Dg categories, Hochschild homology, and modules of finite projective dimension

About This Grant

This project aims to settle various open conjectures in the field of commutative homological algebra. The award will also support graduate students working on affiliated topics. Commutative algebra is the study of formal systems in which the rules for manipulating formulas and equations are the same as one learned in high school algebra, but in a vastly more general setting. The field is thus at the heart of much of pure mathematics and is related to many other areas of study, such as number theory and algebraic geometry. Homological algebra is a branch of algebra related to the field of algebraic topology, which, in turn, is the study of topological spaces, i.e., "shapes". In this project, the principal investigator will carry out research in commutative algebra, broadly defined, with a focus on topics related to so-called "dg categories" and modules of finite projective dimension. More precisely, the principal investigator, in collaboration with Michael Brown, Srikanth Iyengar, Linquan Ma, Keller VandeBogert, and others, will pursue research on the following topics: (1) non-commutative analogues for dg categories of several classical conjectures for smooth varieties; (2) duality in the context of Hochschild homology of commutative rings and schemes; (3) homological properties of modules of finite projective dimension; (4) Ulrich modules and sheaves; and (5) matrix factorization. A central goal of (1) is to study non-commutative versions of the Hodge conjecture and the related Standard Conjectures of Grothendieck. The main goal of (2) is to establish and explore the consequences of what might be thought of as Poincare duality for the Hochschild homology (and the periodic cyclic homology) of commutative rings and schemes. Project (3) will explore, among other things, the length conjecture, which predicts that a module of finite projective dimension must have length at least as large as the multiplicity of the ring. Ulrich modules and sheaves, the topic of (4), have extremely special properties, and the mere existence of a single such module/sheaf for a ring/scheme has dramatic consequences. Although it was recently proven by the investigator and his collaborators that certain local rings do not admit any Ulrich modules, the analogous question for Ulrich sheaves on projective varieties remains open; settling this is a goal of this research. A matrix factorization, the topic of (5), is a pair of square matrices with entries in a commutative ring whose product (in either order) gives a scalar matrix. Under certain assumptions, there is a predicted lower bound on the smallest possible size of such a matrix factorization (known as the Buchweitz-Greuel-Schreyer Conjecture), and this project will aim to settle this conjecture at least in certain cases. 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.