyper-surface with an isolated singularity. Smooth and property dg-categories can be thought of as non-commutative analogues of smooth and proper algebraic varieties. The success of the goals in this area will advance our understanding of the non-commutative analogues of well-known conjectures in classical algebraic geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.