# Firm Frobenius monads and firm Frobenius algebras

@article{Bhm2013FirmFM, title={Firm Frobenius monads and firm Frobenius algebras}, author={G. B{\`o}hm and J. G'omez-Torrecillas}, journal={arXiv: Rings and Algebras}, year={2013} }

Firm Frobenius algebras are firm algebras and counital coalgebras such that the comultiplication is a bimodule map. They are investigated by categorical methods based on a study of adjunctions and lifted functors. Their categories of comodules and of firm modules are shown to be isomorphic if and only if a canonical comparison functor from the category of comodules to the category of non-unital modules factorizes through the category of firm modules. This happens for example if the underlying… Expand

#### 9 Citations

Yetter-Drinfeld modules over weak multiplier bialgebras

- Mathematics
- 2013

We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in [4, 2]. Yetter-Drinfeld modules are defined as modules and comodules,… Expand

Yetter-Drinfeld modules over weak multiplier bialgebras

- Mathematics
- 2013

We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in [4, 2]. Yetter-Drinfeld modules are defined as modules and comodules,… Expand

Weak Frobenius bimonads and Frobenius bimodules

- Mathematics
- 2014

As shown by S. Eilenberg and J.C. Moore (1965), for a monad $F$ with right adjoint comonad $G$ on any catgeory $\mathbb{A}$, the category of unital $F$-modules $\mathbb{A}_F$ is isomorphic to the… Expand

Weak multiplier bialgebras

- Mathematics
- 2013

A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to… Expand

Weak Multiplier Bimonoids

- Mathematics, Computer Science
- Appl. Categorical Struct.
- 2018

Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules over the base object. Expand

Comodules over weak multiplier bialgebras

- Mathematics
- 2013

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full… Expand

Comodules over weak multiplier bialgebras

- Mathematics
- 2013

This is a sequel paper of [2] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction… Expand

Weak Frobenius monads and Frobenius bimodules

- Mathematics
- 2016

As observed by Eilenberg and Moore (1965), for a monad \(F\) with right adjoint comonad \(G\) on any category \(\mathbb{A}\), the category of unital \(F\)-modules \(\mathbb{A}_F\) is isomorphic to… Expand

Object-unital groupoid graded rings, crossed products and separability

- Mathematics
- 2020

Abstract We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal)… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

Frobenius monads and pseudomonoids

- Mathematics
- 2004

Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous… Expand

Modules, Comodules, and Cotensor Products over Frobenius Algebras

- Mathematics
- 1998

Abstract We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left Ae-modules. We show that the category of right (left) comodules over A,… Expand

The Structure of Corings: Induction Functors, Maschke-Type Theorem, and Frobenius and Galois-Type Properties

- Mathematics
- 2000

Given a ring A and an A-coring C, we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint −⊗AC are separable. We then… Expand

Symmetric coalgebras

- Mathematics
- 2003

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any… Expand

Morita Theory of Comodules

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita… Expand

Finiteness Conditions, Co-Frobenius Hopf Algebras, and Quantum Groups

- Mathematics
- 1998

Abstract Some finiteness conditions for infinite dimensional coalgebras, particularly right or left semiperfect coalgebras, or co-Frobenius Hopf algebras are studied. As well, examples of… Expand

Morita Theory for Comodules Over Corings

- Mathematics
- 2009

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita… Expand

Quasi-co-Frobenius Coalgebras. II

- Mathematics
- 2003

Abstract We prove new characterizations of Quasi-co-Frobenius (QcF) coalgebras and co-Frobenius coalgebras. Among them, we prove that a coalgebra is QcF if and only if C generates every left and… Expand

A GENERALIZATION OF THE SMASH PRODUCT OF A GRADED RING

- Mathematics
- 1988

For any group G and G-graded ring R, there exists a ring S = R ♯ G∗, defined analogously to the smash product of R with the dual of the group ring for finite G, such that the categories of unital… Expand

On coseparable and biseparable corings

- Mathematics
- 2002

A relationship between coseparable corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is… Expand