  For C a factorisable and pivotal finite tensor category over an algebraically
closed field of characteristic zero we show: 1) C always contains a simple
projective object; 2) if C is in addition ribbon, the internal characters of
projective modules span a submodule for the projective SL(2,Z)-action; 3) the
action of the Grothendieck ring of C on the span of internal characters of
projective objects can be diagonalised; 4) the linearised Grothendieck ring of
C is semisimple iff C 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 C carries a
unique-up-to-scalars modified trace function. We express the modified trace of
open Hopf links coloured by projectives in terms of S-matrix elements.
Furthermore, we give a Verlinde-like formula for the decomposition of tensor
products of projective objects which uses only the modular S-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.
