F-rational rings and the integral closure of ideals

This is a preprint of an article published in Michigan Mathematical Journal, vol 49 (2001), pp. 3-11.The history of the Briançon-Skoda theorem and its ensuing avatars in commutative algebra has been well-documented in many papers. A theorem proved by Briançon and Skoda for convergent power series over the complex numbers and generalized to arbitrary regular local rings by Lipman and Sathaye states: Theorem 1.2 Let R be a regular local ring and let I be an ideal generated by ℓ elements. Then for all n ≥ ℓ, In ⊆ In−ℓ+1. This was partially extended to the class of pseudo-rational rings by Lipman and Teissier. However, they were unable to recover the full strength of (1.2). The present two authors, as well as Lipman, have pushed the original theorem further by introducing 'coefficients.' The methods used by the present authors have relied on the theory of tight closure. These improvements, however, have been valid only in regular rings, and the question of whether the statement of Theorem 1.2 remains valid in arbitrary pseudo-rational rings has remained open since 1981. Recent progress was made by Hyry and Villamayor, who proved (among other things) that if R is local Gorenstein and essentially of finite type over a field of characteristic 0, then In+ℓ−1 ⊆ In for an arbitrary ideal I with ℓ generators. In this paper we will use tight closure methods to prove (1.2) is valid for F-rational rings

Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type

In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type.Comment: 15 Page

Eruption Age of a Pleistocene Basalt From ^(40)Ar-^(39)Ar Analysis of Partially Degassed Xenoliths

We have applied ^(40)Ar−^(39)Ar dating to potassium-rich granitic xenoliths and host basalt from the Pleistocene Big Pine volcanic field, California. These xenoliths had been partially degassed upon their inclusion in the basaltic lava. Argon released from the xenoliths at extraction temperatures below ∼900°C yielded plateau ages indistinguishable from the total K-Ar age of the basalt. The best estimate of the age of eruption was 1.18±0.05 (2σ) m.y. ^(40)Ar extracted at higher temperatures included radiogenic argon not degassed from the late Cretaceous xenoliths 1.18 m.y. ago, causing an increase in the apparent age for the high-temperature fractions. The agreement of the low-temperature xenolith plateau ages and the basalt K-Ar ages demonstrates that ^(40)Ar−^(39)Ar analysis of xenoliths may be used to measure the age of eruption of very young lavas. This is significant because in many instances ages cannot be reliably determined by analysis of the lavas themselves

Eruption Age of a ~100,000-Year-Old Basalt From ^(40)Ar-^(39)Ar Analysis of Partially Degassed Xenoliths

We have applied ^(40)Ar-^(39)Ar dating, using stepwise thermal extraction of Ar, to five potassium-rich granitic xenoliths and their host basalt from Sawmill Canyon in the Sierra Nevada, California. Previous K/Ar analyses showed the age of the basalt to be roughly 100,000 years or less. The xenoliths, which had accumulated large amounts of radiogenic ^(40)Ar since their crystallization ∼100 m.y. ago, were partially degassed upon their inclusion in the basaltic magma. Ar released from the xenoliths in the laboratory at temperatures substantially below the melting temperature of the basalt, was created since the host magma cooled. Isotopic compositions of Ar released from the xenoliths in several extraction steps at temperatures below ∼900°C were colinear in ^(36)Ar/^(40)Ar versus ^(39)Ar/^(40)Ar diagrams and defined isochrons giving a mean age of degassing of 119,000±7000 (2σ) years. ^(40)Ar extracted at higher temperatures included ancient radiogenic ^(40)Ar that was never diffused from the xenoliths during immersion in the magma. This ^(40)Ar caused an increase in the apparent age for the high-temperature extractions. The high precision of the eruption age determined by this method is comparable to that obtained elsewhere by conventional K/Ar dating of sanidine. ^(40)Ar-^(39)Ar analysis of granitic xenoliths to date young basaltic lava flows may prove to yield results superior to those found from analysis of the lava itself. Establishing the age of eruption of the basalt flow in Sawmill Canyon establishes age limits for two Sierran glaciations which left moraines stratigraphically above and below the lava. Thus the younger glaciation must be Wisconsin; the older must be pre-Wisconsin in age

An Assessment of ^(40)Ar-^(39)Ar Dating of Incompletely Degassed Xenoliths

The possibility of measuring the age of eruption of Pleistocene lavas by ^(40)Ar-^(39)Ar analysis of entrapped ancient potassic xenoliths is demonstrated by a study of model systems. Upon inclusion in the hot magma such xenoliths are commonly only partially degassed of radiogenic ^(40)Ar which has accumulated in them since their original crystallization. The residual ^(40)Ar will increase the apparent K/Ar age of the xenolith. However, if a xenolith is of Cretaceous age or younger, then a plateau in its ^(40)Ar-^(39)Ar age spectrum giving the age of eruption is expected to extend over 25–50% of the total 39Ar released if degassing of the xenolith in the magma exceeded 90% and if the phases in the xenolith are characterized by sufficiently different diffusion dimensions or activation energies. If diffusion was from a bimodal population of spheres, then the radii must differ by a factor of 10 or more (or the diffusion coefficients by a factor of 100 or more); or if the spheres were equal in size (and in diffusion coefficients), then the activation energies must differ by a factor of at least 1.5. That such requirements may be realized in real xenoliths containing K-feldspars is expected from published activation energies for microcline and from data determined on a granitic xenolith which was degassed in an early Pleistocene basalt flow. The experimental results appear to establish that old xenoliths may contain Ar in distinctive phases which degas at sufficiently different temperatures as to permit determination of the age of degassing or eruption

The {\L}ojasiewicz exponent of a set of weighted homogeneous ideals

We give an expression for the {\L}ojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the {\L}ojasiewicz exponent of the gradient of a semi-weighted homogeneous function (\C^n,0)\to (\C,0) with an isolated singularity at the origin.Comment: 15 page

Spallogenic Rare Gases in Iron Meteorites with Isotopically Anomalous Ag

Cosmic-ray produced rare gases have been measured in four meteorites with excess ^(107)Ag^* (Kelly and Wasserburg, 1978; Kaiser et al., 1980a, 1980b) to determine if ^(107)Ag^* is related to cosmic ray exposure. Kaiser et al. (1980a) considered the possibility that ^(107)Ag^* could derive from energetic particle reactions on Pd if the cross section times multiplicity for all reactions were ~ 10^2 b

Ion Microprobe Evidence for the Presence of Excess ^(26)Mg in an Allende Anorthite Crystal

An ion microprobe has been used to observe ^(26)Mg/^(24)Mg in excess of normal values in anorthite crystals from Allende refractory inclusions. In one crystal, ^(26)Mg excesses up to 40% above normal, correlated with Al/Mg, were observed, These excesses are many times larger than those observed by other techniques. The correlation with Al/Mg indicated that the excess ^(26)Mg in the Allende anorthite is uniformly distributed in the anorthite crystal. Analytical procedures and conditions were carefully controlled. Nevertheless, small but significant apparent ^(26)Mg/^(24)Mg anomalies in isotopically normal samples were observed in analyses by one ion microprobe. A repetition of the isotopic analyses using another ion probe produced normal ratios in the normal samples and confirmed large ^(26)Mg excesses in the Allende anorthite

Correlation Between Fission Tracks and Fission-Type Xenon from an Extinct Radioactivity

Meteoritic whitlockite containing excess fission tracks has a large concentration of excess neutron-rich xenon isotopes which is 25 times that calculated from the track density. The isotopic spectrum is identical to that calculated previously for the Pasamonte achondrite. These results uniquely associate this spectrum with in situ fission. Identification of the fissioning nucleus as Pu244 gives Pu244/U238 ≅ 1/30 at the time of xenon retention. Neither "sudden" nor "uniform" nucleosynthetic models give consistent solutions for Pu244/U238 and U235/U238