Proximity inequalities for complete ideals in two-dimensional regular local rings. (English) Zbl 0814.13016

Lipman, Joseph

Heinzer, William J. (ed.) et al., Commutative algebra: syzygies, multiplicities, and birational algebra: AMS-IMS-SIAM summer research conference on commutative algebra, held July 4-10, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 159, 293-306 (1994).

The author refers to an idea which goes back to Enriques and is connected with the classification of singularities of irreducible plane curves. Let \(K\) be a field and consider two-dimensional local rings (“points”) \(\alpha, \beta, \dots\) having fraction field \(K\). To each point \(\alpha\) with maximal ideal \({\mathfrak m}_ \alpha\) associate the valuation \(\text{ord}_ \alpha \) of \(K\) such that \(\text{ord}_ \alpha(x) = \max \{n | x \in {\mathfrak m}^ n_ \alpha\}\) \((0 \neq x \in \alpha)\). Then \(\beta \supsetneq \alpha\) is “proximate to \(\alpha\)” – \(\beta \succ \alpha\) – if the valuation ring of \(\text{ord}_ \alpha\) contains \(\beta\).
Let \(I\) be an ideal in \(\alpha\) of finite colength. Then \(\beta \supset \alpha\) is a “base point” of \(I\) if \(\text{ord}_ \beta(I^ \beta) \neq 0\). \((I^ \beta\) is the \(\beta\)-ideal \(I(I\beta)^{-1}.)\) The “point basis” \({\mathbf B}(I)\) of \(I\) is the family \((\text{ord}_ \beta (I^ \beta))_{\beta \supset \alpha}\) of nonnegative integers (which are zero for all but finitely many \(\beta)\). The following main theorem of the paper describes the set \(\{{\mathbf B}(I)\}\), \(I\) a finite-colength ideal in \(\alpha\), by the “proximity inequalities”. As the author mentions, “it is the ideal-theoretic version of an old result on the existence of plane curves with given effective multiplicities at infinity near points”.
Theorem. Let \((r_ \beta)_{ \beta \supset \alpha}\) be a family of integers which are zero for all but finitely many \(\beta\). Then there exists a finite-colength ideal \(I\) in \(\alpha\) with \({\mathbf B}(I) = (r_ \beta)\) if and only if for each \(\beta \supset \alpha\) the inequality \(r_ \beta \geq \sum_{\gamma \succ \beta} [\gamma/{\mathfrak m}_ \beta:\beta/{\mathfrak m}_ \beta]\cdot \gamma\) holds. If there is such an \(I\) then there is exactly one which is complete (i.e. integrally closed in \(K)\).
The theorem has various corollaries concerning unique factorization of complete ideals and properties of complete ideals which are “simple”, i.e. not a product of two other complete ideals. A further theorem deals with the ‘predecessor’ of a simple complete ideal.
For the entire collection see [Zbl 0790.00007].

Reviewer: U.Vetter (Oldenburg)

MSC:

13H05	Regular local rings
13B22	Integral closure of commutative rings and ideals
14H20	Singularities of curves, local rings

Keywords:

complete ideals; two-dimensional regular local rings; classification of singularities of irreducible plane curves