Kaledin classes and formality criteria

TL;DR

This paper introduces a comprehensive obstruction theory using Kaledin classes to determine the formality of algebraic structures over arbitrary commutative rings, extending previous methods to broader contexts.

Contribution

It generalizes Kaledin's obstruction classes to non-zero characteristic and more algebraic structures, enabling new formality criteria and descent results.

Findings

Developed a general obstruction theory for algebraic structure formality

Extended formality criteria to include torsion coefficients and families

Established new methods for detecting formality via chain-level automorphisms

Abstract

We develop a general obstruction theory to the formality of algebraic structures over any commutative ground ring. It relies on the construction of Kaledin obstruction classes that faithfully detect the formality of differential graded algebras over operads or properads, possibly colored in groupoids. The present treatment generalizes the previous obstruction classes in two directions: outside characteristic zero and including a wider range of algebraic structures. This enables us to establish novel formality criteria, including formality descent with torsion coefficients, formality in families, intrinsic formality, and criteria in terms of chain-level lifts of homology automorphism.