Homotopical methods and cohomological supports in local algebra

This research project investigates singularities in commutative algebra through the lens of various homological constructions. Commutative algebra serves as a local model for algebraic geometry; the latter is a central branch of modern mathematics, where the focus is on the geometric properties of solutions sets to systems of polynomial equations (objects ubiquitous throughout mathematics). In commutative algebra, one examines algebraic structures known as (local) rings, which provide insights into both smooth and singular points on the solution sets explored in algebraic geometry. Since its inception in the 1950's, homological algebra has been instrumental in describing singularities in local commutative algebra, offering valuable ring-theoretic insights. This project aims to leverage tools from homological algebra to gain a deeper understanding of commutative rings, thereby shedding light on singularities in local commutative algebra and algebraic geometry. Moreover, this strategy advances commutative algebra by drawing from the wealth of ideas in algebraic topology and representation theory, areas that have leaned heavily on developing homological methods for application in their respective fields, and (further) revealing connections between commutative algebra and these areas. The award will also be used to fund graduate students interested in the proposed research program. More specifically, the PI will apply an array of homological tools to glean insights in commutative algebra; the two central tools being homotopical methods and cohomological support. Applications of both theories have been far-reaching in commutative algebra, and the proposed research program will further hone this machinery with an eye toward breakthroughs in local algebra. In particular, a primary focus of this project will be on studying the structural properties of certain triangulated categories arising in commutative algebra. Projects in this direction include gaining traction on a long-standing conjecture of Quillen, unifying results in prime characteristic commutative algebra by understanding generators in the bounded derived category, and relating the dimension of certain cohomological supports to classical invariants in commutative algebra with the hopes of making progress on questions of Avramov and Jacobsson. The structural properties of resolutions is another primary focus of the project. In this direction, the PI will extend Koszul duality phenomena in local algebra to include non-quadratic and non-graded algebras. This will be achieved using A-infinity structures on resolutions to introduce and study a class of rings (or more generally ring maps) generalizing the class of classical Koszul algebras. 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.