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Abstract For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated differents. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs. independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, Kähler differentials, and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their ranks or value groups. 

Abstract Birational properties of generically finite morphisms of algebraic varieties can be understood locally by a valuation of the function field ofX. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance. 

Let I = {In} be a Q-divisorial filtration on a two dimensional normal excellent local ring (R, mR). Let R[I] = ⊕n≥0In be the Rees algebra of I and τ : ProjR[I]) → Spec(R) be the natural morphism. The reduced fiber cone of I is the R- algebra R[I]/ p mRR[I], and the reduced exceptional fiber of τ is Proj(R[I]/ p mRR[I]). In [7], we showed that in spite of the fact that R[I] is often not Noetherian, mRR[I] always has only finitely many minimal primes, so τ −1 (mR) has only finitely many ir- reducible components. In Theorem 1.2, we give an explicit description of the scheme structure of Proj(R[I]). As a corollary, we obtain in Theorem 1.3 a new proof of a theorem of F. Russo, showing that Proj(R[I]) is always Noetherian and that R[I] is Noetherian if and only if Proj(R[I]) is a proper R-scheme. In Corollary 1.4 to Theorem 1.2, we give an explicit description of the scheme structure of the reduced exceptional fiber Proj(R[I]/ p mRR[I]) of τ , in terms of the possible values 0, 1 or 2 of the analytic spread l(I) = dim R[I]/mRR[I]. In the case that l(I) = 0, τ −1 (mR) is the emptyset; this case can only occur if R[I] is not Noetherian. At the end of the introduction, we give a simple example of a graded filtration J of a two dimensional regular local ring R such that Proj(R[J ]) is not Noetherian. This filtration is necessarily not divisorial 

We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry. 

We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties. 

We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties. 

We give an example of an extension of two dimensional regular local rings in a tower of two independent defect Artin-Schreier extensions for which strong local monomialization does not hold. 

The page and line numbers refer to the manuscript which is posted on my webpage, www.math.missouri.edu/ ̃dale. This is the published version (Annales de L’Institut Fourier 63 (2013), 865 - 922), but the page and line numbers are different. A case was missed in Lemma 3.7 (Case (A) and a modification of (15) in the restatement of Definition 3.3 below). The consideration of this new case does not introduce any significant change in the proof. I have written out in detail all of the changes which need to be made in the manuscript to incorporate this new case. Numbers indexing equations, theorems, defini- tions etc. are as in the earlier manuscript. New equations, theorems etc. are indexed by letters. I thank Andre Belotto and Ed Bierstone for pointing out that a case was missed in the original Lemma 3.7. 

Abstract In this paper, we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two-dimensional normal excellent local ring R , and that the Hilbert polynomial of the fiber cone of a divisorial filtration on R has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of R . The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.