Cyclic purity versus purity in excellent Noetherian rings
Author:
Melvin Hochster
Journal:
Trans. Amer. Math. Soc. 231 (1977), 463-488
MSC:
Primary 13D99
DOI:
https://doi.org/10.1090/S0002-9947-1977-0463152-5
MathSciNet review:
0463152
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Abstract | References | Similar Articles | Additional Information
Abstract: A characterization is given of those Noetherian rings R such that whenever R is ideally closed ( cyclically pure) in an extension algebra S, then R is pure in S. In fact, R has this property if and only if the completion
of each local ring of R at a maximal ideal has the following two equivalent properties:
(i) For each integer there is an m-primary irreducible ideal
.
(ii) Either and A is Gorenstein or else depth
and there is no
such that
and
is embeddable in A.
It is then shown that if R is a locally excellent Noetherian ring such that either R is reduced (or, more generally, such that R is generically Gorenstein), or such that Ass(R) contains no primes of coheight in a maximal ideal, and R is ideally closed in S, then R is pure in S. Matlis duality and the theory of canonical modules are utilized. Module-theoretic analogues of condition (i) above are, of necessity, also analyzed.
Numerous related questions are studied. In the non-Noetherian case, an example is given of a ring extension such that R is pure in S but
is not even cyclically pure in
.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1977-0463152-5
Article copyright:
© Copyright 1977
American Mathematical Society