A generalisation of the Abhyankar Jung theorem to associated graded rings of valuations. (10th December 2015)

• Record Type: Journal Article
• Authors: CUTKOSKY, STEVEN DALE
• Is Part Of: Mathematical proceedings of the Cambridge Philosophical Society. Volume 160:Part 2(2016:Mar.)
• Page Start: 233
• Page End: 255
• Publication Date: 2015-12-10
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0305004115000687
• Languages: English
• ISSNs: 0305-0041

Abstract

Abstract: Suppose that R → S is an extension of local domains and ν * is a valuation dominating S . We consider the natural extension of associated graded rings along the valuation gr ν * ( R ) → gr ν * ( S ). We give examples showing that in general, this extension does not share good properties of the extension R → S, but after enough blow ups above the valuations, good properties of the extension R → S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalisation of the Abhyankar–Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.