# Completions of complexes of differential modules on singular schemes

## TL;DR

This paper extends Spencer cohomology to singular schemes, constructs a suitable complex for these spaces, and investigates the homological properties of differential operators, providing insights into Vinogradov's conjecture.

## Contribution

It introduces a new Spencer complex for singular schemes, enabling cohomology theory in these contexts and offering a nuanced perspective on differential operators' homological properties.

## Key findings

- Constructed a Spencer complex applicable to a large class of singular schemes.
- Provided a negative answer to Vinogradov's conjecture in its original form.
- Offered a more positive perspective on the conjecture within the new framework.

## Abstract

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.