Abstract algebraic integrals and Frobenius categories. (Q2863011)

A coalgebra \(C\) over a field \(K\) is called left co-Frobenius if \(C\) embeds in \(C^*\) as left \(C^*\)-module, and left quasi-co-Frobenius (QcF) if \(C\) embeds in a free \(C^*\)-module. For a finite-dimensional right \(C\)-comodule \(M\), the space of left integrals \(I_{l,M}\) is defined as \(\Hom_C^*(C^C,M^C)\). The author shows that \(C\) is left co-Frobenius if and only if \(\dim I_{l,M}\) is less than or equal to \(\dim M\) for all finite-dimensional right \(C\)-comodules \(M\), and that \(C\) is co-Frobenius (left and right) if and only if \(\dim I_{l,M}=\dim M\) for all \(M\). This is used to show that \(C\) is co-Frobenius if and only if the two functors \(\Hom_C^*(M,C^*)\) and \(\Hom_K(M,K)\) are isomorphic as vector spaces for every finite-dimensional right \(C\)-comodule \(M\). In this case, the two functors are naturally isomorphic.NEWLINENEWLINE Turning to QcF, the author shows that if \(C\) is left QcF, then \(C\) is right semi-perfect, i.e., the category \(M^C\) of right \(C\)-comodules has enough projectives (it was previously known that \(C\) is left semi-perfect). Thus \(C\) being Qcf (two-sided) is equivalent to the class of projective left \(C\)-comodules coinciding with the class of injective left \(C\)-comodules (equivalently for right \(C\)-comodules). He also gives a functorial characterization of left semi-perfect coalgebras, namely that the forgetful functor from finite-dimensional left \(C\)-comodules to \(K\)-vector spaces be the restriction of a representable functor. Many interesting examples of the results concerning integrals are given. There are also applications to Hopf algebras, in particular the Hopf algebra of representative functions on a compact group.