By Hiroaki Hikikata
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Additional info for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2
Suppose I = Pi Π · · · Π Pr (= the intersection of prime ideals Pi, Pr). s u c ^ that (A/xA)a is Then, there exists a non-zero χ g J and a € A — Ui=i -P* reduced. 3. As A is known to be a nagata ring (cf. Marot's Theorem), it suffices to show that A is a P-ring. 2) A / a is a P-ring for any non-zero ideal a. Let L be a finite algebraic extension field of Κ and Β be a finite A-algebra s with Q(B) — L. We are to show that L
B &n * normal for any η g M a x ( B ) . 4) Β is J-adically complete, and B/b is a P-ring for any non-zero ideal b.
Rotthaus, we considered the case when A is semi-local and Ρ is being reduced, normal or regular: L o c a l Lifting T h e o r e m . (, , ) Let A be a semi-local ring with an ideal I. Suppose A is I-adically complete and A/1 is a P-ring with Ρ = being reduced, normal or regular. Then A is also a P-nng. However, we have an unfavorable example to general Lifting Problem: E x a m p l e . 3)]) R[[X\] is not a G-ring. There exists a one-dimensional G-ring R such that On the other hand, due to J.
J. Math. 91 (1969), 508-528. C. Rotthaus, Komplettierung semilokaler quasiausgezeichneter Ringe, Nagoya Math. J. 76 (1979), 173-180. C. Rotthaus, Zur Komplettierung ausgezeichneter Ringe, Math. Ann. 253 (1980), 213-226.