Suppose that is a 2-dimensional excellent local domain with quotient field , is a finite separable extension of and is a 2-dimensional local domain with quotient field such that dominates . Suppose that is a valuation of such that dominates . Let be the restriction of to . The associated graded ring was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension of valued fields is without defect if and only if there exist regular local rings and such that is a local ring of a blowup of , is a local ring of a blowup of , dominates , dominates and the associated graded ring is a finitely generated -algebra.
We also investigate the role of splitting of the valuation in in finite generation of the extensions of associated graded rings along the valuation. We say that does not split in if is the unique extension of to which dominates . We show that if and are regular local rings, has rational rank 1 and is not discrete and is a finitely generated -algebra, then is a localization of the integral closure of in , the extension is without defect and does not split in . We give examples showing that such a strong statement is not true when does not satisfy these assumptions. As a consequence, we deduce that if has rational rank 1 and is not discrete and if is a nontrivial sequence of quadratic transforms along , then is not a finitely generated -algebra.
"The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation." Algebra Number Theory 11 (6) 1461 - 1488, 2017. https://doi.org/10.2140/ant.2017.11.1461