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 $I$.

  • $\begingroup$ 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 $f \in p_jR_{p_j}$ $\endgroup$ Oct 19, 2016 at 13:18
  • $\begingroup$ Improved, thanks for your corrections. $\endgroup$ 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 $\endgroup$ Oct 19, 2016 at 13:42

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.