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Valuations in function fields of surfaces. (English) Zbl 0716.13003

Let \(R\) denote a noetherian local domain with maximal ideal \(\mathfrak m\), residue field \(k\), and field of fractions \(K\). Consider a valuation \(v\) of \(K\) with value group \(\Gamma\) which is nonnegative on \(R\) and strictly positive on \(\mathfrak m\). Let \(R_ v\) be the valuation ring of \(v\) and \({\mathfrak m}_ v\) the maximal ideal of \(R_ v\). Then \(k\subset R_ v/{\mathfrak m}_ v\) in a natural way. There are two basic invariants associated with \(v\), the rational rank \(\text{rat.rk}(v):=\dim_{\mathbb Q}(\Gamma \otimes_{\mathbb Z}\mathbb Q)\) of \(v\) and the relative \(R\)-dimension \(\dim_ Rv\) of \(v\) which is the transcendence degree of \(R_ v/{\mathfrak m}_ v\) over \(k\).
The invariants are linked up in the inequality (*) \(\text{rat.rk}(v)+\dim_ Rv\leq \dim (R)\) which is due to S. Abhyankar [Am. J. Math. 78, 321–348 (1956; Zbl 0074.26301)]. The author’s aim is to develop the theory of valuations \(v\) for which equality holds in (*). A series of three papers is devoted to this purpose. The one in question deals with a structure theorem for \(v\) in the case in which \(R\) is a 2-dimensional regular local ring and \(k\) is algebraically closed. It is given in terms of generating sequences for \(v\) which are (finite or infinite) sequences \((Q_ i)\) of elements of \({\mathfrak m}\) such that every \(v\)-ideal \(I\) of \(R\) is generated by the products \(\prod_{j}Q_ j^{\gamma_ j} \), \(\gamma_ j\in\mathbb N\), \(\sum_{j}\gamma_ jv(Q_ j) \geq v(I)\) (where \(v(I)=\min \{v(x)| x\in I\})\). In particular \(v\) satisfies equality in (*) if and only if \(v\) has a finite generating sequence.

MSC:

13A18 Valuations and their generalizations for commutative rings
14J99 Surfaces and higher-dimensional varieties
13H99 Local rings and semilocal rings
13E05 Commutative Noetherian rings and modules

Citations:

Zbl 0074.26301
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