Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

Abel范畴中态射乘积诱导的短正合列及其应用

本文给出了Ａｂｅｌ范畴中态射乘积αβγ诱导的短正合列的立体交换图。其中共有７个３×３平面，３０条线。每一条线（两端的零对象省略）都是一个短正合列．这些短正合列用到ｐ－除环上的矩阵，可以导出一系列矩阵秩的恒等式，它不仅推广了域上矩阵秩的结果，还能得到一些新的关系式．     

A ∞-simplicial objects andA ∞-topological groups

The notion ofA _∞-topological group is introduced. It is proved that, if a space is obtained by deformation retraction of a topological groupG, then it has the structure of anA _∞-topological group, and theA _∞-homotopy equivalence holds. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 913–919, December, 1999.     

Biextensions by Indecomposable Modules of Derived Regular Length 2

We establish sufficient conditions for a biextension algebra of a piecewise hereditary algebra of type _n or D_n by indecomposable modules of derived regular length 2 to be of tame representation type.     

p-Banach Spaces and p-Totally Convex Spaces

We introduce the categories Vec _p of p-normed vector spaces, Ban _p of _p -Banach spaces, AC _p of _p -absolutely and TC _p of _p -totally convex spaces (0 < p 1). It will be shown that TC _p (AC _p ) is the Eilenberg–Moore category of Ban _p (Vec _p ). Then congruence relations on TC _p (AC _p )-spaces are studied. There are many differences between TC _p (AC _p )-spaces and totally (absolutely) convex spaces (i.e. p = 1) (Pumplün and Röhrl, 1984, 1985), which will become apparent in Section 4.     

Homology of the universal covering of a co-H-space

The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-H-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional assumptions on the given co-H-structure. In this paper, we show a homological property of co-H-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.

     

Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds

We study invariants of -manifolds derived from finite abelian groups equipped with quadratic forms. These invariants arise in Turaev's theory of modular categories and generalize those of H. Murakami, T. Ohtsuki and M. Okada. The crucial algebraic tool is a new reciprocity formula for Gauss sums, generalizing classical formulas of Cauchy, Kronecker, Krazer and Siegel. We use this reciprocity formula to give an explicit formula for the invariants and to generalize them to higher dimensions.

     

The effect of isodensity surface sampling on ESP derived charges and the effect of adding bondcenters on DMA derived charges

The effect of sampling the electrostatic potential around a molecule on the quality of electrostatic potential derived charges is investigated. In addition, the effect of the number of expansion sites in a Distributed Multipole Analysis (DMA) on the quality of charges fitted to the DMA derived electrostatic potential is investigated. Sampling on constant electron density surfaces gives a better fit between the quantum mechanical potential and the potential derived from the fitted charges, compared to sampling on a van der Waals surface composed of intersecting spheres. The fit between the electrostatic potential derived from point charges and the quantum mechanical potential becomes poorer with increasing quality of the employed basis set. The inclusion of bondcenters into the calculations improves the fit between the Quantum Mechanical (QM) electrostatic potential and the DMA derived potential. The number of expansion sites needed for an accurate approximation of the QM electrostatic potential increases with increasing quality of the used basis set.     

Synthesis and characterization of SnOx/Al2O3 derived gel catalysts

SnO_x/Al₂O₃ catalysts were prepared by the sol-gel method using as starting reactants aluminium-tri-s-butoxide and (i) tetrabutyltin, (ii) tin tetrachloride and (iii) tin tetra-t-amyloxide. The gel derived catalysts show acidities of 0.72, 1.08 and 1.05 mol NH₃/m² and BET areas of 91, 112 and 189 m²/g. The activity and selectivity pattern for isopropanol decomposition were found to depend strongly on the tin precursor used.     

NaNO2-Ceric ammonium nitrate mediated conversion of acrylic esters and Baylis-Hillman derived acrylic esters into corresponding β-nitro acrylic esters

A variety of acrylic esters, including those derived from Baylis-Hillman reactions, react with NaNO₂-ceric ammonium nitrate to form the corresponding β-nitro alcohols 2 and 5 whose dehydration, via their mesylates, leads to β-nitro acrylic esters, in good to excellent yields. Further, β-nitro acrylic esters containing a mesylate group 6, obtained from the Baylis-Hillman products, react with NaN₃ to form 2-cyano-3-substituted acrylic esters 10 in excellent yields.