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).
Key Words: Firm Frobenius monad, firm Frobenius adjunction,
firm module, comodule, separable functor.
2000 Mathematics Subject Classification: Primary: 16L60;
Secondary: 16D90, 18C15, 18C20, 18A40, 16T15.
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