Comparison of two notions of weak crossed product

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.

INVERTIBLE BIMODULES, MIYASHITA ACTION IN MONOIDAL CATEGORIES AND AZUMAYA MONOIDS

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

Gorenstein global dimension and Hopf algebroid actions

Let [Formula: see text] be a Hopf algebroid and [Formula: see text] a left [Formula: see text]-module algebra. In this paper, we mainly present the duality theorem for the smash product [Formula: see text], and making use of integral theory for Hopf algebroids, we investigate the stability of Gorenstein injective pre-envelopes and Gorenstein projective precovers between the category of [Formula: see text]-modules and the category of [Formula: see text]-modules. Moreover, we establish the relationship between Gorenstein global dimension of [Formula: see text] and that of [Formula: see text], and prove that [Formula: see text] has finite representation type, resp. is selfinjective, resp. is CM-finite [Formula: see text]-Gorenstein, if and only if [Formula: see text] has the same property under suitable conditions. As an application, we investigate the representation dimension of the lower triangular matrix Artin algebra [Formula: see text].