Characterisations of purity in a locally finitely presented additive category: A short functorial proof

In this short note, we will give an efficient functorial proof of the equivalence of various characterisations of purity in a locally finitely presented additive category $C$. The complications of the proofs for specific choices of $C$ (e.g. $C=A\text{-Mod}$ for a ring $A$) are contained in the description of fp-injective and injective objects in $(\text{fp}C,\text{Ab})$. For example, the equivalence of many characterisations of purity in a module category $A\text{-Mod}$ is a simple corollary of what we will prove here, once we know that $\text{Fpinj}(A\text{-mod},\text{Ab})$ and $\text{Inj}(A\text{-mod},\text{Ab})$ look like (we do).Comment: 5 page

Duality and contravariant functors in the representation theory of artin algebras

We know that the model theory of modules leads to a way of obtaining definable categories of modules over a ring R as the kernels of certain functors (R-Mod)op→Ab rather than of functors R-Mod→Ab which are given by a pp-pair. This paper will give various algebraic characterizations of these functors in the case that R is an artin algebra. Suppose that R is an artin algebra. An additive functor G:(R-Mod)op→Ab preserves inverse limits and G|(R-mod)op:(R-mod)op→Ab is finitely presented if and only if there is a sequence of natural transformations (−,A)→(−,B)→G→0 for some A,B∈R-mod which is exact when evaluated at any left R-module. Any additive functor (R-Mod)op→Ab with one of these equivalent properties has a definable kernel, and every definable subcategory of R-Mod can be obtained as the kernel of a family of such functors. In the final section, a generalized setting is introduced, so that our results apply to more categories than those of the form R-Mod for an artin algebra R. That is, our results are extended to those locally finitely presented K-linear categories whose finitely presented objects form a dualizing variety, where K is a commutative artinian ring

The defect recollement, the MacPherson-Vilonen construction, and pp formulas

For any abelian category $\mathcal{A}$, Auslander constructed a localisation $w:\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})\to \mathcal{A}$ called the defect, which is the left adjoint to the Yoneda embedding $Y:\mathcal{A}\to\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})$. If $\mathcal{A}$ has enough projectives, then this localisation is part of a recollement called the defect recollement. We show that this recollement is an instance of the MacPherson-Vilonen construction if and only if $\mathcal{A}$ is hereditary. We also discuss several subcategories of $\mathrm{fp}(\mathcal{A}^{\mathrm{op}},\mathrm{Ab})$ which arise as canonical features of the defect recollement, and characterise them by properties of their projective presentations and their orthogonality with other subcategories. We apply some parts of the defect recollement to the model theory of modules. Let $R$ be a ring and let $\phi/\psi$ be a pp-pair. When $R$ is an artin algebra, we show that there is a smallest pp formula $\rho$ such that $\psi\leqslant\rho\leqslant\phi$ which agrees with $\phi$ on injectives, and that there is a largest pp formula $\mu$ such that $\psi\leqslant \mu\leqslant \phi$ and $\psi R=\mu R$. When $R$ is left coherent, we show that there is a largest pp formula $\sigma$ such that $\psi\leqslant\sigma\leqslant \phi$ which agrees with $\psi$ on injectives, and that the pp-pair $\psi/\phi$ is isomorphic to a pp formula if and only if $\psi=\sigma$, and that there is a smallest pp formula $\nu$ such that $\psi\leqslant \nu\leqslant\phi$ and $\phi R=\nu R$. We also show that, for any pp-pair $\phi/\psi$, $w(\phi/\psi)\cong (D\psi)R/(D\phi)R$, where $D$ is the elementary duality of pp formulas. We also give an expression for $w(\phi/\psi)$ in terms of the free realisation of $\phi$ and $\psi$.Comment: Typos corrected. To appear in the Journal of Algebra. 23 pages. https://www.sciencedirect.com/science/article/pii/S0021869319302698?via%3Dihu

The Stereochemistry of 5-chloromethyl-5-methyl-2-oxo-2-phenoxy-1,3,2- dioxaphosphorinan and the Reaction of Phenoxide Ion with 2-chloro-5-chloromethly-5-methyl-2-oxo-1,3,2-dioxaphosphorinan

The primary objective of the research outlined in Part One is to assign the stereochemistry of some phosphorus containing heterocycles. In order to avoid confusion over nomenclature, most of the compounds described in this thesis will be named as derivatives of the 1,3,2-dioxaphosphorinan ring. For compounds described in the Experimental section, the designation of cis and trans will be assigned according to the arrangement of the phosphoryl oxygen at position 2 in relation to the chloromethyl group at C-5. Axial (ax) and equatorial (eq) designations will also refer to the chloromethyl group at 5, unless otherwise stated. Thus, the axial chair conformer of trans-5-chloromethyl-5-methyl-2-oxo-2-(R4)-l,3,2-dioxaphosphorinan refers to (1). Most of the compounds described in the Experimental section will be referred to by Roman numerals as outlined in Table (1). The stereochemistry has been established for compounds (III), (IV), (VI), and (VII). However, the stereochemistry of compounds (VIII)-(XVII) and (V) is not known and therefore, the relationship of substituents at C-5 to P-2 as outlined in Table 1 is arbitrary. The reaction of (IV) with sodium phenoxide yielded a mixture of compounds which were found to be isomers of 5-chloromethyl-5-methyl-2-oxo-2phenoxy-1,3,2-dioxaphosphorinan. NMR signals at 0.96 δ and 1.28 δ were assigned to methyl groups. Chloromethyl groups were assigned to signals at 3.34 δ and 3.77 δ. It is assumed that the product is a mixture of two isomers. When this material was subjected to chromatographic analysis (see Experimental) two compounds were isolated; isomer A (mp 105) and isomer B (mp 136). The nmr spectrum of A included signals at 0.96 δ and 3.77 δ. Likewise, the spectrum of B indicated signals at 1.28 δ and 3.34 δ. Recombination of A and B gave a spectrum similar to the one obtained from the original mixture. The purpose of Part One is to define the stereochemistry of isomer A and isomer B. It should be noted, A is synonymous with (VIII) and B is synonymous with (IX)

An investigation of the relationship between the structure and properties of some stacked perovskites

This thesis discusses a comprehensive study of the structure and properties of selected stacked perovskites using non-ambient powder and single-crystal x-ray diffraction, powder neutron diffraction, birefringence microscopy and dielectric spectroscopy. The major portion of this work focuses on materials forming in Aurivillius phases, with the general chemical formula [Bi2O2]2+[An−1BnO3n+1]2−, and containing n perovskite layers sandwiched between fluorite-like interstices. Comparisons are made between the structure and properties of the isomorphous materials SrBi2Nb2O9 and BaBi2Nb2O9, and the ferroelectric phase transition in SrBi2Nb2O9 is compared to that in SrBi2Ta2O9. The latter shows two distinct phase transitions upon cooling from the high temperature phase: a ferroelastic transition to space group Amam at 848 K, followed by a ferroelectric transition to A21am at 608 K. By contrast, the transition in SrBi2Nb2O9 does not go via the intermediate Amam phase, as is demonstrated using both structural and property measurements. It is also shown that in a material that is both ferroelectric and ferroelastic, the individual contributions to the birefringence of strain and polarization can be isolated using birefringence microsocopy by applying the principles of Landau theory to extract the two contributions. The structure and properties of a number of other materials within the general class of ferroelectrics are investigated using similar techniques, in particular, single-crystal x-ray diffraction is used to reinvestigate the room-temperature structure of BaTiO3 following prominent reports of monoclinicity of this phase. Quantitative investigation of the (101) ferroelastic twinning in these crystals was also undertaken. In addition, evidence for the existence of a ferroelectric phase transition in CsBiNb2O7 is derived from the results of powder neutron diffraction as a function of temperature. The effect of substituting fluorine on to the anion site on both the structure and properties of ferroelectric stacked perovskites is also investigated, and is shown to have a radical effect on the the ferroelectric properties of the material, as well as on their phase formation

Restless gossamers: antibody clearance by hydrodynamic flow forces generated at the surface of motile trypanosome parasites

Trypanosomes evade antibody-mediated lysis via antigenic variation and rapid antibody removal from their cell surface. Recently, in Cell, Engstler et al. (2007) have discovered the mechanism for antibody clearance. Hydrodynamic forces generated by trypanosome swimming create a current, causing surface-bound antibodies to act as “molecular sails.” Consequently, they are swept to the cell posterior, internalized via the flagellar-pocket, and degraded. Hydrodynamic sorting is a novel biological process, possibly applicable in other contexts

Functional analysis of a family of proteins implicated in trypanosoma brucei lifecycle progression

Bloodstream trypanosomes initiate differentiation to procyclic forms in response to a citrate/ cis-aconitate (CCA) signal. A cell line was previously selected (“defective in differentiation-clone 1”; DiD1) that was unable to differentiate to procyclic forms (Tasker et al. (2000)). Additionally, expression profiling of this line in comparison to the parental line by macroarray hybridisation identified two differentially-expressed transcripts from an 8 gene cluster of highly homologous genes we named PAD genes (Proteins Associated with Differentiation). Members of this family show distinct expression profiles throughout the trypanosome lifecycle at both the mRNA and protein level, and are localised to the cell surface membrane of the cell. At least 1 member of the family (PAD1) shows stumpy form specific RNA and protein expression, representing the first useful molecular marker for this stage, and exhibits biochemical specificity for citrate. Additionally, another member of this family (PAD2) is upregulated in response to low temperature, a condition reported to cause hypersensitivity to CCA. Finally, RNAi mediated ablation of the PAD gene transcripts compromised the capacity of stumpy form trypanosomes to differentiate to the procyclic form in response to CCA. These combined expression, cytological, reverse-genetic and biochemical data make PAD proteins excellent candidates for recognition of the signal to initiate differentiation in response to CCA