Some exact sequences associated with adjunctions in bicategories. Applications (original) (raw)
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To a B-coring and a (B, A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projective on one side, together with a map of corings over the same base ring, lead to the notion of a module-morphism, which extends the notion of a morphism of corings (over different base rings). A module-morphism of corings induces functors between the categories of comodules. These functors are termed pull-back and push-out functors respectively and thus relate categories of comodules of different corings. We study when the pullback functor is fully faithful and when it is an equivalence. A generalised descent associated to a morphism of corings is introduced. We define a category of modulemorphisms, and show that push-out functors are naturally isomorphic to each other if and only if the corresponding module-morphisms are mutually isomorphic. All these topics are studied within a unifying language of bicategories and the extensive use is made of interpretation of corings as comonads in the bicategory Bim of bimodules and module-morphisms as 1-cells in the associated bicategories of comonads in Bim.
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Mathematics and Mathematics Education, 2002
It is well known that the category M C of right comodules over an A-coring C, A an associative ring, is a subcategory of the category of left modules * C M over the dual ring * C. The main purpose of this note is to show that M C is a full subcatgeory in * C M if and only if C is locally projective as a left A-module.
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Algebras and Representation Theory, 2010
We give a bicategorical version of the main result of Masuoka (Tsukuba J Math 13:353-362, 1989) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings R ⊆ S, the relative Picard group Pic(S/R) is isomorphic to the Amitsur 1-cohomology group H 1 (S/R, U) with coefficients in the units functor U.
Gorenstein projective objects in comma categories
Periodica Mathematica Hungarica, 2021
Let A and B be abelian categories and F : A → B an additive and right exact functor which is perfect, and let (F, B) be the left comma category. We give an equivalent characterization of Gorenstein projective objects in (F, B) in terms of Gorenstein projective objects in B and A. We prove that there exists a left recollement of the stable category of the subcategory of (F, B) consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in B and A. Moreover, this left recollement can be filled into a recollement when B is Gorenstein and F preserves projectives.