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Homological freeness criterion for operadic modules and application to Cohen-Macalayness of posets

Paul Laubie

TL;DR

This work extends the homological freeness criterion to Koszul operads by linking freeness to diagonal-concentrated homology of chained (co)bar constructions, enabling explicit identification of generators for free modules. The authors then apply the criterion to decorated partition posets associated with several operads, proving Cohen–Macaulayness and computing poset homology in terms of species like $\mathrm{CycLie}$ and $\mathrm{Mag}$. Key results include freeness and decomposition statements for $\mathrm{PreLie}$, $\mathrm{Lie_2}$, and $\mathrm{PostLie}$-related structures, and concrete Cohen–Macaulay conclusions for posets $\Pi^{\mathrm{Perm}}_+$, $^\mathrm{Com_2}\Pi^+$, $\Pi^{\mathrm{Com_2}}_+$, and $\Pi^{\mathrm{ComTriAss}}_+$. The framework answers open questions in OpPOS and provides a versatile toolkit for studying homological properties of decorated operadic posets, with potential extensions to other operads like $\mathrm{Dup}$ and $\mathrm{TriAss}$.

Abstract

We show a variation of the usual homological freeness criterion for operadic modules over a Koszul operad. We then apply this result to decorated partition posets for some operads, showing that their augmentation is Cohen-Macaulay and computing its homology. This work answers several open questions asked by Bérénice Delcroix-Oger and Clément Dupont in a recent article.

Homological freeness criterion for operadic modules and application to Cohen-Macalayness of posets

TL;DR

This work extends the homological freeness criterion to Koszul operads by linking freeness to diagonal-concentrated homology of chained (co)bar constructions, enabling explicit identification of generators for free modules. The authors then apply the criterion to decorated partition posets associated with several operads, proving Cohen–Macaulayness and computing poset homology in terms of species like and . Key results include freeness and decomposition statements for , , and -related structures, and concrete Cohen–Macaulay conclusions for posets , , , and . The framework answers open questions in OpPOS and provides a versatile toolkit for studying homological properties of decorated operadic posets, with potential extensions to other operads like and .

Abstract

We show a variation of the usual homological freeness criterion for operadic modules over a Koszul operad. We then apply this result to decorated partition posets for some operads, showing that their augmentation is Cohen-Macaulay and computing its homology. This work answers several open questions asked by Bérénice Delcroix-Oger and Clément Dupont in a recent article.

Paper Structure

This paper contains 13 sections, 17 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recollection on species and operads
  3. Linear species
  4. Operadic twisting morphisms
  5. (Co)Bar complexes
  6. Main result
  7. The chained (co)bar constructions
  8. Freeness theorem
  9. Application of the main result to decorated partition posets
  10. The permutative operad
  11. The double commutative operad
  12. The commutative triassociative operad
  13. Other operads

Key Result

Theorem 1

crl:main Let $\mathcal{P}\to \mathcal{Q}$ a morphism of graded operads with $\mathcal{P}$ and $\mathcal{Q}$ Koszul. This morphism induces a free left (resp. right) $\mathcal{P}$-module structure on $\mathcal{Q}$ if and only if the homology of $\mathbf{\Omega}(\mathcal{P}^\textnormal{¡},\mathcal{Q}^\

Figures (1)

  • Figure 1: Summary table of posets considered in the article.

Theorems & Definitions (25)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: AlgOp Lemma 6.4.12
  • Theorem 2.4: AlgOp Lemma 6.5.14
  • Theorem 2.5: AlgOp Theorem 6.6.5
  • Theorem 2.6: AlgOp Proposition 7.3.2
  • ...and 15 more