big finitistic dimensions for categories of quiver representations

Assume that a is a grothendieck category and r is the category of all arepresentations 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 nperfect then we show that ffd(r) < +∞  if and only if fpd(r) < +∞.