Free Resolutions

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A core goal in the mathematical areas Algebraic Geometry and Commutative Algebra deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. The solutions form a geometric object. The main idea is to study the rich and beautiful interplay between its geometric and algebraic properties. Closely related to this is the concept of a free resolution, which was introduced by the famous mathematician David Hilbert in two papers in 1890 and 1893. Constructing a free resolution amounts to repeatedly solving systems of polynomial equations. The study of these objects flourished in the second half of the twentieth century, and has seen spectacular progress in the last ten years. The field is very broad, with strong connections and applications to other mathematical areas and string theory. Recent computational methods have made it possible to compute free resolutions by computer. The main research goal in this project is to make significant progress in understanding the structure of free resolutions and their numerical invariants.Free resolutions provide a method for describing the structure of finitely generated modules over a commutative noetherian ring. In the local and the graded cases there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution of the resolved module. Hilbert proved that every finitely generated graded module over a polynomial ring has a finite minimal free resolution. There has been a lot of progress on the properties of finite free resolutions. Much less is known about the properties of infinite free resolutions, which occur abundantly over graded non-linear quotient rings of a polynomial ring. This project deals with infinite minimal free resolutions. It focusses on complete intersections and edge rings. The objectives are: (1) to build Boij-Soderberg Theory over graded complete intersections; (2) to explore applications of matrix factorizations; (3) to construct explicit minimal free resolutions over Clements-Lindstrom rings; (4) to provide formulas for the high differentials of a minimal free resolution over a complete intersection; (5) to study infinite minimal free resolutions over edge rings. The methods to be employed in (1), (2), and (4) are those recently introduced by Eisenbud and Peeva in their work on matrix factorizations for complete intersections. (3) is a continuation of Peeva's work on Clements-Lindstrom rings. The broader impacts of the proposed activities include advising students, organizing conferences, and the PI will continue to serve as an editor of the journal Proceedings of the American Mathematical Society.