Homological Behavior of Modules over Commutative Local Rings

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This project aims to contribute towards understanding the homological behavior of modules over commutative rings. Modules are the objects on which a given ring acts: geometrically, they correspond to bundles and, more generally, sheaves on a space. Sega will study certain classes of commutative local rings which extend in several directions the well-studied class of complete intersection rings, and will investigate to what extent the known properties of complete intersections can be translated to a larger context. Earlier work of the PI, in collaboration with D. Jorgensen, has shown that vanishing of cohomology over non-complete intersection rings can have rather non-rigid behavior. The accent in the proposed work is shifted towards finding a common bridge between complete intersections and other good classes of rings. Attention will be paid to vanishing of (co)homology, properties of the minimal free resolutions of modules, Betti numbers and other homologically defined invariants. An important aspect is an effort to show that the classes of rings and the modules considered occur abundantly in a variety of situations of geometric interest.Commutative algebra allows one to encode information regarding solution sets of polynomial equations into objects such as rings and modules, and further understand their structure, using algebraic tools and a recent infusion of techniques from other fields that deal with physical spaces, such as topology. This method of encoding information and its further study is relevant to almost any area of mathematics, and to theoretical physics. In particular, the classes of rings and the properties considered in this project are relevant in the field of algebraic geometry, and some of the methods and outcomes make connections with the field of non-commutative algebra.