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Register a new userGeometric partial comodules over flat coalgebras in Abelian categories are globalizable
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The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literature. Finally, we introduce Hopf partial comodules over a bialgebra as geometric partial comodules in the monoidal category of (global) modules. By applying our globalization theorem we obtain an analogue of the fundamental theorem for Hopf modules in this partial setting.
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We discuss globalization for geometric partial comodules in a monoidal category with pushouts and we provide a concrete procedure to construct it, whenever it exists. The mild assumptions required by our approach make it possible to apply it in a number of contexts of interests, recovering and extending numerous ad hoc globalization constructions from the literature in some cases and providing obstruction for globalization in some other cases.
It will be seen that if $H$ is a weak Hopf algebra in the definition of coaction of weak bialgebras on coalgebras cite{Wang}, then a definition property is suppressed giving rise to the (global) coactions of weak Hopf algebras on coalgebras. The next step will be introduce the more general notion of partial coactions of weak Hopf algebras on coalgebras as well as a family of examples via a fixed element on the weak Hopf algebra, illustrating both definitions: global and partial. Moreover, it will also be presented how to obtain a partial comodule coalgebra from a global one via projections, giving another way to find examples of partial coactions of weak Hopf algebras on coalgebras. In addition, the weak smash coproduct cite{Wang} will be studied and it will be seen under what conditions it is possible to generate a weak Hopf algebra structure from the coproduct and the counit defined on it. Finally, a dual relationship between the structures of partial action and partial coaction of a weak Hopf algebra on a coalgebra will be established.
We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We show that this approach does not only allow to recover all known results in these settings, but it allows to treat new cases of interest, too.
In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
A finite algebra $bA=alg{A;cF}$ is emph{dualizable} if there exists a discrete topological relational structure $BA=alg{A;cG;cT}$, compatible with $cF$, such that the canonical evaluation map $e_{bB}colon bBto Hom( Hom(bB,bA),BA)$ is an isomorphism for every $bB$ in the quasivariety generated by $bA$. Here, $e_{bB}$ is defined by $e_{bB}(x)(f)=f(x)$ for all $xin B$ and all $fin Hom(bB,bA)$. We prove that, given a finite congruence-modular Abelian algebra $bA$, the set of all relations compatible with $bA$, up to a certain arity, emph{entails} the whole set of all relations compatible with $bA$. By using a classical compactness result, we infer that $bA$ is dualizable. Moreover we can choose a dualizing alter-ego with only relations of arity $le 1+alpha^3$, where $alpha$ is the largest exponent of a prime in the prime decomposition of $card{A}$. This improves Kearnes and Szendrei result that modules are dualizable, and Bentz and Mayrs result that finite modules with constants are dualizable. This also solves a problem stated by Bentz and Mayr in 2013.
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