[Paper Review] Firm Frobenius monads and firm Frobenius algebras

Gabriella Böhm, José Gómez-Torrecillas|arXiv (Cornell University)|Apr 24, 2013

Rings, Modules, and Algebras19 references6 citations

TL;DR

This paper introduces firm Frobenius monads and firm Frobenius algebras as non-unital generalizations of classical Frobenius algebras, using categorical methods to show that the category of firm modules and the category of comodules are isomorphic if and only if a canonical comparison functor from comodules to non-unital modules factors through firm modules. This occurs, for example, when the underlying algebra has local units or when the comultiplication splits the multiplication (coseparable case), extending Abrams' classical theorem to the non-unital setting with new characterizations via generalized Casimir elements and multiplier algebras.

ABSTRACT

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 algebra possesses local units, e.g. the firm Frobenius algebra arises from a co-Frobenius coalgebra over a base field; or if the comultiplication splits the multiplication (hence the underlying coalgebra is coseparable).

Motivation & Objective

• To generalize the classical Frobenius algebra concept to non-unital settings using categorical and monadic methods.
• To define firm modules and firm monads as a non-unital analog of unital Eilenberg-Moore categories.
• To establish conditions under which the category of firm modules and the category of comodules are isomorphic.
• To extend Abrams' theorem on module-comodule duality to non-unital, firm Frobenius algebras.
• To provide new characterizations of firm Frobenius algebras via generalized Casimir elements in multiplier algebras.

Proposed method

• Categorical framework using non-unital monads and comonads, with focus on adjunctions and lifted functors.
• Definition of firm modules as non-unital modules where the action is an epimorphism and a specific fork is a coequalizer.
• Introduction of firm monads as non-unital monads for which free modules are firm.
• Construction of a canonical comparison functor from the category of comodules to the category of non-unital modules.
• Analysis of when this functor factors through the category of firm modules, leading to the main isomorphism result.
• Application of results to algebras over commutative rings, including co-Frobenius and coseparable coalgebras.

Experimental results

Research questions

• RQ1 Under what conditions is the category of firm modules over a non-unital monad isomorphic to the category of comodules over its constituent comonad?
• RQ2 How can the classical duality between modules and comodules for Frobenius algebras be extended to the non-unital case?
• RQ3 What role does the existence of local units or a splitting comultiplication play in establishing the isomorphism between firm module and comodule categories?
• RQ4 How can a generalized Casimir element in the multiplier algebra of R ⊗ R be used to characterize firm Frobenius algebras?
• RQ5 In what cases does a non-unital Frobenius monad arise from a separable adjunction, and what are the implications for module-comodule equivalence?

Key findings

• The category of firm modules and the category of comodules are isomorphic if and only if the canonical comparison functor from comodules to non-unital modules factors through the category of firm modules.
• Firm Frobenius algebras arising from co-Frobenius coalgebras over a field satisfy the isomorphism condition, extending [11, Theorem 2.3] and [6, Proposition 2.7].
• When the comultiplication splits the multiplication (i.e., the coalgebra is coseparable), the comparison functor factors through firm modules, ensuring the isomorphism of categories.
• A firm algebra with non-degenerate multiplication is a firm Frobenius algebra if and only if there exists a generalized Casimir element in the multiplier algebra of R ⊗ R.
• For any G-graded unital k-algebra A, the smash product A♯G∗ is isomorphic to the firm algebra A⊗R, and the category of graded A-modules is equivalent to the category of firm modules over A⊗R.
• The category of firm modules over a firm Frobenius algebra with local units is isomorphic to its category of comodules, as shown via the action m·r := m₀ǫ(m₁r).