Let $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\mathfrak{p})$ the residue field of the local ring $R_\mathfrak{p}$. We prove that if an R-module M satisfies $\operatorname{Ext}_R^{n}(k(\mathfrak{p}),M)=0$ for some $n\geqslant\dim R$, then $\operatorname{Ext}_R^i(k(\mathfrak{p}),M)=0$ holds for all $i \geqslant n$. This improves a result of Christensen, Iyengar and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

To assess the role of dietary creatine on myofibre characteristics and protein synthesis in muscle, we fed grass carp (Ctenopharyngodon idellus, initial body weight: 88·47 ± 1·44 g) creatine-supplemented diets (1·84, 5·91, 8·48 and 15·44 g/kg diet) for 8 weeks. Creatine supplementation did not affect growth performance, but significantly increased creatine contents in muscle and liver. At 8·48 g/kg, creatine decreased the activities of alanine transaminase and aspartate aminotransferase in serum and improved hardness and chewiness of muscle due to shorter myofibre mean diameter, higher myofibre density and the frequencies of the diameters of classes I and III and collagen content, longer sarcomere length and upregulated mRNA levels of slow myosin heavy chains. Creatine supplementation upregulated the mRNA expressions of myogenic regulatory factors. The 8·48 g/kg creatine-supplemented diet significantly increased the contents of protein, total amino acids (AA), essential AA and free flavour AAs in muscle, the protein levels of insulin-like growth factor I, myogenic differentiation antigen and PPAR-γ coactlvator-1α in muscle and stimulated the phosphorylation of target of rapamycin (TOR) pathway in muscle. In summary, 8·48 mg/kg creatine improved fish health and skeletal muscle growth and increased hardness and protein synthesis in muscle of grass carp by affecting myofibre characteristics and the TOR signalling pathway. A second-order regression model revealed that the optimal dietary creatine supplementation of grass carp ranges between 8·48 and 12·04 g/kg.

Let 𝔭 be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies ${\rm Tor}_n^R $(k (𝔭), M) = 0 for some n ⩾ R𝔭, where k(𝔭) is the residue field at 𝔭, then ${\rm Tor}_i^R $(k (𝔭), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $(k (𝔭), M) are proved, and applications to the theory of homological dimensions are explored.

A trispanning orphan receptor (TOR) has been described in Schistosoma haematobium and S. mansoni. Here we report the complete molecular organization of the S. mansoni TOR gene, also known as SmCRIT (complement C2 receptor inhibitor trispanning). The SmTOR gene consists of 4 exons and 3 introns as shown by cloning the single exons from S. mansoni genomic DNA and the corresponding cDNA from the larval stage (cercaria) and the adult worm. The SmTOR ORF consists of 1260 bp and is longer than previously reported, with a fourth trans-membrane domain (proposed new name: Tetraspanning Orphan Receptor) and with, however, an unchanged C2-binding domain on the extracellular domain 1 (ed1). This domain differs in S. japonicum. A protein at the approximate expected molecular weight (55 kDa) was detected in adult worm extracts with polyclonal and monoclonal antibodies, and was found to be expressed on the tegumental surface of cercariae.

Let $R$ be a commutative ring. Let $M$ respectively $A$ denote a Noetherian respectively Artinian $R$-module, and $\mathfrak{a}$ a finitely generated ideal of $R$. The main result of this note is that the sequence of sets $(\mathrm{Att}_R\mathrm{Tor}_1^R((R/\mathfrak{a}^{n}),A))_{n\in\mathbb{N}}$ is ultimately constant. As a consequence, whenever $R$ is Noetherian, we show that $\mathrm{Ass}_R\mathrm{Ext}_R^1((R/\mathfrak{a}^{n}),M)$ is ultimately constant for large $n$, which is an affirmative answer to the question that was posed by Melkersson and Schenzel in the case $i=1$.

AMS 2000 Mathematics subject classification: Primary 13E05; 13E10