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Quillen (co)homology of divided power algebras over an operad

Ioannis Dokas, Martin Frankland, Sacha Ikonicoff

TL;DR

The paper develops a unified operadic framework for Quillen cohomology of divided power algebras over a operad $P$. It identifies Beck modules with $A$-modules via a universal enveloping algebra $\mathbb{U}_{\Gamma(P)}(A)$, defines derivations and a module of Kähler differentials $\Omega_{\Gamma(P)}(A)$, and constructs the cotangent complex to realize Quillen cohomology HQ$^*(A;M)$. It then specializes to the prototypical operads $\mathrm{Com}$ and $\mathrm{Lie}$, recovering classical divided power and restricted Lie algebra theories within the operadic setting, including their enveloping algebras, derivations, and differential theory. The work further develops comparison maps arising from adjunctions between $\Gamma(P)$-algebras and $P$-algebras, and proves a general good-triple comparison theorem in characteristic zero, recovering Cartan–Eilenberg type results and extending them to new operadic contexts such as dendriform/brace structures. Overall, the article provides a cohesive, characteristic-aware framework that links divided power/cohomology theories with classical algebraic cohomology via universal envelopes and cotangent complexes, enabling systematic comparisons across operadic settings.

Abstract

Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example André--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad $P$: the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad $P$, identifying Beck modules, derivations, and Kähler differentials in that setup. We also compare the cohomology of divided power algebras over $P$ with that of $P$-algebras, and work out some examples.

Quillen (co)homology of divided power algebras over an operad

TL;DR

The paper develops a unified operadic framework for Quillen cohomology of divided power algebras over a operad . It identifies Beck modules with -modules via a universal enveloping algebra , defines derivations and a module of Kähler differentials , and constructs the cotangent complex to realize Quillen cohomology HQ. It then specializes to the prototypical operads and , recovering classical divided power and restricted Lie algebra theories within the operadic setting, including their enveloping algebras, derivations, and differential theory. The work further develops comparison maps arising from adjunctions between -algebras and -algebras, and proves a general good-triple comparison theorem in characteristic zero, recovering Cartan–Eilenberg type results and extending them to new operadic contexts such as dendriform/brace structures. Overall, the article provides a cohesive, characteristic-aware framework that links divided power/cohomology theories with classical algebraic cohomology via universal envelopes and cotangent complexes, enabling systematic comparisons across operadic settings.

Abstract

Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example André--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad : the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad , identifying Beck modules, derivations, and Kähler differentials in that setup. We also compare the cohomology of divided power algebras over with that of -algebras, and work out some examples.
Paper Structure (21 sections, 59 theorems, 223 equations)

This paper contains 21 sections, 59 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Quillen cohomology
  3. Recollections on divided power algebras
  4. Abelian Gamma(P)-algebras, A-modules
  5. Beck modules over A
  6. Let us prove that the data of a Beck module over $A$ yields an $A$-module.
  7. Let us now prove that the data of an $A$-module yields a Beck module over $A$.
  8. Universal enveloping algebra
  9. Derivations, Module of Kähler differentials
  10. Quillen cohomology
  11. Example: The operad Com
  12. Classical definition and state of the art
  13. Operadic point of view
  14. Example: The operad Lie
  15. Classical definition and state of the art
  16. ...and 6 more sections

Key Result

Proposition 1

The structure of a $\Gamma(\mathcal{P})$-algebra is entirely determined by monomial operations whose monomial degrees are powers of the characteristic of the base field.

Theorems & Definitions (120)

  • Proposition : see Proposition \ref{['prop:powersp']}
  • Theorem : see Theorem \ref{['thm:Amodule']}
  • Theorem : See Theorem \ref{['thm:BeckMod']}
  • Theorem : See Theorem \ref{['thm:Abelianization']}
  • Theorem : See Theorem \ref{['thm:AssLie']}
  • Theorem : See Theorem \ref{['thm:good']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 110 more