RUI: Representation theory and homological algebra over local rings

Fairfield University, Fairfield CT

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI will concentrate her attention on characterizing local Noetherian rings in terms of the behavior of certain subcategories of modules, extending her research to methods from homological algebra and representation theory. The idea stems from a successful description of rings of finite Cohen-Macaulay type, which flourished in the mid 1980s. Results about finite-dimensional algebras are generalized to rings of higher dimension. Such a description was recently extended to rings with a finite Gorenstein type. The PI?s main goal is to understand which categories of modules have the right properties to provide a description of the ring when finiteness conditions are available. To do so, the PI proposes to investigate further the relationship between the properties of the category of totally reflexive modules and those of the base ring. For example, the PI proposes to understand the long standing open problem to characterize which properties of the ring force every totally reflexive module to be free. As totally reflexive modules come from infinite complexes, the investigation naturally includes methods from homological algebra, where the properties of modules are linearized by a possibly infinite approximation. The project is devoted to investigating problems in commutative algebra with the main goal being to understand the set of solutions of polynomial equations in many variables. In the effort to describe such a set of solutions, algebraists study the set of functions defined on the set of solution when a visual intuition is not available. As functions can be added and multiplied, the set of functions has the structure of a commutative ring, of which the most familiar example is the ring of polynomials. It is well established that understanding a ring is tantamount to understanding the category of its modules, which are generalization of vector spaces over fields.