Purity and torsors under quasi-split group schemes over Pr\"ufer bases

We establish an analogue of the Zariski--Nagata purity theorem for finite \'etale covers on smooth schemes over Pr\"ufer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. Building on Gabber--Ramero's upper bound of projective dimensions over Pr\"ufer bases, we derive an Auslander--Buchsbaum formula, which provides a useful tool for studying the algebraic structures involved in our work. Through analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from EGAIV4 on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck--Serre conjecture for torsors under a quasi-split reductive group scheme over schemes smooth over Pr\"ufer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi-split reductive group schemes in our Pr\"uferian context, inspired by the recent work by Cesnavicius.Comment: 33 pages, comments are welcome! we split the constant case of the Grothendieck--Serre out of this draf

Progress in Commutative Algebra 2

This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and more

Unramified Grothendieck-Serre for isotropic groups

The Grothendieck-Serre conjecture predicts that every generically trivial torsor under a reductive group $G$ over a regular semilocal ring $R$ is trivial. We establish this for unramified $R$ granted that $G$ is totally isotropic, that is, has a "maximally transversal" parabolic $R$-subgroup. We also use purity for the Brauer group to reduce the conjecture for unramified $R$ to simply connected $G$--a much less direct such reduction of Panin had been a step in solving the equal characteristic case of Grothendieck-Serre. We base the group-theoretic aspects of our arguments on the geometry of the stack $\mathrm{Bun}_G$, instead of the affine Grassmannian used previously, and we quickly reprove the crucial weak $\mathbb{P}^1$-invariance input: for any reductive group $H$ over a semilocal ring $A$, every $H$-torsor $\mathscr{E}$ on $\mathbb{P}^1_A$ satisfies $\mathscr{E}|_{\{t = 0\}} \simeq \mathscr{E}|_{\{t = \infty\}}$. For the geometric aspects, we develop reembedding and excision techniques for relative curves with finiteness weakened to quasi-finiteness, thus overcoming a known obstacle in mixed characteristic, and show that every generically trivial torsor over $R$ under a totally isotropic $G$ trivializes over every affine open of $\mathrm{Spec}(R) \setminus Z$ for some closed $Z$ of codimension $\ge 2$.Comment: 22 page

Formation of Si Nanocrystals for Single Electron Transistors by Ion Beam Mixing and Self-Organization – Modeling and Simulation

The replacement of the conventional field effect transistor (FET) by single electron transistors (SET) would lead to high energy savings and to devices with significantly longer battery life. There are many production approaches, but mostly for specimens in the laboratory. Most of them suffer from the fact that they either only work at cryogenic temperatures, have a low production yield or are not reproducible and each unit works in a unique way. A room temperature (RT) operating SET can be configured by inserting a small (few nm diameters) Si-Nanocrystal (NC) into a thin (<10 nm) SiO2 interlayer in Si. Industrial production has so far been excluded due to a lack of manufacturing processes. Classical technologies such as lithography fail to produce structures in this small scale. Even electron beam lithography or extreme ultraviolet lithography are far from being able to realize these structures in mass production. However, self-organization processes enable structures to be produced in any order of magnitude down to atomic sizes. Earlier studies realized similar systems using a layer of Si-NCs to fabricate a non-volatile memory by using the charge of the NCs for data storage. Based on this, it is very promising to use it for the realization of the SET. The self-organization depends only on the start configuration of the system and the boundary conditions during the process. These macroscopic conditions control the self-formed structures. In this work, ion beam irradiation is used to form the initial configuration, and thermal annealing is used to drive self-organization. A Si/SiO2/Si stack is irradiated and transforms the stack into Si/SiOx/Si by ion beam mixing (IBM) of the two Si/SiO2 interfaces. The oxide becomes metastable and the subsequent thermal treatment induces selforganization, which might leave a single Si-NC in the SiO2 layer for a sufficiently small mixing volume. The transformation of the planar SiOx layer (restriction only in one dimension) into a small SiOx volume (restriction in all three dimensions) is done by etching nanopillars with a diameter of less than 10nm. This forms a small SiOx plate embedded between two Si layers. The challenge is to control the self-organization process. In this work, simulation was used to investigate dependencies and parameter optimization. The ion mixing simulations were performed using binary collision approximation (BCA), followed by kinetic Monte Carlo (KMC) simulations of the decomposition process, which gave good qualitative agreement with the structures observed in related experiments. Quantitatively, however, the BCA simulation seemed to overestimate the mixing effect. This is due to the neglect of the positive entropy of the Si-SiO2 system mixing, i.e. the immiscibility counteracts the collisional mixing. The influence of this mechanism increases with increasing ion fluence. Compared to the combined BCA and KMC simulations, a larger ion mixing fluence has to be applied experimentally to obtain the predicted nanocluster morphology. To model the ion beam mixing of the Si/SiO2 interface, phase field methods have been applied to describe the influence of chemical effects during the irradiation of buried SiO2 layers by 60 keV Si+ ions at RT and thermal annealing at 1050°C. The ballistic collisional mixing was modeled by an approach using Fick’s diffusion equation, and the chemical effects and the annealing were described by the Cahn Hilliard equation. By that, it is now possible to predict composition profiles of Si/SiO2 interfaces during irradiation. The results are in good agreement with the experiment and are used for the predictions of the NCs formation in the nanopillar. For the thermal treatment model extensions were also necessary. The KMC simulations of Si-SiO2 systems in the past were based on normed time and temperature, so that the diffusion velocity of the components was not considered. However, the diffusion of Si in SiO2 and SiO2 in Si differs by several orders of magnitude. This cannot be neglected in the thermal treatment of the Si/SiO2 interface, because the processes that differ in speed in this order of magnitude are only a few nanometers apart. The KMC method was extended to include the different diffusion coefficients of the Si-SiO2 system. This allows to extensively investigate the influence of the diffusion. The phase diagram over temperature and composition was examined regarding decomposition (nucleation as well as spinodal decomposition) and growing of NCs. Using the methods and the knowledge gained about the system, basic simulations for the individual NC formation in the nanopillar were carried out. The influence of temperature, diameter, and radiation fluence was discussed in detail on the basis of simulation results

Atomic structure and structural stability of Fe90Sc10 nanoglasses

Nanoglasses are non-crystalline solids whose internal structure is characterized by fluctuations of the free volume. Due to the typical dimensions of the structural features in the nanometer-range and the disordered atomic structure of the interfacial regions, the atomic structure and the structural stability of nanoglasses is not yet completely understood. Nanoglasses are typically produced by consolidation of glassy nanoparticles. Consequently, the basis for the understanding of the atomic structure of nanoglasses lies in the atomic structure of the primary glassy nanoparticles. Using electron energy loss spectroscopy, the elemental distribution in the Fe90Sc10 primary glassy nanoparticles and in the corresponding nanoglasses produced by consolidation of these glassy nanoparticles have been studied. Due to surface segregation, Fe has been found to be enriched at the surface of the primary Fe90Sc10 glassy nanoparticles. This behavior was found to be consistent with theoretical results based on a monolayer model for surface segregation behavior of the binary liquid alloys. In addition, the heterogeneous structure of Fe90Sc10 nanoglasses with Fe enriched interfaces was also directly observed, and may be attributed to the segregation of the primary glassy nanoparticles on the surface. Furthermore, the electron density of the isolated and loosely compacted primary glassy nanoparticles was investigated using small- and wide- angle X-ray scattering. The results indicate that the surface shells of glassy nanoparticles have an electron density that is lower than the electron density in the cores of the glassy nanoparticles. The lower electron density seems to result mainly from a lower atomic packing density of the surface shells rather than from compositional variations due to the surface segregation. During the consolidation of the glassy nanoparticles, the inhomogeneous elemental distribution and the short-range order in the shells of Fe90Sc10 glassy nanoparticles can be transferred into the interfaces of the resulting bulk Fe90Sc10 nanoglasses. The free volume within the shells of the Fe90Sc10 glassy nanoparticles may delocalize into the interfaces between the Fe90Sc10 glassy nanoparticles resulting in interfacial regions of lower atomic packing density in the Fe90Sc10 nanoglasses. The structural stability of Fe90Sc10 nanoglasses has been studied by means of low temperature annealing in situ in a transmission electron microscope, and ex situ in an ultra-high-vacuum tube-furnace. The analysis of both experiments showed similar results. The structure of the Fe90Sc10 nanoglasses was stable for up to 2 hours when annealed at 150 °C. Annealing of nanoglasses at higher temperatures resulted in the formation of a metastable nanocrystalline bcc-Fe(Sc) with Sc-enriched interfaces. The crystallization process of Fe90Sc10 nanoglasses was clarified and a plausible mechanism for the structural stability was proposed

Valuative lattices and spectra

The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical methods uncovers a hidden computational content for numerous abstract objects of classical mathematics, which seem a priori inaccessible constructively, e.g., the algebraic closure of a (discrete) field. When a proof in classical mathematics uses these abstract objects and results in a concrete outcome, dynamical methods generally make possible to discover an algorithm for this concrete outcome. The second part of the article applies this dynamical method to the theory of divisibility. We compare two notions of valuative spectra present in the literature and we introduce a third notion, which is implicit in an article devoted to the dynamical theory of algebraically closed discrete valued fields. The two first notions are respectively due to Huber \& Knebusch and to Coquand. We prove that the corresponding valuative lattices are essentially the same. We establish formal Valuativestellens\"atze corresponding to these theories, and we compare the various resulting notions of valuative dimensions.Comment: This file contains also a French version of the paper. English version appears in the Proceedings of Graz Conference on Rings and Factorizations 2021. Title: Algebraic, Number Theoretic, and Topological Aspects of Ring Theory. Editors: Jean-Luc Chabert, Marco Fontana, Sophie Frisch, Sarah Glaz, Keith Johnson. Springer 2023 ISBN 978-3-031-28846-3 DOI 10.1007/978-3-031-28847-