Sheaf cohomology and free resolutions over exterior algebras

We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part' of a resolution over the exterior algebra.

We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree 1$"> in the complex yields a new complex which is eventually exact.

As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.

     

Unirationality of the Hurwitz space $$\varvec{\mathcal {H}_{9,8}}$$

In this paper we prove that the Hurwitz space \(\mathcal {H}_{9,8}\), which parameterizes 8-sheeted covers of \({\mathbb P }^1\) by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.     

Milling Down to Nanometers: A General Process for the Direct Dry Synthesis of Supported Metal Catalysts

Supported catalysts are among the most important classes of catalysts. They are typically prepared by wet‐chemical methods, such as impregnation or co‐precipitation. Here we disclose that dry ball milling of macroscopic metal powder in the presence of a support oxide leads in many cases to supported catalysts with particles in the nanometer size range. Various supports, including TiO₂, Al₂O₃, Fe₂O₃, and Co₃O₄, and different metals, such as Au, Pt, Ag, Cu, and Ni, were studied, and for each of the supports and the metals, highly dispersed nanoparticles on supports could be prepared. The supported catalysts were tested in CO oxidation, where they showed activities in the same range as conventionally prepared catalysts. The method thus provides a simple and cost‐effective alternative to the conventionally used impregnation methods.     

IDPi Catalysis

High acidity and structural confinement are pivotal elements in asymmetric acid catalysis. The recently introduced imidodiphosphorimidate (IDPi) Brønsted acids have met with remarkable success in combining those features, acting as powerful Brønsted acid catalysts and “silylium” Lewis acid precatalysts in numerous thus far inaccessible transformations. Substrates as challenging to activate as simple olefins were readily transformed, ketones were employed as acceptors in aldolizations allowing sub‐ppm level catalysis, whereas enolates of the smallest donor aldehyde, acetaldehyde, did not polymerize but selectively added a single time to a variety of acceptor aldehydes.     

Field induced chirality in the helix structure of Dy/Y multilayer films and experimental evidence for Dzyaloshinskii-Moriya interaction on the interfaces

Polarized neutron scattering experiments have demonstrated that Dy/Y multilayer structures possess a coherent spin helix with a preferable chirality induced by the magnetic field. The average chirality, being proportional to the difference in the left- and right-handed helix population numbers, is measured as a polarization-dependent asymmetric part of the magnetic neutron scattering. The magnetic field applied in the plane of the sample upon cooling below T(N) is able to repopulate the otherwise equal population numbers for the left- and right-handed helixes. The experimental results strongly indicate that the chirality is caused by Dzyaloshinskii-Moriya interaction due to the lack of the symmetry inversion on the interfaces.     

High‐Temperature Stable Zirconia Particles Doped with Yttrium,Lanthanum, and Gadolinium

Zirconia microspheres synthesized by a wet‐chemical sol–gel process are promising building blocks for various photonic applications considered for heat management and energy systems, including highly efficient reflective thermal barrier coatings and absorbers/emitters used in thermophotovoltaic systems. As previously shown, pure zirconia microparticles deteriorate at working temperatures of ≥1000 °C. While the addition of yttrium as a dopant has been shown to improve their phase stability, pronounced grain growth at temperatures of ≥1000 °C compromises the photonic structure of the assembled microspheres. Here, a new approach for the fabrication of highly stable ceramic microparticles by doping with lanthanum, gadolinium, and a combination of those with yttrium is introduced. The morphological changes of the particles are monitored by scanning electron microscopy, ex situ X‐ray diffraction (XRD), and in situ high‐energy XRD as a function of dopant concentration up to 1500 °C. While the addition of lanthanum or gadolinium has a strong grain growth attenuating effect, it alone is insufficient to avoid a destructive tetragonal‐to‐monoclinic phase transformation occurring after heating to >850 °C. However, combining lanthanum or gadolinium with yttrium leads to particles with both efficient phase stabilization and attenuated grain growth. Thus, ceramic microspheres are yielded that remain extremely stable after heating to 1200 °C.     

Doping-induced temperature evolution of a helicoidal spin structure in the MnGe compound

The helicoidal magnetic structure of a MnGe compound doped with 25% Fe is studied by means of small-angle neutron scattering in a wide temperature range of 10–300 K. Analysis of the scattering-function profile demonstrates that magnetic structures inherent to both pure MnGe and its doped compounds are unstable. The doping of manganese monogermanide is revealed to lead to higher destabilization of the magnetic system. In passing from MnGe to Mn_0.75Fe_0.25Ge, the magnetic-ordering temperature T _N decreases from 130 to 95 K, respectively. It is demonstrated that, at temperatures close to 0 K, the intensity of the contribution to scattering from stable spin helices decreases and the intensity of scattering by spin helix fluctuations increases with increasing impurity-metal concentration. An increased intensity of anomalous scattering caused by spin excitations existing in the system is observed. Helicoidal fluctuations and spin excitations corresponding to low temperatures indicate the quantum nature of the instability in the doped compound. However, MnGe doping with Fe atoms has no influence on the compound’s magnetic properties at temperatures of higher than T _N. The temperature range of short-range ferromagnetic correlations is independent of concentrations and is restricted by temperatures T ranging from 175 to 300 K.