Differential graded algebras with divided powers and homotopy Lie algebras

Given a commutative algebra $A$ and a quotient $A$-algebra $A/I$, we construct a resolution of $A/I$ as an $A$-module such that it is also a differential graded (dg) algebra with divided powers (PD). This construction makes use of symmetric tensors in the symmetric tensor category of dg $A$-modules and does not require a Noetherian assumption on $A$. Moreover, the resolution has many lifting properties which we leverage to study the homotopy Lie algebra associated to the pair $(A,A/I)$, which is defined as the cohomology of the PD derivations of this PD dg algebra. Finally we investigate the complete intersection case in more details as well as connect it to the finite generation of the Yoneda algebra.

💡 Research Summary

The paper develops a comprehensive framework for constructing differential graded (dg) algebras equipped with divided power (PD) structures and uses these objects to define and study homotopy Lie algebras associated with a pair ((A, A/I)), where (A) is an arbitrary commutative ring and (I) an ideal. The authors begin by formalising the notion of a PD dg ring: a strictly graded‑commutative dg ring together with an ideal (I) and a family of maps (\gamma_n: I_{\mathrm{ev}}\to I_{\mathrm{ev}}) satisfying the classical divided‑power axioms (including compatibility with the differential). They show that the category of PD dg rings is symmetric monoidal, although the tensor product does not coincide with the coproduct unless suitable flatness (freeness) conditions hold.

A central technical contribution is the construction of a functor \