Categories of modules, comodules and contramodules over representations

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all the adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.Comment: Several update

Comodule theories in Grothendieck categories and relative Hopf objects

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category $\mathfrak S$. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category ${_A}\mathfrak S^H$ of relative $(A,H)$-Hopf modules in $\mathfrak S$, where $H$ is a Hopf algebra and $A$ is a right $H$-comodule algebra. We study the cohomological theory in ${_A}\mathfrak S^H$ by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in ${_A}\mathfrak S^H$. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in ${_A}\mathfrak S^H$.Comment: Minor update

Determination of susceptible growth stage and efficacy of fungicidal management of Curvularia leaf spot of maize caused by Curvularia lunata (Wakker) Boedijn

Maize is an important food security crop along with rice and wheat globally. Losses caused by biotic stresses in maize are substantial and Curvularia leaf spot is important among them. Further management of Curvularia leaf spot is done primarily through chemicals, therefore an attempt was made to evaluate the efficacy of com- monly used systemic and non-systemic fungicides against the pathogen, and most susceptible growth stage for disease development was identified. In vitro evaluation of four systemic and four non-systemic fungicides was done at different concentrations, for checking the growth of pathogen. The data revealed that Carboxin (at 25 ppm) completely inhibited growth of pathogen. Further among non-systemic fungicides treatment of Mancozeb showed maximum growth inhibition (98. 24% at 200 ppm). Under glass house conditions mancozeb was found to be more effective than Carboxin for controlling the disease severity. Further to determine the time of application of fungicides, developmental stage most susceptible to Curvularia lunata was studied. Three growth stages (Knee height stage, Silking stage and Tasseling stage) were compared. Maximum disease index and severity was found at Silking stage (47% and 53.75%, respectively) followed by Tasseling stage (42.5% and 18.4%, respectively) and Knee height stage (37% and 30%, respectively), indicating that the disease progresses with the maturity of the plant and is maximum at the Silking stage. Results suggest that susceptible maize varieties may give higher yield with the optimisation of the time of application of the fungicides and higher economic and environmental gains can be achieved with judicious use of fungicides