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Rees valuations

In: Commutative Algebra

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  • Irena Swanson

    (Reed College, Department of Mathematics)

Abstract

This expository paper contains history, definitions, constructions, and the basic properties of Rees valuations of ideals. A section is devoted to one-fibered ideals, that is, ideals with only one Rees valuation. Cutkosky [17] proved that there exists a two-dimensional complete Noetherian local integrally closed domain in which no zero-dimensional ideal is one-fibered. However, no concrete ring of this form has been found. An emphasis in this paper is on bounding the number of Rees valuations of ideals. A section is on the projective equivalence of ideals, with the discussion of “rational powers” of ideals. The last section is about the Izumi–Rees Theorem, which establishes comparability of Rees valuations with the same center. Several examples are computed explicitly. More on Rees valuations can be done via the projective equivalence of ideals, and there have been many articles along that line. See the latest article by Heinzer, Ratliff, and Rush, in this volume.

Suggested Citation

  • Irena Swanson, 2011. "Rees valuations," Springer Books, in: Marco Fontana & Salah-Eddine Kabbaj & Bruce Olberding & Irena Swanson (ed.), Commutative Algebra, pages 421-440, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-6990-3_16
    DOI: 10.1007/978-1-4419-6990-3_16
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