交换局部环和半局部环

COMMUTATIVE LOCAL RINGS AND SEMI-LOCAL RINGS

本文是在W·V·Vasconcelos研究的基础上,将Auslander,Vasconcelos等人的一些结果进行推广,对局部环和半局部环上的模及其同调性质作了进一步刻划。Auslander-Buchsbaum-Nagata用同调代数的方法证明了用其它方法还无法证明的著名定理:每个正则局部环是唯一分解整环。本文采用同调代数与环论相结合方法证得定理:若A是一个不可分的凝聚半局部环,且每个主理想有有限投射维数,则A是最大公因子整环。从某种意义上讲此定理推广了Auslander-Buchsbaum-Nagata定理的结果。

This paper, based on W·V·Vasconcelos' studies, generalized some results of Auslander, Vasconcelos etc and further characterized modules on local rings and semi-local ring and their homological properties. Auslander-Buchsbaum Nagata have proved a famous theorem using homological method, which cannot be proved by other mathematical method yet : Every regular local ring is a UFD. In this paper, it is proved that if A is a indecomposable coherent semilocal ring, and every principal ideal has finite projective demension, then A is a GCD. by using homological method combined with method in ring theory. This theorem generalized Auslander-Buchsbaum- Nagata theorem in some extent.