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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cyclic purity versus purity in excellent Noetherian rings

Author: Melvin Hochster
Journal: Trans. Amer. Math. Soc. 231 (1977), 463-488
MSC: Primary 13D99
MathSciNet review: 0463152
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Abstract: A characterization is given of those Noetherian rings R such that whenever R is ideally closed ($ \equiv$ 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 $ (A,m)$ of each local ring of R at a maximal ideal has the following two equivalent properties:

(i) For each integer $ N > 0$ there is an m-primary irreducible ideal $ {I_N} \subset {m^N}$.

(ii) Either $ \dim \;A = 0$ and A is Gorenstein or else depth $ A \geqslant 1$ and there is no $ P \in {\text{Ass}}(A)$ such that $ \dim (A/P) = 1$ and $ (A/P) \oplus (A/P)$ 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 $ \leqslant 1$ 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 $ R \to S$ such that R is pure in S but $ R[[T]]$ is not even cyclically pure in $ S[[T]]$.

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Article copyright: © Copyright 1977 American Mathematical Society