Finiteness conditions in the stable module category

We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg–Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end, we determine the structure of completely finitary Gorenstein projective modules over group rings. The methods are inspired by representation theory and make use of the stable module category, in which morphisms are defined through complete cohomology. In order to carry out these methods, we need to restrict ourselves to certain classes of hierarchically decomposable groups.     

S-alkyl and S-alkylaryl cysteine complexes with platinum(II) and their interactions with nucleosides

The reactions of K₂PtCl₄ with the aminoacids, S-methyl-L-cysteine, S-ethyl-L-cysteine, S-benzyl-L-cysteine, S-para-nitro-benzyl-L-cysteine, S-diphenyl-methyl-L-cysteine, S-tribenzyl-L-cysteine and S-4′,4′-dimethoxy-diphenylmethyl-L-cysteine were studied in neutral or acidic aqueous solutions. Complexes of the formulae PtLCl₂, [PtL₂]Cl₂ and Pt(L-H⁺)₂, where L = aminoacid, were isolated in the solid state and their structures investigated with elemental analysis, conductivity measurements, IR, ¹H NMR and ¹³CNMR spectra. The results show that the coordination sites of Pt(II) with the amino-acids are the N and S atoms, producing two diastereoisomers around the chiral sulphur atom, which were identified in the ¹H NMR and ¹³CNMR spectra. The complexes PtLCl₂ further react with the nucleosides guanosine and inosine. The complexes [PtL(nucl)₂]Cl₂ were isolated from these reactions and studied with the same methods. They showed a PtN₇ bonding with the nucleosides and retained the N, S bondings with the aminoacids. As a result of the higher trans influence of S than N, the nucleoside molecule coordinated to the metal through N₇ and trans to S has a weaker bond strength than the other, as it is revealed from the ¹H NMR and ¹³C NMR spectra of these complexes.     

On Periodic (Co)Homology of Groups

Here we show that a countable group G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H ^q (G, ?) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H ^q (G, ?). In [2 Asadollahi , J. , Hajizamani , A. , Salarian , Sh. Periodic flat resolutions and periodicity in group (co)homology. To appear in Forum Mathematicum.  [Google Scholar]] Asadollahi, Hajizamani, and Salarian showed that, if a group G is such that every flat ?G-module has finite projective dimension, then G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H ^q (G, ?) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H ^q (G, C), where C is the cotorsion envelope of the trivial ?G-module ?.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

Synthesis, structural characterization, and biological studies of new antimony(III) complexes with thiones. The influence of the solvent on the geometry of the complexes

Five new antimony(III) complexes with the heterocyclic thiones 2-mercapto-benzimidazole (MBZIM), 5-ethoxy-2-mercapto-benzimidazole (EtMBZIM), and 2-mercapto-thiazolidine (MTZD) of formulas {[SbCl(2)(MBZIM)4]+.Cl-.2H(2)O. (CH(3)OH)} (1), {[SbCl(2)(MBZIM)4]+.Cl-.3H(2)O.(CH3CN)} (2), [SbCl(3)(MBZIM)2] (3), [SbCl(3)(EtMBZIM)(2)] (4), and [SbCl(3)(MTZD)2] (5) have been synthesized and characterized by elemental analysis, FT-IR, far-FT-IR, differential thermal analysis-thermogravimetry, X-ray diffraction, and conductivity measurements. Complex {[SbCl2(tHPMT)(2)]+Cl-}, (tHPMT = 2-mercapto-3,4,5,6-tetrahydro-pyrimidine), already known, was also prepared, and its X-ray crystal structure was solved. It is shown that the complex is better described as {[SbCl3(tHPMT)(2)]} (6). Crystal structures of all other complexes (1-5) have also been determined by X-ray diffraction at ambient conditions. The crystal structure of the hydrated ligand, EtMBZIM.H2O is also reported. Compound [C(28)H(24)Cl(2)N(8)S(4)Sb.2H(2)O.Cl.(CH(3)OH)] (1) crystallizes in space group P2(1), with a = 7.7398(8) A, b = 16.724(3) A, c = 13.717(2) A, beta = 98.632(11) degrees, and Z = 2. Complex [C(28)H(24)Cl(2)N(8)S(4)S(b).Cl.3H(2)O.(CH(3)CN)] (2) corresponds to space group P2(1), with a = 7.8216(8) A, b = 16.7426(17) A, c = 13.9375(16) A, beta = 99.218(10) degrees , and Z = 2. In both 1 and 2 complexes, four sulfur atoms from thione ligands and two chloride ions form an octahedral (Oh) cationic [SbS(4)Cl(2)]+ complex ion, where chlorides lie at axial positions. A third chloride counteranion neutralizes it. Complexes 1 and 2 are the first examples of antimony(III) compounds with positively charged Oh geometries. Compound [C(14)H(12)Cl(3)N(4)S(2)S(b)] (3) crystallizes in space group P, with a = 7.3034(5) A, b = 11.2277(7) A, c = 12.0172(8) A, alpha = 76.772(5) degrees, beta = 77.101(6) degrees, gamma = 87.450(5) degrees, and Z = 2. Complex [C(18)H(20)Cl(3)N(4)O(2)S(2)S(b)] (4) crystallizes in space group P1, with a = 8.6682(6) A, b = 10.6005(7) A, c = 13.0177(9) A, alpha = 84.181(6) degrees, beta = 79.358(6) degrees, gamma = 84.882(6) degrees, and Z = 2, while complex [C(6)H(10)Cl(3)N(2)S(4)S(b)] (5) in space group P2(1)/c shows a = 8.3659(10) A, b = 14.8323(19) A, c = 12.0218(13) A, beta = 99.660(12) degrees, and Z = 4 and complex [C(8)H(16)Cl(3)N(4)S(2)S(b)] (6) in space group P1 shows a = 7.4975(6) A, b = 10.3220(7) A, c = 12.1094(11) A, alpha = 71.411(7) degrees, beta = 84.244(7) degrees, gamma = 73.588(6) degrees, and Z = 2. Crystals of complexes 3-6 grown from acetonitrile solutions adopt a square-pyramidal (SP) geometry, with two sulfur atoms from thione ligands and three chloride anions around Sb(III). The equatorial plane is formed by two sulfur and two chloride atoms in complexes 3-5, in a cis-S, cis-Cl arrangement in 3 and 5 and a trans-S, trans-Cl arrangement in 4. Finally, in the case of 6, the equatorial plane is formed by three chloride ions and one sulfur from the thione ligand while the second sulfur atom takes an axial position leading to a unique SP conformation. The complexes showed a moderate cytostatic activity against tumor cell lines.