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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. 

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.