Free Resolutions

Cornell University, Ithaca NY

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project is in the field of Commutative Algebra. The proposed research deals with the structure of graded free resolutions, their numerical invariants, and applications. The general research goal is to study graded free resolutions using a variety of methods from Commutative Algebra, Computational Algebra, and Topological Combinatorics. The project focuses on: Borel ideals and applications of mapping cones over Clements-Lindstrom rings, resolutions of monomial edge ideals, and infinite cellular resolutions. The proposed research involves interdisciplinary approaches and connects Commutative Algebra with the fields of Combinatorics, Computational Algebra, and Topology. The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in two famous papers in 1890 and 1893. He proved that over a polynomial ring (over a field) every finitely generated module has a finite free resolution. If the ring and the module are graded then there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution. The minimal free resolution is graded, and its properties are closely related to the invariants of the module. From another point of view: in essence constructing a free resolution consists of repeatedly solving systems of linear equations. Recent computational methods have made it possible to compute graded free resolutions by computer. For many years, free resolutions have been both central objects and fruitful tools in Commutative Algebra; they have many applications in other mathematical fields.