On higher order analogues of de Rham cohomology

TL;DR

This paper introduces higher order analogues of de Rham cohomology for commutative rings and algebras, showing their cohomology is invariant under certain conditions, with detailed exposition of differential calculus functors.

Contribution

It defines higher order de Rham complexes for commutative algebras and proves their cohomology invariance under smoothness assumptions, expanding classical de Rham theory.

Findings

Cohomology of higher order de Rham complexes is independent of the order sequence σ.

Provides a detailed exposition of differential calculus functors on commutative algebras.

Establishes invariance results under smoothness assumptions.

Abstract

If K is a commutative ring and A is a K-algebra, for any sequence $\sigma $ of positive integers there exists an higher order analogue dR($\sigma $) of the standard de Rham complex dR(1,...,1,...), which can also be defined starting from suitable ("differentially closed") subcategories of (A-mod). The main result of this paper is that the cohomology of dR($\sigma $) does not depend on $\sigma $, under some smoothness assumptions on the ambient category. Before proving the main theorem we give a rather detailed exposition of all relevant (to our present purposes) functors of differential calculus on commutative algebras. This part can be also of an independent interest.