The work of Jan-Erik Roos on the cohomology of commutative rings

Abstract

This is an attempt to present one aspect of the work of Jan-Erik Roos. A glance at the list of publications reveals three clearly deÞned periods in his life as an algebraist. During the Þrst one he studied abelian categories, obtaining fundamental results on derived functors of inverse limits. They are contained in (3), (5)-(9), (11)-(17), (19)-(21). In the second period he focused on the homological theory of non-commutative rings, producing methods and results of lasting interest, among them a truly classic theorem—the determination of the global dimension of Weyl algebras. The papers (4), (18), (22)-(26), and (31) (treating related questions from commutative ring theory) contain the results of that period. Bjork (Bj) has given an overview in the context of contemporary and subsequent research. The work discussed here starts in the mid-1970s, when Jan-Erik turned to homo- logical problems on Þnitely generated modules over commutative noetherian local (or graded) rings. He has produced fascinating results on the structure of free res- olutions of modules of inÞnite projective dimension, and has investigated deep and mysterious links between homological properties of commutative rings and topo- logical spaces. His study of numerical invariants encoded in Poincare series, and of algebraic invariants determined by Yoneda products and by homology products, brings an unusual degree of integration between these components. This highly original and technically dicult work also brings to mind other qual- ities, such as elegance and optimism. A quick look at the many rings appearing on the following pages shows that there is nothing contrived about his 'examples': they are deÞned by the kind of simple expressions in few variables that one scrib- bles on a piece of paper to have something 'concrete' to play with. Appearances notwithstanding, some of these rings have been craftily constructed to posess a desired property. Others have been found by sifting, with the determination of a gold prospector, through computations of (literally!) thousands of examples. The purpose in this survey is to provide a guide to some of Jan-Erik's Þnds. A dierent perspective of work completed by 1985 can be found in the article of Anick and Halperin (AH). Connection with topology, which were discussed early on by Lemaire (Le) in a Bourbaki talk, and recently by Hess (He) in historical context,