The Koszul complex is the cotangent complex

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.

💡 Research Summary

The paper extends Koszul duality, traditionally formulated for associative algebras, to the broader setting of algebras over an arbitrary operad 𝒪. The authors begin by recalling that in the classical associative case the Koszul complex is the central chain complex governing both homological calculations and the duality between an algebra and its Koszul dual cooperad. They then construct, for any 𝒪‑algebra A, a free 𝒪‑algebra F(V) together with its relation operad R, and define the bar construction B(A) and the Koszul complex C(A) in the operadic context.

The core contribution is the identification of the operadic cotangent complex T_A as a natural generalization of the Koszul complex. T_A is defined as the module of Kähler differentials of A equipped with a differential that combines the 𝒪‑action and the internal differential of the free 𝒪‑algebra. The authors construct an explicit morphism φ from the Koszul complex C(A) to the cotangent complex T_A and prove that φ is a quasi‑isomorphism provided A satisfies an operadic Koszulness condition (𝒪‑Koszul). This condition ensures that the operadic quadratic relations of A behave analogously to the classical quadratic relations in associative Koszul algebras.

To substantiate the theory, three families of operads are examined in detail. For the commutative operad Comm, the cotangent complex coincides with the classical de Rham complex, which is known to be the Koszul complex of a commutative Koszul algebra. For the associative operad Ass, the bar‑Koszul construction reproduces the standard Hochschild complex, and the cotangent complex is shown to be isomorphic to it. For the Poisson operad, which mixes commutative and Lie structures, the cotangent complex carries an L∞‑structure; nevertheless, under the Poisson Koszulness hypothesis it remains quasi‑isomorphic to the operadic Koszul complex.

The paper argues that this identification has several important consequences. First, it unifies the homological tools used in operadic deformation theory: the cotangent complex, previously introduced to define operadic cohomology, now subsumes the Koszul complex, eliminating the need for separate constructions. Second, because the cotangent complex naturally encodes higher homotopical information (e.g., L∞‑operations), it provides a more flexible framework for studying derived deformation problems, obstruction theories, and homotopy transfer. Third, the result opens the door to a systematic study of 𝒪‑Koszul modules, extensions, and derived categories, mirroring the well‑developed theory for associative algebras.

In conclusion, the authors present a clear pathway for future research: applying the operadic cotangent/Koszul complex to derived algebraic geometry, to the deformation quantization of Poisson structures, and to the homotopical analysis of higher‑categorical operads. By demonstrating that the cotangent complex is precisely the operadic analogue of the Koszul complex, the paper bridges a conceptual gap and equips mathematicians with a unified, powerful tool for operadic homological algebra.