The investigator and his collaborators address a number of specific problems in the homological theory of modules over commutative rings, and develop some tools intended for further research in this topic. One example of the first aspect of the project is the proposed study of the asymptotic behaviour of Betti numbers of the Frobenius endomorphism. The paradigm for the results sought is Kunz's theorem: local ring of non-zero characteristic is regular precisely when the Frobenius endomorphism is flat. A second example concerns Koszul algebras: the investigator will build on recent work with Herzog that was motivated by results of Eisenbud, Floystad, and Schreyer on the relationship between linear strands of resolutions over a polynomial ring and modules over its Koszul dual exterior algebra. Some of the techniques proposed are derived from rational homotopy theory, and a guiding light in this endeavour is the "Looking glass principle" of Avramov and Halperin. However, many of the crucial results that govern it have not been pinned down in the desired level of detail. Avramov and the investigator propose a manuscript that fills this gap in the literature. The project will also investigate the role of cellular approximations, staple to topologists, in the context of commutative rings. In the second half of the 19th century, David Hilbert discovered a close relationship between geometric objects called varieties, and certain types of functions defined onthem. The latter form algebraic gadgets called commutative rings. Over the last century, commutative rings have arisen in combinatorics, topology, and other branches of mathematics. They have also found applications in diverse fields like cryptography, pattern recognition, and theoretical physics. This project seeks to apply techinques from 'homological algebra' to study commutative rings. Topology has been a major force in the development of homological algebra, although some of its roots can be traced to geometry. Homological methods have proved remarkably efficacious in tackling problems in commutative algebra; in turn, this has infused new ideas into the subject.
