This project is in the field of commutative algebra. The main research goal is to understand the structure of minimal free resolutions over complete intersection ring. The idea is to describe the asymptotic structure of the generic Eisenbud operators, and then use it in order to obtain the differentials. The proposed research also includes a topic involving interdisciplinary approaches connecting commutative algebra with the fields of combinatorics and topology. The PI will co-organize two conferences in commutative algebra, which will provide a forum for discussion of recent developments and foster research in new directions.
Research on free resolutions is a core and beautiful area in commutative algebra. It contains a number of challenging and important conjectures and open problems. 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 resolution. 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. Its properties are closely related to the invariants of the resolved module. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra; they also have applications in other mathematical fields.
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 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. Recent computational methods have made it possible to compute finite free resolutions by computer. 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 quotient rings of a polynomial ring. Such resolutions can have very complicated structure. For example, Anick's solution to the Serre-Kaplansky Problem (which was one of the central problems in Commutative Algebra for many years) provides an example of a resolution with irrational generating function. Thus, one has to impose some conditions on the ring in order to obtain structural results. The next natural class to consider after polynomial rings are the quotient rings that are complete intersections. These rings play important role in Algebra. The original interest in free resolutions over a (local or graded) complete intersection R came from group cohomology. Perhaps with this motivation Tate gave in 1957 an elegant description of the minimal R-free resolution of the residue field. In particular, the generating function of that resolution is rational. In 1974 Gulliksen proved the rationality of the generating function of the minimal R-free resolution of any finitely generated R-module. Since then several results on the numerical invariants (the Betti numbers) of such resolutions have been obtained. All of them indicate that the resolutions are highly structured. The first natural case to consider is when R is a hypersurface, which is the simplest type of complete intersection. In 1980 Eisenbud introduced the concept of matrix factorization for one element f in a regular local ring S and gave a description of the asymptotic structure of all minimal free resolutions over the hypersurface S/f. Matrix factorizations have applications in many fields of mathematics: for the study of cluster algebras, Cohen-Macaulay modules, knot theory, moduli of curves, quiver and group representations, and singularity theory. A major advance was made by Orlov, who showed that matrix factorizations could be used to study Kontsevich's homological mirror symmetry by giving a new description of singularity categories. Starting with Kapustin and Li, physicists discovered amazing connections with string theory. The structure of minimal free resolutions over complete intersections remained mysterious for many years. The main outcome of this project is that Eisenbud and Peeva introduce the concept of matrix factorization for complete intersection rings (of any codimension) and prove that it suffices to describe the asymptotic structure of all minimal free resolutions over complete intersections. They introduce some completely new ideas, and also use the previously known technique of ci-operators.Their proofs are constructive and corresponding algorithms are already implemented in the algebra software system Macaulay2. The obtained results and new ideas make significant progress on understanding infinite minimal free resolutions. The results have been reported at major international conferences. Broader Impacts include: (1) Peeva organized a book-project on writing expository papers by 41 authors. The papers were collected in the book ``Commutative Algebra" edited by Peeva and published by Springer in 2013. (2) The Principal Investigator is an editor of the journal Proceedings of the American Mathematical Society. She was also an editor of a special volume in honor of J. Herzog of the Journal of Commutative Algebra. (3) Peeva served as the Faculty Advisor for undergraduate students at Cornell University. The grant provided summer support to graduate students working on their research projects. (4) The Principal Investigator coorganized the following three conferences. Jointly with L. Ein, D. Eisenbud, and G.Farkas, she organized ``Syzygies in Algebraic Geometry, with an exploration of a connection with String Theory" at Banff IRS (Banff International Research Station). Jointly with L. Avramov and D. Eisenbud, she organized ``Representation Theory, Homological Algebra, and Free Resolutions" at MSRI (Mathematical Sciences Research Institute). She also organized a workshop on Syzygies for junior mathematicians, which took place at Cornell University. (5) Peeva gave several plenary talks at international mathematical conferences in Canada, France, and USA.
