MARUYAMA Masaki Kyoto University. Faculty of Science. Professor, 理学部, 教授 (50025459)
UENO Kenji Kyoto University. Faculty of Science. Professor, 理学部, 教授 (40011655)
MATSUZAWA Jun-ichi Kyoto University. Faculty of Science. Assistant, 理学部, 助手 (00212217)
YOSHIDA Hiroyuki Kyoto University. Faculty of Science. Professor, 理学部, 教授 (40108973)
HIJIKATA Hiroaki Kyoto University. Faculty of Science. Professor, 理学部, 教授 (00025298)
|Budget Amount *help
¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1993 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1992 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1991 : ¥700,000 (Direct Cost : ¥700,000)
Construction of Counter-Examples of Noetherian Rings
Through famous examples due to Akizuki and to Nagata, in commutative Noetherian ring theory, it is well-known that constructing counter-examples is no less important than showing positive results. However, thier methods of construction were complicated and hard to get a general principle.
For these twenty years, the new construction method, originated by Rotthaus, have been and developed and simplified by Ogoma and by Heitmann. This new construction enables us not only to reconstruct easily known examples but to obtain new unknown examples, which give answers to a number of open problems.
Here, we improve this new tool, combining ideas of Nagata, and get the following examples :
1)3-dimensional factorial local domain which is not universally catenaty.
2)2-dimensional normal local domain of characteristic 0 which is not analytically unramified.
3)3-dimensional local domain of characteristic 0 whose derived normal ring is not NOetherian.
Chain Conditions on Ideal-adically Complete Nagata Rings
Greco has constructed the following surprising example :
Example. There exists a semi-local domain (A,m_1, m_2) with an ideal I=P_1 * P_2 (= the intersection of two prime ideals) such that 1) A is complete in I-adic topology, and 2) A/I is excellent, hence universally catenary. But A itself is not universally catenary.
On the other hand, we get the following :
Theorem 1. Let (A,m) be a local ring with an ideal I.Suppose that 1) A is complete in I-adictopology, and 2) A/I is a universally catenary Nagata ring. Then, A itself is universally catenary.
Theorem 2. Let A be a Noetherian domain with a prime ideal P.Suppose that 1) A is complete in P-adic topology, and 2) A/P is a universally catenary Nagata ring. Then, A itself is universally catenary.