Cofiniteness conditions, projective covers and the logarithmic tensor product theory

We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C1-cofinite in the sense of Li. 2. There exists a positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra AN(V) is finite dimensional. This result shows that the category of grading-restricted generalized V-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V also satisfies Condition 3 that irreducible V-modules are R-graded and C1-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under operations ⧅P(z) for z∈C×. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when V is of positive energy and C2-cofinite, Conditions 1-3 are satisfied and thus all the conclusions hold.