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PD Operads and Explicit Partition Lie Algebras

Lukas Brantner, Ricardo Campos, Joost Nuiten

Abstract

Infinitesimal deformations are governed by partition Lie algebras. In characteristic $0$, these higher categorical structures are modelled by differential graded Lie algebras, but in characteristic $p$, they are more subtle. We give explicit models for partition Lie algebras over general coherent rings, both in the setting of spectral and derived algebraic geometry. For the spectral case, we refine operadic Koszul duality to a functor from operads to divided power operads, by taking refined linear duals of $Σ_n$-representations. The derived case requires a further refinement of Koszul duality to a more genuine setting.

PD Operads and Explicit Partition Lie Algebras

Abstract

Infinitesimal deformations are governed by partition Lie algebras. In characteristic , these higher categorical structures are modelled by differential graded Lie algebras, but in characteristic , they are more subtle. We give explicit models for partition Lie algebras over general coherent rings, both in the setting of spectral and derived algebraic geometry. For the spectral case, we refine operadic Koszul duality to a functor from operads to divided power operads, by taking refined linear duals of -representations. The derived case requires a further refinement of Koszul duality to a more genuine setting.

Paper Structure

This paper contains 28 sections, 105 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Outline
  4. Acknowledgements
  5. Functors on pro-coherent modules
  6. Pro-coherent modules
  7. Extended functors
  8. Monoidal structures
  9. $\mathscr{O}$-monoidal structures
  10. PD operads and refined Koszul duality
  11. A reminder on $\infty$-operads
  12. The levelwise tensor product
  13. Pro-coherent symmetric sequences and PD operads
  14. Refined Koszul duality
  15. Derived operads and derived PD operads
  16. ...and 13 more sections

Key Result

Proposition 1.2

Let $R$ be a coherent $\mathbb{E}_\infty$-ring. Then $\mathop{\mathrm{sSeq}}\nolimits^\vee_R$ admits a monoidal structure $\circ$, the pro-coherent composition product, which preserves small colimits in the first and sifted colimits in the second variable. If $X, Y$ are 'continuous duals' of symmetr

Figures (1)

  • Figure :

Theorems & Definitions (340)

  • Proposition 1.2: Pro-coherent composition product
  • Remark 1.3
  • Theorem 1: \ref{['thm:chain model for exotic composition']} (Point-set model for pro-coherent ${\circ}$ )
  • Remark 1.4
  • Definition 1.5: Divided power operads
  • Theorem 2: \ref{['thm:koszul duality for PD operads']} (Refined Koszul duality for operads)
  • Example 1.6: Partition Lie algebras
  • Theorem 3: \ref{['thm:koszul duality for algebras']} (Refined Koszul duality for algebras)
  • Theorem 4: \ref{['chainkoszul']} (Chain models for Koszul duality)
  • Definition 1: \ref{['explicitone']} (Spectral partition $L_\infty$-algebra)
  • ...and 330 more