For a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1) always contains a simple projective object;(2)if is in addition ribbon, the internal characters of projective modules span a submodule for the projective-action;(3)the action of the Grothendieck ring of on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of is semisimple if and only if is semisimple.Results (1)-(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.