Poincar\'e's lemma for formal manifolds

TL;DR

This paper extends Poincaré's lemma to de Rham complexes on formal manifolds, advancing the understanding of formal Lie algebra homologies and cohomologies within the framework of formal geometry.

Contribution

It establishes Poincaré's lemma for de Rham complexes with coefficients in formal functions and distributions on formal manifolds, generalizing classical results to formal geometric settings.

Findings

Proves Poincaré's lemma for formal functions and distributions.

Extends de Rham cohomology theory to formal manifolds.

Provides foundational results for formal Lie algebra homologies.

Abstract

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In two previous papers, we develop the basic theory of formal manifolds, including generalizations of vector-valued distributions and generalized functions on smooth manifolds to the setting of formal manifolds. In this paper, we establish Poincar\'e's lemma for de Rham complexes with coefficients in formal functions, formal generalized functions, compactly supported formal densities, or compactly supported formal distributions.