The focus of this project is derived categories arising in commutative algebra. The investigator and his collaborators bring to bear a point of view on triangulated categories fashioned in algebraic topology to establish new results on, and unearth surprising connections among, classical invariants of modules over commutative rings. Another aspect of the proposed research concerns derived categories as a whole, and is inspired by recent spectacular developments in the representation theory of finite dimensional algebra, due to Rouquier: extending earlier work of Konstevich, Bondal and Van den Bergh, and others, on derived categories of coherent sheaves, Rouquier settled a long standing question concerning Auslander's representation dimension, and also an important conjecture of Benson concerning modular representations of finite groups. To this end, Rouquier introduced a notion of a dimension for derived categories, and demonstrated its relevance to representation theory and algebraic geometry. A crucial step Rouquier's work is a lower bound on the derived category of the exterior algebra on a vector space over a field of characteristic two, which is an example of a zero-dimensional commutative complete intersection ring. The project will investigate the dimension of derived categories of general commutative rings, with emphasis on locally complete intersections. This brings in, and contributes to the study of, Hochschild cohomology of commutative algebra. The results obtained will serve to clarify that relationship between dimension and classical invariants of rings. Techniques developed here would have an impact on developments in representation theory and algebraic geometry.
It has long been recognized that derived categories, which were introduced by Verdier in the mid-sixties and in the context of algebraic geometry and commutative algebra, provide a convenient milieu for doing homological algebra. However, it has been realized over the last fifteen years that derived categories, and more generally, triangulated categories, are also exceptionally well-suited for expressing (and proving!) results in diverse subjects in mathematics and mathematical physics. This has had the effect that methods developed in one field have often profoundly influenced a host of others, and generated new interactions among them. One of the main components of the research outlined in this proposal seeks to exploit a elementary notion arising in algebraic topology, namely, the idea of `building' objects out of a given collections of objects in a triangulated category, and bring them to be bear on problems in commutative algebra. The project promises to have an impact on the study of derived categories in general, and, in particular, on geometry, representation theory, and algebraic topology.
