The Non-Commutative Hodge Conjecture and Multiplicities of Modules and Complexes

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This research project concerns topics in commutative and homological algebra and related fields. In commutative algebra, one studies formal systems in which the rules for manipulating equations are the same as in high school algebra but done in more general settings. The field is related to many other areas of pure mathematics, such as number theory, the study of properties of the integers, and algebraic geometry, the study of geometric properties of solutions to systems of polynomial equations. Homological algebra is a branch of algebra related to the field of algebraic topology and is the study of topological spaces, that is, "shapes." A central object of study in homological algebra is that of a complex of modules, which can be thought of an abstraction of the notion of a topological space. This project aims to settle various open conjectures, including one on the possible values of Euler characteristics of certain types of complexes. Here, the Euler characteristic of a complex is a generalization of the integer invariant for polyhedra. The grant will also support graduate students working on affiliated topics.The project involves four main topics: (1) lengths of modules of finite projective dimension and Dutta multiplicities of "tiny complexes," (2) Ulrich modules and lim Ulrich sequences of modules, (3) cones of Betti tables and cohomology tables, (4) the nc-Hodge-conjecture for matrix factorizations. The central goal of (1) is to prove the conjecture that a module of finite projective dimension over a local ring must have length at least as large as the multiplicity of the module. This conjecture admits a generalization involving Euler and Dutta multiplicities of "tiny complexes". Part (2) concerns a primary tool used in tackling these conjectures: Ulrich modules, which are maximal Cohen-Macaulay modules whose multiplicities equal their minimal numbers of generates, and lim Ulrich sequences of modules—sequences of modules that asymptotically approximate the former. The central goal is to construct such things for a larger class of rings than previously known. Part (3) concerns the cones of Betti tables of modules over local rings and cones of cohomology tables of coherent sheaves on projective varieties. Ulrich sheaves and lim Ulrich sequences of sheaves—sheaf theoretic analogues of the module versions of these notions—play an essential role here. The central aim of Part (4) is to prove the non-commutative analogue of the classical Hodge conjecture for the category of matrix factorizations of a hypersurface with an isolated singularity. Each part will be pursued in collaboration with other researchers.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.