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1ocal domain additional Algebra altitude answer asked Assume Avoidance Conjecture Banach Spaces Catenary Chain Conjecture catenary local domain chain condition Chain Conjecture holds chain of prime Chapter characterizations clear closed complete concerning considered contains Corollary definitions Depth Conjecture depth q Edited example exists a height exists a mcpil fact Finally finite following statements GB-domain given gives H-Conjecture H-domain height one maximal height one prime hence Henselian local domain holds and let hypothesis implies infinitely integral closure integral extension domain known Lemma listed Math maximal ideal mcpil n mentioned minimal prime ideal named Noetherian domain noted principal problem Proceedings proof Proposition prove quasi-local quasi-unmixed question readily REMARK respectively result satisfies the c.c. satisfies the o. h.c. c. satisfies the s.c. c. saturated chain shown shows Spec statements are equivalent taut Theorem Theory unmixed Upper Conjecture