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Simple complete ideals in two-dimensional regular local rings. (English) Zbl 0878.13015

Let \((R,m)\) be a two-dimensional regular local ring with algebraically closed residue field \(k\) and let \(J\neq m\) be a simple \(m\)-primary complete (= integrally closed) ideal of \(R\) whose order is \(r\) (that is, \(m^r \supseteq J\) and \(m^{r+1} \not\supseteq J\)). Then the main result is stated as follows:
(1) there exists a unique complete ideal of order \(r+1\) adjacent to \(J\) from below,
(2) there are infinitely many simple complete ideals of order \(r\) adjacent to \(J\) from below,
(3) any complete ideal adjacent to \(J\) from below is one of the ideals in either (1) or (2). (An ideal \(I\) is said to be adjacent to \(J\) from below if \(I \subset J\) and \(\text{length}(J/I)=1\).)
It is known that there are one-to-one correspondences between the sets \( A = \{\)simple \(m\)-primary complete ideals of \(R\},\;B = \{\)infinitely near points to \(R\}\) and \( C = \{\)prime divisors of \(R\}\). Let \(S\in B\) and \(w\in C\) be the corresponding objects to \(J\). Then, as a corollary, we obtain one-to-one correspondences \[ \begin{aligned} &A(J)=\{\text{simple complete ideals adjacent to \(J\) from below}\}\\ \leftrightarrow \quad &B(S)=\{\text{first quadratic transformations of }S\}\\ \leftrightarrow \quad &C(w)=\{\text{first neighborhood prime divisors of }w\}. \end{aligned} \] {}.

MSC:

13H05 Regular local rings
13B22 Integral closure of commutative rings and ideals
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