Flatness of Tensor Products and Semi-Rigidity for C2-cofinite Vertex Operator Algebras. II

Because of the associativity of fusion products ⊠, the above hypothesis is equivalent to the condition that for each irreducible module W , there is an irreducible module W̃ such that HomV (W̃ ⊠ W,V ) 6= 0. In the previous paper [6], we have introduced a concept of ”Semi-Rigidity” and then proved that if W is semi-rigid, then W is flat for the fusion products ⊠, that is, 0 → W ⊠ A → W ⊠ B → W ⊠ C → 0 is still exact for any exact sequence 0 → A → B → C → 0 of V -modules. Here, we call a V -module W semi-rigid if for a given epimorphism eW : W̃ ⊠ W → V and a canonical isomorphism μ : (W ⊠ W̃ ) ⊠ W → W ⊠ (W̃ ⊠ W ), there are homomorphisms ef W : W ⊠ W̃ → V and ρ : P → W ⊠ W̃ such that ef Wρ(P ) = V and (idW ⊠ eW )(μ(ρ ⊠ idW )(P ⊠W )) = W ⊠ V , where P is a projective cover of V . The aim of this paper is to show the following theorem.