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

We study properties of a C2-cofinite vertex operator algebra V = ⊕ ∞ i=0Vi of CFT type. If it is also rational (i.e. all modules are completely reducible) and V ′ ∼= V , then the rigidity of the tensor category of modules has been proved by Huang [11], where V ′ denotes the restricted dual of V . However, when we treat irrational C2cofinite VOAs, the rigidity is too strong, because it is almost equivalent to be rational as we see. We introduce a weaker condition ”semi-rigidity”. We expect that all C2cofinite VOAs satisfy this condition. Under the assumption of the semi-rigidity and the existence of canonical homomorphisms, we prove the following results. We show that if P is a projective cover of a V -module V , then for any finitely generated V module M , its projective cover is a direct summand of the tensor product P ⊠ M (defined by logarithmic intertwining operators) of M and P . Using this result, we prove the flatness property of finitely generated modules for the tensor products, that is, if 0 → A → B → C → 0 is exact then so is 0 → D ⊠ A → D ⊠ B → D ⊠ C → 0 for any finitely generated V -modules A, B, C and D. As a corollary, we have that if a semi-rigid C2-cofinite V contains a rational subVOA with the same Virasoro element, then V is rational.