Let (T, M, K) be a quasilocal domain with maximal ideal M and residue field K, phi: T --> K the natural surjection, and R the pullback phi-1(D), where D is a subring of K. It is shown that R[[X]] is catenarian if and only if T[[X]] and D[[X]] are each catenarian. We also construct a non-Noetherian domain R such that dim(R) > 1 and R[[X1,...,X(n)]] is catenarian for each integer n greater-than-or-equal-to 1. This work leads to the question of determining the field extensions k subset-of K such that Spec(K[[X1,...,X(n)]]) --> Spec(k[[X1,...,X(n)]]) is a homeomorphism for each integer n greater-than-or-equal-to 1. It is shown that any such extension must be purely inseparable; the converse holds if K is a finitely generated extension of k.
Anderson, D.f., Dobbs, D.e., Fontana, M., Khalis, M. (1992). CATENARITY OF FORMAL POWER-SERIES RINGS OVER A PULLBACK. JOURNAL OF PURE AND APPLIED ALGEBRA, 78(2), 109-122 [10.1016/0022-4049(92)90089-X].
CATENARITY OF FORMAL POWER-SERIES RINGS OVER A PULLBACK
FONTANA, Marco;
1992-01-01
Abstract
Let (T, M, K) be a quasilocal domain with maximal ideal M and residue field K, phi: T --> K the natural surjection, and R the pullback phi-1(D), where D is a subring of K. It is shown that R[[X]] is catenarian if and only if T[[X]] and D[[X]] are each catenarian. We also construct a non-Noetherian domain R such that dim(R) > 1 and R[[X1,...,X(n)]] is catenarian for each integer n greater-than-or-equal-to 1. This work leads to the question of determining the field extensions k subset-of K such that Spec(K[[X1,...,X(n)]]) --> Spec(k[[X1,...,X(n)]]) is a homeomorphism for each integer n greater-than-or-equal-to 1. It is shown that any such extension must be purely inseparable; the converse holds if K is a finitely generated extension of k.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
