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Let $(R, \mathfrak m)$ be a local Cohen-Macaulay domain which is analytically unramified (i.e. the $\mathfrak m$-adic completion of $R$ is reduced). Let $\bar R$ be the integral closure of $R$ in the fraction field of $R$ (so $\bar R$ is module finite over $R$).

Let $C_R:=\{r\in R: r\bar R \subseteq R\}$.

My question is: Can $C_R$ ever be contained in a parameter ideal ?

(An ideal $I$ is called a parameter ideal if $\sqrt I=\mathfrak m$ and $\mu(I)=\dim R$ ).

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  • $\begingroup$ The proof is likely unrelated, but seems related in form to a result in prime characteristic methods about the parameter test ideal -- see Karen E. Smith, Test Ideals in Local Rings, Proposition 6.1 (ams.org/journals/tran/1995-347-09/S0002-9947-1995-1311917-0/…). $\endgroup$
    – walkar
    Jun 3, 2020 at 0:29
  • $\begingroup$ Actually, the proof technique may be fruitful for you. Often, in Cohen-Macaulay rings properties hold for one system of parameters implies they hold for any/all. Can you show that if $C_R$ is contained in one parameter ideal generated by the regular sequence $x_1,...,x_d$, $d=\dim(R)$, then it also holds for $x_1^n,...,x_d^n$ for each $n$? In that case, the conductor must be zero. $\endgroup$
    – walkar
    Jun 3, 2020 at 17:33

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