Pure exactness of the principal cover of vertex operator algebras

Let V be a vertex operator algebra. The fusion products in this paper are defined by logarithmic intertwining operators. Under this setting, we prove the pure exactness of any extension of the V-module V. Namely, if 0→Q→ϵP→ρV→0 is a short exact sequence of V-modules and W is a V-module, then 0→Q⊠W→ϵ⊠1WP⊠W→ρ⊠1WV⊠W→0 is also exact, where we view it as a sequence of g(V)-modules.