Cyclic purity versus purity in excellent Noetherian rings

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 $P \in {\operatorname {Ass}}(A)$ 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 $S[[T]]$.