Big Finitistic Dimensions for Categories of Quiver Representations

Assume that A is a Grothendieck category and R is the category of all A-representations of a given quiver Q. If Q is left rooted and A has a projective generator, we prove that the big finitistic flat (resp. projective) dimension FFD(A) (resp. FPD(A)) of A is finite if and only if the big finitistic flat (resp. projective) dimension of R is finite. When A is the Grothendieck category of left modules over a unitary ring R, we prove that if FPD(R) < +∞ then any representation of Q of finite flat dimension has finite projective dimension. Moreover, if R is n-perfect then we show that FFD(R) < +∞ if and only if FPD(R) < +∞.