### Abstract

Let A be a ring and M_{A} the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor - ⊗_{A} B : M_{A} → M_{A} is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor - ⊗_{A} C : M_{A} → M_{A} is a comonad (or cotriple). The related categories of modules (or algebras) of - ⊗_{A} B and comodules (or coalgebras) of - ⊗_{A} C are well studied in the literature. On the other hand, the right adjoint endofunctors Hom_{A} (B, -) and Hom_{A} (C, -) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of Hom_{A} (B, -)-comodules is isomorphic to the category of B-modules, while the category of Hom_{A} (C, -)-modules (called C-contramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and Hom_{A} (C, -)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of Hom_{A} (C, -)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor Hom_{R} (H, -) and the category of mixed Hom_{R} (H, -)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H^{*} is a Hopf algebra.

Original language | English |
---|---|

Pages (from-to) | 1719-1747 |

Number of pages | 29 |

Journal | Journal of Algebra |

Volume | 322 |

Issue number | 5 |

DOIs | |

Publication status | Published - Sep 1 2009 |

### Fingerprint

### Keywords

- (Co)modules
- (Co)monads
- (Co)rings
- Contramodules
- Hopf algebras
- Hopf monads

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*322*(5), 1719-1747. https://doi.org/10.1016/j.jalgebra.2009.06.003

**Monads and comonads on module categories.** / Böhm, G.; Brzeziński, Tomasz; Wisbauer, Robert.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 322, no. 5, pp. 1719-1747. https://doi.org/10.1016/j.jalgebra.2009.06.003

}

TY - JOUR

T1 - Monads and comonads on module categories

AU - Böhm, G.

AU - Brzeziński, Tomasz

AU - Wisbauer, Robert

PY - 2009/9/1

Y1 - 2009/9/1

N2 - Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor - ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor - ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of - ⊗A B and comodules (or coalgebras) of - ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA (B, -) and HomA (C, -) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA (B, -)-comodules is isomorphic to the category of B-modules, while the category of HomA (C, -)-modules (called C-contramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and HomA (C, -)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of HomA (C, -)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor HomR (H, -) and the category of mixed HomR (H, -)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H* is a Hopf algebra.

AB - Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor - ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor - ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of - ⊗A B and comodules (or coalgebras) of - ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA (B, -) and HomA (C, -) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA (B, -)-comodules is isomorphic to the category of B-modules, while the category of HomA (C, -)-modules (called C-contramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and HomA (C, -)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of HomA (C, -)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor HomR (H, -) and the category of mixed HomR (H, -)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H* is a Hopf algebra.

KW - (Co)modules

KW - (Co)monads

KW - (Co)rings

KW - Contramodules

KW - Hopf algebras

KW - Hopf monads

UR - http://www.scopus.com/inward/record.url?scp=67650535259&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67650535259&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2009.06.003

DO - 10.1016/j.jalgebra.2009.06.003

M3 - Article

VL - 322

SP - 1719

EP - 1747

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 5

ER -