Valuations. (English) Zbl 1003.13001

Hauser, H. (ed.) et al., Resolution of singularities. A research textbook in tribute to Oscar Zariski. Based on the courses given at the working week in Obergurgl, Austria, September 7-14, 1997. Basel: Birkhäuser. Prog. Math. 181, 539-590 (2000).
This is a rather nice survey article on valuations, especially on their application to the resolution of singularities. It includes some historical remarks, and proofs of many of the stated results along with numerous examples.
The first two sections give some equivalent definitions for a valuation domain, define a valuation, and discuss the value group. Section 3 discusses totally ordered abelian groups and the relationship between the isolated subgroups of the value group and the prime ideals of the valuation domain. Section 4 covers composition of valuation domains. Section 5 surveys prolongations of valuations. Section 6 concerns the center of a valuation domain and gives Krull’s result that the integral closure of a domain is the intersection of its valuation overrings. Section 7 covers the abstract Riemann surface of a field as expounded in the book by O. Zariski P. Samuel, “Commutative algebra”. Section 8 concerns valuation domains and the resolution of singularities. Section 10 titled “Valuations and Noetherian rings” gives Abhyankar’s theorem. The final section consists of examples.
For the entire collection see [Zbl 0932.00042].


13A18 Valuations and their generalizations for commutative rings
14E15 Global theory and resolution of singularities (algebro-geometric aspects)