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In this paper, we introduce an X -precover completing domain X − 1 (L) , with L being a class of modules and not necessarily a single module, and then investigate when every module in L has an X − 1 (L) -preenvelope. Epic and monic X − 1 (L) -preenvelopes are also investigated. This study plays a key role in setting a general framework for several classical results. Then, for a class of finitely generated modules M , we introduce the notion of M -R-Mittag-Leffler modules as a natural extension of R-Mittag-Leffler modules. This enabled us to find easier proofs of some known results and also establish new ones. [ABSTRACT FROM AUTHOR]

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander's conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander's conditions hold. In this paper, we investigate the relative counterpart of Auslander's conditions. So, we extend Ringel and Zhang's work and introduce other concepts. Namely, for a semidualizing module C , we introduce weakly C -Gorenstein and partially C -Gorenstein rings as rings representing relations between the relative counterpart of Auslander's conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are C -Frobenius categories in characterizing weakly C -Gorenstein and partially C -Gorenstein rings. [ABSTRACT FROM AUTHOR]

Recently, several authors have adopted new alternative approaches in the study of some classical notions of modules. Among them, we find the notion of subprojectivity which was introduced to measure in a way the degree of projectivity of modules. The study of subprojectivity has recently been extended to the context of abelian categories, which has brought to light some interesting new aspects. For instance, in the category of complexes, it gives a new way to measure, among other things, the exactness of complexes. In this paper, we prove that the subprojectivity notion provides a new sight of null-homotopic morphisms in the category of complexes. This will be proven through two main results. Moreover, various results which emphasize the importance of subprojectivity in the category of complexes are also given. Namely, we give some applications by characterizing some classical rings and establish various examples that allow us to reflect the scope and limits of our results. [ABSTRACT FROM AUTHOR]

In the last few years, López-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as a tool to measure, somehow, the projectivity level of such a module (so not just to determine whether or not the module is projective). In this paper we develop a new treatment of the subprojectivity in any abelian category which shed more light on some of its various important aspects. Namely, in terms of subprojectivity, some classical results are unified and some classical rings are characterized. It is also shown that, in some categories, the subprojectivity measures notions other than the projectivity. Furthermore, this new approach allows, in addition to establishing nice generalizations of known results, to construct various new examples such as the subprojectivity domain of the class of Gorenstein projective objects, the class of semi-projective complexes and particular types of representations of a finite linear quiver. The paper ends with a study showing that the fact that a subprojectivity domain of a class coincides with its first right Ext-orthogonal class can be characterized in terms of the existence of preenvelopes and precovers. [ABSTRACT FROM AUTHOR]

Homology theory relative to classes of objects other than those of projective or injective objects in abelian categories has been widely studied in the last years, giving a special relevance to Gorenstein homological algebra. We prove the existence of Gorenstein flat precovers in any locally finitely presented Grothendieck category in which the class of flat objects is closed under extensions, the existence of Gorenstein injective preenvelopes in any locally noetherian Grothendieck category in which the class of all Gorenstein injective objects is closed under direct products, and the existence of special Gorenstein injective preenvelopes in locally noetherian Grothendieck categories with a generator lying in the left orthogonal class to that of Gorenstein injective objects. [ABSTRACT FROM AUTHOR]

Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call themRugged Modules.Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only ifR = S×T, whereSis semisimple Artinian andTis either Morita equivalent to a right PCI-domain, orTis right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. IfRis a ring whose regular left modules are semisimple, then a right moduleMis rugged if and only if its character left moduleM+is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules. [ABSTRACT FROM AUTHOR]

In this paper we are interested in studying the stability question of subcategories of an abelian category $\mathcal {A}$ constituted of all objects that admit (proper) coproper resolutions (resp. (coproper) proper coresolutions) with terms in a subcategory $\mathcal {E}$ of $\mathcal {A}$ . Using a new approach, we give an affirmative answer to the stability question on these categories under the condition that $\mathcal {E}$ is closed under finite direct sums. This result generalizes Huang's affirmative answer to the well-known stability question of Gorenstein categories raised by Sather-Wagstaff, Sharif and White. We end the paper with an example showing that the condition imposed on $\mathcal {E}$ cannot be dropped. [ABSTRACT FROM AUTHOR]

We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module . We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add and the -projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of is always bounded above by the -projective global dimension of , these two dimensions being equal when the class of all -Gorenstein projective -modules is contained in the Bass class of relative to . Of course we also give the dual results concerning the relative Gorenstein injective global dimension. [ABSTRACT FROM AUTHOR]

The relation between the Auslander (resp., Bass) class and the class of modules with finite Gorenstein projective (resp., injective) dimension is well known when these mentioned classes are built with a dualizing module over Noetherian n-perfect rings. Basically, the results are necessary conditions to ensure that both classes coincide. In this article we try to extend and sometimes improve some of these results by weakening the condition of being dualizing. Among other results, we prove that aWakamatsu tilting module with some extra conditions is precisely a module RC such that the Bass class BC(R) coincides with the class ofmodules of finite Gorenstein injective dimension. [ABSTRACT FROM AUTHOR]

We study the concepts of the 𝒫C-projective and the ℐC-injective dimensions of a module in the noncommutative case, weakening the condition ofCbeing semidualizing. We give the relations between these dimensions and theC-relative Gorenstein dimensions (GC-projective and GC-injective dimensions) of the module. Finally, we compare, in some circumstances, the global 𝒫C-projective dimension of a ring and the global dimension of the endomorphisms ring ofC. [ABSTRACT FROM PUBLISHER]

In the last years (Gorenstein) homological dimensions relative to a semidualizing module C have been subject of several works as interesting extensions of (Gorenstein) homological dimensions. In this paper, we extend to the noncommutative case the concepts of G-projective module and dimension, weakening the condition of C being semidualizing as well. We prove that indeed they share the principal properties of the classical ones and relate this new dimension with the classical Gorenstein projective dimension of a module. The dual concepts of G-injective modules and dimension are also treated. Finally, we show some interesting interactions between the class of G-projective modules and the Bass class associated to C on one side, and the class of G $${_{C^{\vee}}}$$ -injective modules ( C = Hom ( C, E) where E is an injective cogenerator in R-Mod) and the Auslander class associated to C in the other. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the concept of right and left semidualizing adjoint pair of functors and study its main properties. This concept generalizes the concept of semidualizing module and allows one to consider semidualizing comodules, graded modules, etc. We also study tilting adjoint pair of functors as a particular case. We show generalized tilting theorem in this general setting and give some applications to tilting theory in the category of comodules over a coalgebra. [ABSTRACT FROM AUTHOR]

We introduce the concept of homological Frobenius functors as the natural generalization of Frobenius functors in the setting of triangulated categories, and study their structure in the particular case of the derived categories of those of complexes and modules over a unital associative ring. Tilting complexes (modules) are examples of homological Frobenius complexes (modules). Homological Frobenius functors retain some of the nice properties of Frobenius ones as the ascent theorem for Gorenstein categories. It is shown that homological Frobenius ring homomorphisms are always Frobenius. [ABSTRACT FROM PUBLISHER]

The relative importance of the processes that generate and maintain biodiversity is a major and controversial topic in evolutionary biology with large implications for conservation management. The Atlantic Forest of Brazil, one of the world's richest biodiversity hot spots, is severely damaged by human activities. To formulate an efficient conservation policy, a good understanding of spatial and temporal biodiversity patterns and their underlying evolutionary mechanisms is required. With this aim, we performed a comprehensive phylogeographic study using a low-dispersal organism, the land planarian species Cephaloflexa bergi (Platyhelminthes, Tricladida). Analysing multi-locus DNA sequence variation under the Approximate Bayesian Computation framework, we evaluated two scenarios proposed to explain the diversity of Southern Atlantic Forest (SAF) region. We found that most sampled localities harbour high levels of genetic diversity, with lineages sharing common ancestors that predate the Pleistocene. Remarkably, we detected the molecular hallmark of the isolation-by-distance effect and little evidence of a recent colonization of SAF localities; nevertheless, some populations might result from very recent secondary contacts. We conclude that extant SAF biodiversity originated and has been shaped by complex interactions between ancient geological events and more recent evolutionary processes, whereas Pleistocene climate changes had a minor influence in generating present-day diversity. We also demonstrate that land planarians are an advantageous biological model for making phylogeographic and, particularly, fine-scale evolutionary inferences, and propose appropriate conservation policies. [ABSTRACT FROM AUTHOR]

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories. [ABSTRACT FROM AUTHOR]

The Brazilian Atlantic Forest is one of the richest biodiversity hotspots of the world. Paleoclimatic models have predicted two large stability regions in its northern and central parts, whereas southern regions might have suffered strong instability during Pleistocene glaciations. Molecular phylogeographic and endemism studies show, nevertheless, contradictory results: although some results validate these predictions, other data suggest that paleoclimatic models fail to predict stable rainforest areas in the south. Most studies, however, have surveyed species with relatively high dispersal rates whereas taxa with lower dispersion capabilities should be better predictors of habitat stability. Here, we have used two land planarian species as model organisms to analyse the patterns and levels of nucleotide diversity on a locality within the Southern Atlantic Forest. We find that both species harbour high levels of genetic variability without exhibiting the molecular footprint of recent colonization or population expansions, suggesting a long-term stability scenario. The results reflect, therefore, that paleoclimatic models may fail to detect refugia in the Southern Atlantic Forest, and that model organisms with low dispersal capability can improve the resolution of these models. [ABSTRACT FROM AUTHOR]

Chromosomal inversion polymorphism play a major role in the evolutionary dynamics of populations and species because of their effects on the patterns of genetic variability in the genomic regions within inversions. Though there is compelling evidence for the adaptive character of chromosomal polymorphisms, the mechanisms responsible for their maintenance in natural populations is not fully understood. For this type of analysis, Drosophila subobscura is a good model species as it has a rich and extensively studied chromosomal inversion polymorphism system. Here, we examine the patterns of DNA variation in two natural populations segregating for chromosomal arrangements that differentially affect the surveyed genomic region; in particular, we analyse both nucleotide substitutions and insertion/deletion variations in the genomic region encompassing the odorant-binding protein genes Obp83a and Obp83b (Obp83 region). We show that the two main gene arrangements are genetically differentiated, but are consistent with a monophyletic origin of inversions. Nevertheless, these arrangements interchange some genetic information, likely by gene conversion. We also find that the frequency spectrum-based tests indicate that the pattern of nucleotide variation is not at equilibrium; this feature probably reflects the rapid increase in the frequency of the new gene arrangement promoted by positive selection (that is an adaptive change). Furthermore, a comparative analysis of polymorphism and divergence patterns reveals a relaxation of the functional constraints at the Obp83b gene, which might be associated with particular ecological or demographic features of the Canary island endemic species D. guanche [ABSTRACT FROM AUTHOR]

Pleistocene climatic oscillations are known to influence the patterns of genetic diversity and the distribution of traits that are the target of selection. Here, we combine phylogeographical and ecological niche modelling (ENM) approaches to explore the influence of historical factors (Pleistocene climatic shifts) and natural selection on the evolution of distyly (two floral morphs) from tristyly (three floral morphs) of Oxalis alpina in the Sky Islands of the Sonoran Desert. Molecular data and ENM indicate that historical factors have had a strong influence on the genetic structure and the geographical distribution of reproductive systems of O. alpina. Moreover, genetic results suggest the possibility that distylous populations do not represent a monophyletic group. We propose that the combined effects of natural selection and genetic drift have influenced the tristyly-distyly transition. [ABSTRACT FROM AUTHOR]

Odorant-binding proteins (OBPs) and chemosensory proteins (CSPs) are two families of small water-soluble proteins, abundant in the aqueous fluid surrounding olfactory receptor neurons in insect antennae. OBPs are involved in the first step of olfactory signal transduction, carrying airborne semiochemicals to the odorant receptors and can be classified into three groups: Classic OBPs, Plus-C OBPs and Atypical OBPs. Here, we identified and annotated genes encoding putative OBPs and CSPs in the pea aphid Acyrthosiphon pisum using bioinformatics. This identified genes encoding 13 Classic and two Plus-C OBPs and 13 CSPs. Homologous OBP sequences were also identified in nine other aphid species, allowing us to compare OBPs across several aphid and non-aphid species. We show that, although OBP sequences are divergent within a species and between different orders, there is a high similarity between orthologs within a range of aphid species. Furthermore, the phylogenetic relationships between OBP orthologs reflect the divergence of aphid evolution lineages. Our results support the ‘birth-and-death’ model as the major mechanism explaining aphid OBP sequence evolution, with the main force acting on the evolution being purifying selection. [ABSTRACT FROM AUTHOR]

Chemoreception is a crucial biological process that is essential for the survival of animals. In insects, olfaction allows the organism to recognise volatile cues that allow the detection of food, predators and mates, whereas the sense of taste commonly allows the discrimination of soluble stimulants that elicit feeding behaviours and can also initiate innate sexual and reproductive responses. The most important proteins involved in the recognition of chemical cues comprise moderately sized multigene families. These families include odorant-binding proteins (OBPs) and chemosensory proteins (CSPs), which are involved in peripheral olfactory processing, and the chemoreceptor superfamily formed by the olfactory receptor (OR) and gustatory receptor (GR) families. Here, we review some recent evolutionary genomic studies of chemosensory gene families using the data from fully sequenced insect genomes, especially from the 12 newly available Drosophila genomes. Overall, the results clearly support the birth-and-death model as the major mechanism of evolution in these gene families. Namely, new members arise by tandem gene duplication, progressively diverge in sequence and function, and can eventually be lost from the genome by a deletion or pseudogenisation event. Adaptive changes fostered by environmental shifts are also observed in the evolution of chemosensory families in insects and likely involve reproductive, ecological or behavioural traits. Consequently, the current size of these gene families is mainly a result of random gene gain and loss events. This dynamic process may represent a major source of genetic variation, providing opportunities for FUTURE specific adaptations.Heredity (2009) 103, 208–216; doi:10.1038/hdy.2009.55; published online 13 May 2009 [ABSTRACT FROM AUTHOR]

In this note we show the interlacing between homological algebra and the category of relative Hopf modules. We show that many well-known concepts related with differential graded algebras can be performed as purely Hopf algebraic phenomenons. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of a flat cover and a cotorsion envelope for any quasi-coherent sheaf over the projective line p¹ (R), where R is any commutative ring. We first prove a general result that guarantees the existence of F-covers and F⊥-envelopes in the general setting of a Grothendieck category (not necessarily with enough projectives) provided that the class F satisfies some "standard" conditions. This will generalize some results of the earlier work. [Aldrich, S. T., Enochs, E., García Rozas, J. R., Oyonarte, L. (2001). Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications. [ABSTRACT FROM AUTHOR]

This article represents another step in our programme of obtaining a Galois theory and a coGalois theory when we have a category C and a given enveloping (for Galois) or covering (for coGalois) class. More precisely, in this paper, we study what should be understood by a conormal morphism between two objects of a given category and we characterize conormal morphisms between finite abelian groups when the covering class under consideration is that of torsion-free abelian groups. [ABSTRACT FROM PUBLISHER]

The category of quasi-coherent sheaves on the projective line P[SUP1](k) (k is a field) is equivalent to certain representations of the quiver • → • ← •. Many of the techniques which are used to study these sheaves apply to more general categories. We give the definitions of these more general categories and then consider one particular such category in depth. In this particular category we prove that there are no (nonzero) projective representations but that the category admits flat covers (or, equivalently in this situation, torsion free covers) and cotorsion envelopes. [ABSTRACT FROM AUTHOR]

In this paper we provide a classification theorem and a structure theorem for exact differential graded algebras, and we use the classification theorem to show that a differential graded algebra A is semisimple (as a differential graded algebra) precisely when the graded algebra Z(A) is semisimple (as a graded algebra) and A is an exact complex. We also relate exact differential graded algebras with a graded version of Hochschild cohomology. [ABSTRACT FROM AUTHOR]

An ~1.6-kb fragment spanning the rp49 gene was sequenced in 16 lines of Drosophila subobscura from Madeira and in 22 lines of the endemic species D. madeirensis. Nucleotide diversity in D. subobscura from Madeira (π=0.0081) was similar to that in lines from Spain carrying the O3+4 chromosomal arrangement (π=0.0080). No significant genetic differentiation was detected between insular and continental O3+4 lines of D. subobscura. These results are compatible both with a rather recent and massive colonization, and with multiple colonization events from the continent. Nucleotide diversity in D. madeirensis (π=0.0076) was similar to that in D. subobscura, which deviates from the expectation, under strict neutrality, of a lower level of variation in an insular species with a small population size. The observed numbers of shared polymorphisms and of fixed differences between D. madeirensis and D. subobscura are compatible with the isolation model of speciation, where shared polymorphisms are due to common ancestry. [ABSTRACT FROM AUTHOR]

In this paper we extend the concept of the group of covering automorphisms associated to a universal covering space Æ: U ’ X (where X is a connected topological manifold), to the case of left (or right) minimal approximations. In the case of torsion-free coverings of abelian groups we exhibit a topology on these groups which makes them into topological groups and we give necessary and sufficient conditions for these groups to be compact. Finally we prove that when these groups are compact they are pronilpotent (Theorem 5·3). We also characterize when these groups are torsion-free (Proposition 5·4). [ABSTRACT FROM AUTHOR]

We introduce Shur and projective Schur subgroup of the Brauer group of a cocommutative coalgebra by means of twisted cogroup coalgebras and we study their properties. In particular we show that these subgroups are always torsion (in contrast with the whole Brauer group). Moreover, when C is coreflexive and irreducible both subgroups coincide with the coradical ones. We illustrate the theory with several examples. [ABSTRACT FROM AUTHOR]

Motivation: The comparative analysis of gene gain and loss rates is critical for understanding the role of natural selection and adaptation in shaping gene family sizes. Studying complete genome data from closely related species allows accurate estimation of gene family turnover rates. Current methods and software tools, however, are not well designed for dealing with certain kinds of functional elements, such as microRNAs or transcription factor binding sites.Results: Here, we describe BadiRate, a new software tool to estimate family turnover rates, as well as the number of elements in internal phylogenetic nodes, by likelihood-based methods and parsimony. It implements two stochastic population models, which provide the appropriate statistical framework for testing hypothesis, such as lineage-specific gene family expansions or contractions. We have assessed the accuracy of BadiRate by computer simulations, and have also illustrated its functionality by analyzing a representative empirical dataset.Availability: BadiRate software and documentation is available from http://www.ub.edu/softevol/badirate.Contact: jrozas@ub.eduSupplementary information: Supplementary data are available at Bioinformatics online. [ABSTRACT FROM PUBLISHER]