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We construct projective covers of irreducible V-modules in the category ofgrading-restricted generalized V-modules when V is a vertex operator algebrasatisfying the following conditions: 1. V is C_{1}-cofinite in the sense of Li.2. There exists a positive integer N such that the differences between the realparts of the lowest conformal weights of irreducible V-modules are bounded by Nand such that the associative algebra A_{N}(V) is finite dimensional. Thisresult shows that the category of grading-restricted generalized V-modules is afinite abelian category over C. Using the existence of projective covers, weprove that if such a vertex operator algebra V satisfies in addition Condition3, that irreducible V-modules are R-graded and C_{1}-cofinite in the sense ofthe author, then the category of grading-restricted generalized V-modules isclosed under P(z)-tensor product operations for z in C^{\times}. We also provethat other conditions for applying the logarithmic tensor product theorydeveloped by Lepowsky, Zhang and the author hold. Consequently, for such V,this category has a natural structure of braided tensor category. Inparticular, when $V$ is of positive energy and C_{2}-cofinite, Conditions 1--3are satisfied and thus all the conclusions hold.

Cofiniteness conditions, projective covers and the logarithmic tensor product theory