This is a proposal in commutative algebra from the point of view of cohomology and homotopy theory, with strong connections to representation theory of finite dimensional algebras. Many of the proposed topics of research are inspired by homotopy theory, which also serves as a source of intuition, and of techniques, for dealing with them. Broadly speaking, the recurrent theme in the project is the interplay of three topics: commutative algebra; differential graded algebras and modules; and triangulated categories. While differential graded algebras and triangulated category methods have long been used successfully in commutative algebra, the PI and his collaborators, among others, have been adapting classical constructions and methods from commutative algebra (for example, local cohomology, and derived completions) to the context of differential graded algebras and triangulated categories, with striking returns in the representation theory of finitely dimensional algebra, notably, group algebras, and in commutative algebra itself. This proposal seeks to further develop these, and in new directions.
A number of rather diverse algebraic structures---this means, loosely speaking, not directly involving any notions from calculus like continuity or rate of change---have been developed in mathematics to model phenomenon in the (physical) world. Two examples particularly relevant to this proposal are groups and their representations, that are remarkably well-adapted to capture phenomenon involving symmetry, and rings, especially those that arise as rings of functions on various geometric objects like manifolds or solution sets of polynomial equations. However, it was only a few years ago that it was realized that if we enhance a ring by the most primitive structure from calculus, namely, a derivative, then it becomes possible to uniformly capture much of the information encoded in these various algebraic structures. Interestingly, these still rather mysterious hybrid structures, called differential graded algebras, emerged as important tools in algebraic topology already in the early 1950s. Differential graded algebras can be seen as bridges that relate various algebraic and geometric contexts in mathematics and mathematical physics. This has had the effect that methods developed in one field have profoundly influenced a host of others, and new connections among them have been discovered. Differential graded algebras are, in general, rather complicated, but there are interesting classes that appear to be amenable to methods developed in commutative ring theory, a classical topic with a large and well-developed body of tools and techniques. The broad aim of this proposal is to investigate differential graded algebras from this perspective. Besides deepening our knowledge of them, the proposed research is expected to have impact on commutative algebra, representation theory, and related fields.
