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LSpice
LSpice
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Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of (https://en.m.wikipedia.org/wiki/Symbolic_power_of_an_idealsymbolic power) of $I$. It is well-known that $I^n \subseteq I^{(n)}$.

Is it true that $I/I^2$ is $R$-free if and only if $I/I^{(2)}$ is $R$-free?

Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of (https://en.m.wikipedia.org/wiki/Symbolic_power_of_an_ideal) $I$. It is well-known that $I^n \subseteq I^{(n)}$.

Is it true that $I/I^2$ is $R$-free if and only if $I/I^{(2)}$ is $R$-free?

Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.

Is it true that $I/I^2$ is $R$-free if and only if $I/I^{(2)}$ is $R$-free?

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uno
uno
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$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$

Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of (https://en.m.wikipedia.org/wiki/Symbolic_power_of_an_ideal) $I$. It is well-known that $I^n \subseteq I^{(n)}$.

Is it true that $I/I^2$ is $R$-free if and only if $I/I^{(2)}$ is $R$-free?