-rings and differential polynomials over universal fields
We introduce the notions of silting comodules and finitely silting comodules in quasi-finite category, and study some properties of them. We investigate the torsion pair and dualities which are related to finitely silting comodules, and give the equivalences among silting comodules, finitely silting comodules, tilting comodules and finitely tilting comodules.
We introduce the notion of Gorenstein star modules and obtain some properties and a characterization of them. We mainly give the relationship between -Gorenstein star modules and -Gorenstein tilting modules, see L. Yan, W. Li, B. Ouyang (2016), and a new characterization of -Gorenstein tilting modules.
The homology theory of colored posets, defined by B. Everitt and P. Turner, is generalized. Two graph categories are defined and Khovanov type graph cohomology are interpreted as Ext* groups in functor categories associated to these categories. The connection, described by J. H. Przytycki, between the Hochschild homology of an algebra and the graph cohomology, defined for the same algebra and a cyclic graph, is explained from the point of view of homological algebra in functor categories.