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My question is whether the below notion of powers of ideals, which may be regarded as a weakening of the notion of symbolic power, has been already defined in the literature and, in that case, I ask for a reference.

Let $R$ be a commutative Noetherian regular local ring, and let $I\subseteq R$ be a radical ideal. Fix a non-negative integer $n$.

If $I=\cap p_j$ is the primary decomposition of $I$, then set $$ I^{<n>}:=\{f\in R:\ f^n\in p_j^n R_{p_j} \text{ for all } j\}. $$ Geometrically, I think this corresponds to all the sections that vanish generically on $V(I)$ at order $n$, where $V(I)$ denotes the algebraic set defined by $I$.

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  • $\begingroup$ As it is written now, $I^{\langle n \rangle}$ is simply $I$: if $f \in p_jR_{p_j}$, then clearly $f^n \in p_jR_{p_j}$ as well. $\endgroup$
    – pinaki
    Oct 19, 2016 at 12:49
  • $\begingroup$ Unless I am confused, the definition doesn't seem to depend on $n$: since $p_jR_{p_j}$ is prime, $f^n\in p_jR_{p_j}$ iff $f \in p_jR_{p_j}$ $\endgroup$
    – Denis Nardin
    Oct 19, 2016 at 13:18
  • $\begingroup$ Improved, thanks for your corrections. $\endgroup$
    – Alberto Fernandez Boix
    Oct 19, 2016 at 13:34
  • $\begingroup$ This notion has to be related somehow with the complete integral closure of an ideal, see worldscientific.com/doi/abs/10.1142/S0219498811004884 $\endgroup$
    – Alberto Fernandez Boix
    Oct 19, 2016 at 13:42

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Sorry, this $I^{<n>}$ is just the $n$-th symbolic power of the ideal, as defined in Vasconcelos book Computational Methods in Commutative Algebra and Algebraic Geometry, see Definition 3.5.1.

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