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For C a factorisable and pivotal finite tensor category over an algebraicallyclosed field of characteristic zero we show: 1) C always contains a simpleprojective object; 2) if C is in addition ribbon, the internal characters ofprojective modules span a submodule for the projective SL(2,Z)-action; 3) theaction of the Grothendieck ring of C on the span of internal characters ofprojective objects can be diagonalised; 4) the linearised Grothendieck ring ofC 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 aunique-up-to-scalars modified trace function. We express the modified trace ofopen Hopf links coloured by projectives in terms of S-matrix elements.Furthermore, we give a Verlinde-like formula for the decomposition of tensorproducts of projective objects which uses only the modular S-transformationrestricted to internal characters of projective objects.  We compute the modified trace in the example of symplectic fermioncategories, and we illustrate how the Verlinde-like formula for projectiveobjects can be applied there.

Projective objects and the modified trace in factorisable finite tensor categories