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Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof: For all $ n\geq 1$, $I^{(n)}=(...

Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$. If $a$ that is not a zero divisor of $R/I$ we have ...