Claudia Menini

When is R-gr equivalent to the category of modules?

Abstract In the first part of this paper, we characterize graded rings R =⊕ σ∈ G R σ for which the category R - gr is equivalent with a category of modules over a certain ring. In the second part, sufficient conditions are given for the following implication to hold: if R - gr is equivalent with R 1 - mod (1 is the unit element of G ), then R is a strongly graded ring.

Cocycle Deformations for Hopf Algebras with a Coalgebra Projection

Abstract Let H be a Hopf algebra over a field K of characteristic 0 and let A be a bialgebra or Hopf algebra such that H is isomorphic to a sub-Hopf algebra of A and there is an H-bilinear coalgebra projection π from A to H which splits the inclusion. Then A ≅ R # ξ H where R is the pre-bialgebra of coinvariants. In this paper we study the deformations of A by an H-bilinear cocycle. If γ is a cocycle for A, then γ can be restricted to a cocycle γ R for R, and A γ ≅ R γ R # ξ γ H . As examples, we consider liftings of B ( V ) # K [ Γ ] where Γ is a finite abelian group, V is a quantum plane and B ( V ) is its Nichols algebra, and explicitly construct the cocycle which twists the Radford biproduct into the lifting.

BRAIDED BIALGEBRAS OF TYPE ONE

Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly ℕ-graded both as an algebra and as a coalgebra.

Strongly rational comodules and semiperfect Hopf algebras over QF rings

Abstract Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or — equivalently — the (left) integrals form a free R-module of rank 1.

Cotensor Coalgebras in Monoidal Categories

We introduce the concept of cotensor coalgebra for a given bicomodule over a coalgebra in an Abelian monoidal category ℳ. If ℳ is also cocomplete, complete, and AB5, we show that such a cotensor coalgebra exists and satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration is filled by considering a direct limit of a filtration consisting of wedge products. We prove that this coalgebra is formally smooth whenever the comodule is relative injective and the coalgebra itself is formally smooth.

Descent theory and Amitsur cohomology of triples

Abstract For a given triple (monad) U : C → C in the category C , we develop a theory of descent for U. We start by introducing the basic constructions associated to a triple: descent data, symmetry operators, and flat connections. The main result of this section asserts that the sets of these objects are bijectively equivalent. Next we construct a monoidal category C (U) such that U is an algebra in C (U) . If C is abelian, we define Amitsur cohomology of U with coefficients in a functor F : C (U)→ D . As an application of this construction, in the case where U is faithfully exact, we describe those morphisms that descend with respect to U. In the last part of the paper we classify all U-forms of a given object C 0 ∈ C . We show that there is a one-to-one correspondence between the set of equivalence classes of U-forms and a certain noncommutative Amitsur cohomology. Let A/B be an extension of associative unitary rings and let C be the category of right B-modules. Then (−)⊗ B A : C → C is a triple which is faithfully exact if and only if the extension A/B is faithfully flat. Specializing our results to this particular setting, we recover faithfully flat descent theory for extensions of (not necessarily commutative) rings.

Realization theorems for categories of graded modules over semigroup-graded rings

We present a collection of finitely generated projective generators for the category of graded modules over a unital semigroup-graded ring. Consequences of the existence of such a collection, as well as other module-theoretic re-sults, will arise from more general constructions involving TTF classes inside Grothendieck categories. For instance, we obtain a description of the category of graded modules whose support lies within a given subset of the semigroup.

BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras

A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β : A → A such that α(a)(bc) = (ab)β(c), for all a,b,c ∈ A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).

Weak projections onto a braided Hopf algebra

Abstract We show that, under some mild conditions, a bialgebra in an abelian and coabelian braided monoidal category has a weak projection onto a formally smooth (as a coalgebra) sub-bialgebra with antipode; see Theorem 1.14. In the second part of the paper we prove that bialgebras with weak projections are cross product bialgebras; see Theorem 2.12. In the particular case when the bialgebra A is cocommutative and a certain cocycle associated to the weak projection is trivial we prove that A is a double cross product, or biproduct in Madjids terminology. The last result is based on a universal property of double cross products which, by Theorem 2.15, works in braided monoidal categories. We also investigate the situation when the right action of the associated matched pair is trivial.

Jacobson's conjecture, Morita duality and related questions

In [J], Jacobson asked a question that gave rise to the so-called Jacobson’s Conjecture: “If R is a left noetherian ring with Jacobson radical J, then nn E mr J” = 0”. Herstein [H] and Jategaonkar [Jl ] constructed counterexamples to show that this conjecture is not true. In [J3] Jategaonkar asked if Jacobson’s conjecture holds for left noetherian rings with a left Morita duality. The main purpose of this paper is to give a negative answer to this question. In Section 1 we give the basic tools for our investigations. These are elementary facts concerning trivial extensions and generalized triangular matrix rings. Many of them are already known. In section 2 we give counterexamples to the above-mentioned question. Using some Commutative Algebra and the results in Section 1, we prove (Proposition 2.5) that a large class of generalized trangular matrix rings provides counterexamples to this question. The ring R = (8