  In Commutative Algebra structure results on minimal free resolutions of
Gorenstein modules are of classical interest. We define Gorenstein modules of
finite length over the weighted polynomial ring via symmetric matrices in
divided powers. We show that their graded minimal free resolution is selfdual
in a strong sense. Applications include a proof of the dependence of the monoid
of Betti tables of Cohen-Macaulay modules on the characteristic of the base
field. Moreover we give a new proof of the failure of the generalization of
Green's Conjecture to characteristic 2 in the case of general curves of genus
$2^n -1$.
