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Discrete valuations centered on local domains. (English) Zbl 1094.13502

Summary: We study applications of discrete valuations to ideals in analytically irreducible domains, in particular, applications to zero divisors modulo powers of ideals. We prove a uniform version of S. Izumi’s theorem [Publ. Res. Inst. Math. Sci. 21, 719–735 (1985; Zbl 0587.32016)] and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-fiberedness, and a valuative criterion for when the normal cone of an ideal in an integrally closed domain is reduced.

MSC:

13A18 Valuations and their generalizations for commutative rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13F40 Excellent rings

Citations:

Zbl 0587.32016
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References:

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