Homology of Local Rings

Description

Kingston. Ontario, Queen's University, 1969. — 192 p. English. Interactive menu.[Queen's Papers in Pure and Applied Mathematics. No. 20].Preface. This monograph is a survey of some of the "classical" results and techniques as well as more recent developments in the theory of homology of local rings. The homological approach to commutative ring theory officially begins in 1890 with Hilbert's tlleory of syzygies, and is extended through the work of Koszul and Cartan in the early 1950's. These pioneering efforts opened the way for the theory of homological dimension.Table of Contents. Preface. Introduction. Notations and Conventions. Defferential Graded Algebras. Basic definitions. Augmented R-algebras. Construction of R-algebra resolutions. The process of adjoining variables to kill cycles. Derivations and the exact sequence associated with the adjunction of a variable. The Koszul complex, R-sequences, Co-dimension and regular rings. Local complete intersections and the Tate-Zariski resolution. Minimal R-algebra resolutions. Divided power algebras. Augmented (R)-algebras. Acyclic closure of augmented (R)-algebras. The structure and duality of TorR(k,k) and ExtR(k,k). AIgebras, co-algehras, and Hopf algebras. TorR(k,k) as a Hopf algebra. The Duality of ExtR{k,k)and TorR(k,k) Examples. The Poincare Series of a Local Rings. The Poincare series P(R) and the deviations Eq(R). Vq(R,-R) as homology groups. Applications to the deviations Eq(R). Behaviour of the Poincare series upon reduction of the ring modulo a principal ideal. Characterizations of local complete intersections. On The Rationality of the Poincare Series. Eagon's resolution. Massey operations and a theorem of Golod. Applications of Go1od's theorem. An exact couple. Bibliography.