Completions of complexes of differential modules on singular schemes

Spencer cohomology theory studies the cohomology of chain complexes of modules over the ring of differential operators $\mathscr{D}$ of a smooth analytic space. In this paper we give a generalisation of Spencer cohomology suitable for singular schemes of finite type over a field. Our motivation was a conjecture of Vinogradov concerning the homological properties of differential operators on singular affine varieties; namely, that complexes of certain such operators are acyclic if and only if the variety is smooth. We will provide a negative answer to Vinogradov's conjecture as stated. In principle Vinogradov's conjecture can also be posed for the Spencer complex of a general $\mathscr{D}$-module -- however the answer is trivial, since singularities prohibit a definition of Spencer cohomology with any good properties. Our main result will be the construction of a Spencer complex on a large class of singular schemes which is suitable as a cohomology theory for the space. Following this we are able to ask the same question as Vinogradov in this case, where we give a more positive answer. Our main technique draws from Hartshorne's construction of de Rham cohomology by formal completion.