## 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

Zbl 0074.26301
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