Errata and Addenda

Key Words:

• Frobenius extensions
• Hopf algebras
• Integral

1991 Mathematics Subject Classification:

• 16W30

“Frobenius Functors of the Second Kind” by S. Caenepeel, E. De Groot, and G. Militaru, Communications in Algebra, 31(11), pp. 5359–5391, 2002

Let k be a commutative ring, and H a finitely generated projective Hopf algebra over k. It follows from the Fundamental Theorem for Hopf algebras that the integral space of H is finitely generated and projective of rank one. It is stated in Caenepeel et al. (2002, Cor. 3.5) that the integral spaces in H and on H are isomorphic. This is not true: is isomorphic to the dual of . The mistake is caused by an inaccuracy in the preceding Theorem 3.4. We present the correct versions, and give an alternative proof of Corollary 3.5.

Theorem 3.4

Let H be Hopf algebra over a commutative ring k and I a projective k-module of rank one. Then the following assertions are equivalent:

1. H/K is I-Frobenius;

2. H is finitely generated and projective and H*/k is I*-Frobenius;

3. H is finitely generated and projective and ;

4. H is finitely generated and projective and ;

5. H is finitely generated and projective and ;

6. H is finitely generated and projective and ;

Proof

v) ⇒ ii) It follows from the Fundamental Theorem for Hopf algebras that we have an isomorphism of left H*-modules (cf. Caenepeel et al., 2002, Theorem 3.2)

This induces an isomorphism

β is left H*-linear, so it follows from ii) of Caenepeel et al. (2002, Theorem 3.1) (with H replaced by H*) that H*/k is I*-Frobenius.

The proof of the other implications is presented correctly in Caenepeel et al. (2002).

Corollary 3.5

Let H be a finitely generated projective Hopf algebra over a commutative ring k. Then we have isomorphisms of k-modules

Proof

This follows immediately from Theorem 3.4. Let us present an alternative proof, taken from the forthcoming Caenepeel and Guédénon (2005). Let and be the spaces of left integrals on and in H. Consider the isomorphism

with ⟨ϕ · h, k⟩ = ⟨ ϕ, kh⟩, coming from the Fundamental Theorem. If t ∈ J, then

so α restricts to a monomorphism α¯: I ⊗ J → kϵ. If I and J are free of rank one, then α¯ is an isomorphism, as there exist ϕ ∈ I and t ∈ J such that ⟨ ϕ, t ⟩ = 1 (see for example, Caenepeel et al., 2002, Theorem 31). Hence α¯ is an isomorphism after we localize at a prime ideal p of k, and this implies that α¯ is itself an isomorphism. Consequently, J* ≅ I. □

The referee pointed out to us that the correct version of Corollary 3.5 is already stated in Kadison and Stolin (2002, Remark 4.2), and is attributed to the first author. Probably this resulted from discussions to Lars Kadison's visit to the Vrije Universiteit Brussel in the fall of 2000.

Research supported by the bilateral project “Hopf Algebras in Algebra, Topology, Geometry and Physics” of the Flemish and Romanian governments.

Communicated by R. Wisbauer.