completion and localization

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Let $S$ be the polynomial ring $k[x_1,..,x_n]$ over a field $k$, $M=langle x_1,...,x_n rangle$, and let $R=S/I$ for some ideal $I$ of $S$. We know that the completion of $S$ with respect to $M$ is the formal power series ring $mathcalS=k[![x_1,...,x_n]!]$ and the completion of $R$ is $mathcalR=mathcalS/ImathcalS$. Now let $a$ be a non zero element of $R$ and $R_a$ be the localization of $R$ with respect to $a$. Is it true that $widehatR_a$ is the lozalization of $mathcalR$ with respect to $a$, i.e. $widehatR_acongmathcalR_a$ ?




commutative-algebra




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    Let $S$ be the polynomial ring $k[x_1,..,x_n]$ over a field $k$, $M=langle x_1,...,x_n rangle$, and let $R=S/I$ for some ideal $I$ of $S$. We know that the completion of $S$ with respect to $M$ is the formal power series ring $mathcalS=k[![x_1,...,x_n]!]$ and the completion of $R$ is $mathcalR=mathcalS/ImathcalS$. Now let $a$ be a non zero element of $R$ and $R_a$ be the localization of $R$ with respect to $a$. Is it true that $widehatR_a$ is the lozalization of $mathcalR$ with respect to $a$, i.e. $widehatR_acongmathcalR_a$ ?




    commutative-algebra




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      Let $S$ be the polynomial ring $k[x_1,..,x_n]$ over a field $k$, $M=langle x_1,...,x_n rangle$, and let $R=S/I$ for some ideal $I$ of $S$. We know that the completion of $S$ with respect to $M$ is the formal power series ring $mathcalS=k[![x_1,...,x_n]!]$ and the completion of $R$ is $mathcalR=mathcalS/ImathcalS$. Now let $a$ be a non zero element of $R$ and $R_a$ be the localization of $R$ with respect to $a$. Is it true that $widehatR_a$ is the lozalization of $mathcalR$ with respect to $a$, i.e. $widehatR_acongmathcalR_a$ ?




      commutative-algebra




      share|cite|improve this question













      Let $S$ be the polynomial ring $k[x_1,..,x_n]$ over a field $k$, $M=langle x_1,...,x_n rangle$, and let $R=S/I$ for some ideal $I$ of $S$. We know that the completion of $S$ with respect to $M$ is the formal power series ring $mathcalS=k[![x_1,...,x_n]!]$ and the completion of $R$ is $mathcalR=mathcalS/ImathcalS$. Now let $a$ be a non zero element of $R$ and $R_a$ be the localization of $R$ with respect to $a$. Is it true that $widehatR_a$ is the lozalization of $mathcalR$ with respect to $a$, i.e. $widehatR_acongmathcalR_a$ ?





      commutative-algebra





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      edited Aug 4 at 20:22









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      asked Jul 31 at 20:24









      greenspider

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