Divisions by the fractional powers of an ideal generated by a regular sequence in the ring of germs of analytic functions at the origin of \(\mathbb{C}^2\) (Q2778344)

Let \(A\) be an analytic algebra and \(I\) an ideal in \(A\). Let \(\overline{\nu_I}\) be the following function (D. Rees): \(\overline{\nu_I}:A \to Q_+\), \(\overline{\nu_I}(h):= \sup\{p/q\mid h\in I^p, h\in A\}\). This function characterises the integral closure of \(I^n\), \(n\) integer, namely, \(\overline{I^n}=\{h\in A\mid \overline{\nu_I}(h)\geq n\}\). In the above characterisation if we replace \(n\) by \(\alpha \in Q_+\), one obtains the definition of a fractional power of \(I\). If \(I\) is an ideal generated by a regular sequence in a Noetherian ring, we have the following formula \(\overline{I^{n+1}}\cap I^n=\overline{I}I^n\). NEWLINENEWLINENEWLINEThe author extends this formula in the two dimensional case, for \(A\) the ring of germs of holomorphic functions at the origin of \(\mathbb{C}^2\). Namely, if \(I=(f,g)\) is an ideal in \(A\), generated by a regular sequence and \(0< \alpha \leq 1, \alpha \in Q_+\), \(n\) a positive integer, the author shows that \(\overline{I^{n+\alpha}}\cap I^n=\overline{I^{\alpha}}I^n\).