In fact, the abstract definition in scheme theory of invertible sheaf can be replaced by the condition of being "locally free of rank 1". That is, the condition of a tensor inverse then implies, locally on "X", that "S" is the sheaf form of a free rank 1 module over a commutative ring. Examples come from fractional ideals in algebraic number theory, so that the definition captures that theory. More generally, when "X" is an affine scheme "Spec(R)", the invertible sheaves come from projective modules over "R", of rank 1.