Injective dimension of D-modules: a characteristic-free approach

We give a characteristic-free proof of the fact that if A is a ring of formal power series in a finite number of variables over a field k and M is any module over the ring of k-linear differential operators of A, then in the category of A-modules, the injective dimension of M is bounded above by the dimension of the support of M. This is applied to give a characteristic-free proof of the same inequality between the injective dimension and the dimension of the support for local cohomology modules HiI(R) where R is any regular Noetherian ring containing a field and I R is any ideal. This result for local cohomology modules had been proven before in characteristic 0 and characteristic p>0 by two methods that were completely different from each other.