Adjacent integrally closed ideals in dimension two (Q1803857)

Let \(R,m\) be a 2-dimensional regular local ring with \(R/m\) algebraically closed. Call ideals \(I \supset J\) adjacent, or say that \(I\) is right above \(J\), if length\((I/J)=1\). Denote the set of all Rees valuations of \(I\) by \(T(I)\). One of the main results is that if \(I \supset J\) are adjacent \(m\)-primary ideals, and \(J_ 1^{s_ 1}J_ 2^{s_ 2} \ldots J_ n^{s_ n}\) is the (Zariski) unique factorisation of the integral closure of \(J\) as a product of simple complete (= integrally closed) ideals, then the difference of multiplicities \(e(J)-e(I)\) is \(\sum s_ j \bigl( w_ j(J)-w_ j(I) \bigr)\) summed over all \(w_ j\) in \(T(J) \backslash T(I)\). The proof uses a multiplicity formula of \textit{J. Lipman} [see Algebraic geometry and commutative algebra, Vol. I, 203-231 (1988; Zbl 0693.13011)]. Other topics include, when \(J\) is simple and complete, the number of complete ideals right above \(J^ n\) (one for \(n=1\), infinite for \(n \geq 2)\) and conditions for \(I\) to be complete when \(I\) is right above \(m^ nJ\) or \(mJ^ n\).