The Koszul complex is the cotangent complex

TL;DR

This paper generalizes the classical Koszul complex to the cotangent complex within the framework of operad algebras, extending Koszul duality theory beyond associative algebras.

Contribution

It introduces a new perspective by showing the cotangent complex generalizes the Koszul complex for operad algebras, broadening the scope of Koszul duality theory.

Findings

The cotangent complex generalizes the Koszul complex.

Koszul duality theory is extended to operad algebras.

Provides a new framework for algebraic cohomology over operads.

Abstract

We extend the Koszul duality theory of associative algebras to algebras over an operad. Recall that in the classical case, this Koszul duality theory relies on an important chain complex: the Koszul complex. We show that the cotangent complex, involved in the cohomology theory of algebras over an operad, generalizes the Koszul complex.