Tameness and Artinianness of Graded Generalized Local Cohomology Modules

Let $R=\bigoplus_{n\geq 0}R_n$, \fa\supseteq \bigoplus_{n> 0}R_n and $M$ and $N$ be a standard graded ring, an ideal of $R$ and two finitely generated graded $R$-modules, respectively. This paper studies the homogeneous components of graded generalized local cohomology modules. First of all, we show that for all $i\geq 0$, H^i_{\fa}(M, N)_n, the $n$-th graded component of the $i$-th generalized local cohomology module of $M$ and $N$ with respect to \fa, vanishes for all $n\gg 0$. Furthermore, some sufficient conditions are proposed to satisfy the equality \sup\{\en(H^i_{\fa}(M, N))| i\geq 0\}= \sup\{\en(H^i_{R_+}(M, N))| i\geq 0\}. Some sufficient conditions are also proposed for tameness of H^i_{\fa}(M, N) such that i= f_{\fa}^{R_+}(M, N) or i= \cd_{\fa}(M, N), where f_{\fa}^{R_+}(M, N) and \cd_{\fa}(M, N) denote the $R_+$-finiteness dimension and the cohomological dimension of $M$ and $N$ with respect to \fa, respectively. We finally consider the Artinian property of some submodules and quotient modules of H^j_{\fa}(M, N), where $j$ is the first or last non-minimax level of H^i_{\fa}(M, N).Comment: 18pages, with some revisions and correction

SDF1 Gene Variation Is Associated with Circulating SDF1 alpha Level and Endothelial Progenitor Cell Number-The Bruneck Study

BACKGROUND: Stromal cell-derived factor-1 (SDF1) and its receptor CXC chemokine receptor 4 (CXCR4) play a critical role in progenitor cell homing, mobilization and differentiation. It would be interesting to assess the predictive value of SDF-1alpha level for EPC number, and to ascertain whether there is a relationship between SDF1 gene variation, plasma SDF-1alpha level, and the number and function of circulating EPCs. We also tested whether EPC number and function was related to CXCR4 gene variation. METHODOLOGY AND PRINCIPAL FINDINGS: We genotyped a cohort of individuals who participated in the Bruneck Study for single nucleotide polymorphisms (SNPs) in the SDF1 and CXCR4 genes, and measured blood SDF1alpha level as well as EPC number and function. SDF1alpha levels were correlated with age, gender, alcohol consumption, circulating reticulocyte numbers, and concentrations of matrix metalloproteinase-9, C-reactive protein, cystatin C, fibrinogen and homocytein. In blood samples taken in 2005, EPC number was inversely associated with SDF1alpha level (p<0.001). EPC number in 2005 was also inversely associated with SDF1alpha level in 2000 (p = 0.009), suggesting a predictive value of plasma SDF1alpha level for EPC number. There was an association between the SDF1 gene rs2297630 SNP A/A genotype, increased SDF1alpha level (p = 0.002) and lower EPC number (p = 0.006). CONCLUSIONS: Our data indicate that a SDF1 gene variation (rs2297630) has an influence on SDF1alpha level and circulating EPC number, and that plasma SDF1alpha level is a predictor of EPC number