The Divisor Class Group of a Krull Domain

ISBN-13: 9783642884078 / Angielski / Miękka / 2012 / 150 str.

Robert M. Fossum

Opis

There are two main purposes for the wntmg of this monograph on factorial rings and the associated theory of the divisor class group of a Krull domain. One is to collect the material which has been published on the subject since Samuel's treatises from the early 1960's. Another is to present some of Claborn's work on Dedekind domains. Since I am not an historian, I tread on thin ice when discussing these matters, but some historical comments are warranted in introducing this material. Krull's work on finite discrete principal orders originating in the early 1930's has had a great influence on ring theory in the suc ceeding decades. Mori, Nagata and others worked on the problems Krull suggested. But it seems to me that the theory becomes most useful after the notion of the divisor class group has been made func torial, and then related to other functorial concepts, for example, the Picard group. Thus, in treating the group of divisors and the divisor class group, I have tried to explain and exploit the functorial properties of these groups. Perhaps the most striking example of the exploitation of this notion is seen in the works of I. Danilov which appeared in 1968 and 1970."

Spis treści

I. Krull Domains.- § 1. The Definition of a Krull Ring.- § 2. Lattices.- § 3. Completely Integrally Closed Rings.- § 4. Krull’s Normality Criterion and the Mori-Nagata Integral Closure Theorem.- § 5. Divisorial Lattices and the Approximation Theorem.- II. The Divisor Class Group and Factorial Rings.- § 6. The Divisor Class Group and its Functorial Properties.- § 7. Nagata’s Theorem.- § 8. Polynomial Extensions.- § 9. Regular Local Rings.- § 10. Graded Krull Domains and Homogeneous Ideals.- §11. Quadratic Forms.- §12. Murthy’s Theorem.- III. Dedekind Domains.- § 13. Dedekind Domains and a Generalized Approximation Theorem.- § 14. Every Abelian Group is an Ideal Class Group.- § 15. Presentations of Ideal Class Groups of Dedekind Domains.- IV. Descent.- § 16. Galois Descent.- § 17. Radical Descent.- V. Completions and Formal Power Series Extensions.- § 18. The Picard Group.- § 19. Completions, Formal Power Series and Danilov’s Results..- Appendix I: Terminology and Notation.- Appendix II: List of Results.