On higher order analogues of de Rham cohomology

Abstract

In [A.M. Vinogradov, Some homological systems associated with the differential calculus in commutative algebras, Russian Math. Surveys 34 (6) (1979) 250–255] for any commutative K-algebra A, K being a commutative ring, any sequence σ of positive integers and any differentially closed (see Section 3) subcategory D of A− Mod, higher analogues dR σ D of the standard de Rham complex dR D ≡ dR (1,…,1,…) D and Spencer complexes were defined. In this paper a detailed exposition of all related functors of differential calculus over general commutative algebras is given for the first time together with some useful working techniques. In the second part of the paper, these techniques are then applied to prove that all complexes dR σ D are quasi-isomorphic under a smoothness assumption on the differentially closed subcategory D . This extends to arbitrary smooth categories of modules the quasi-isomorphism theorem for smooth manifolds and “regular” dR σ complexes proved in [G. Vezzosi, A.M. Vinogradov, Infinitesimal Stokes' formula for higher-order de Rham complexes, Acta Appl. Math. 49 (3) (1997) 311–329].