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Koszulness and confluence conjecture for the family of associative–coassociative properads

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Let EE be the S\mathbb{S}-bimodule generated by an associative product and a coassociative coproduct. For a=(a1,a2,a3,a4)∈C4a=(a_1,a_2,a_3,a_4)\in\mathbb{C}^4, let Pa\mathcal{P}_a be the properad defined by the associative and coassociative relations together with the quadratic relation ≬a\between_a, and let φa:A⊠C→Pa\varphi_a:\mathcal{A}\boxtimes\mathcal{C}\to\mathcal{P}_a be the induced morphism, where A\mathcal{A} is the properad of associative algebras and C\mathcal{C} is the properad of coassociative coalgebras.

Conjecture. For every a∈C4a\in\mathbb{C}^4, the following are equivalent: the properad Pa\mathcal{P}_a induces a confluent system; the morphism φa\varphi_a is a bijection in weight 33; and Pa\mathcal{P}_a is Koszul.

The conjecture proposes that confluence, the weight-three behavior of the canonical morphism, and Koszulness coincide throughout this family. The supplied text gives no resolution status beyond presenting it as a conjecture.

References

Primary source

Silvère Nédélec, “Non-Koszulness in a family of properads”, arXiv:2504.02366 (2025).

koszul-algebrasproperadsrewriting-systems
Posted · edited

Progress summary

Refreshed
Open

A 2025 paper settled the confluence cases and some special cases, but the full equivalence remains open.

The conjecture asserts that three properties of the properads Pa\mathcal{P}_a coincide for every parameter a∈C4a\in\mathbb{C}^4: confluence, bijectivity of φa\varphi_a in weight 33, and Koszulness. Silvère Nédélec’s 2025 paper explicitly says its methods do not decide the conjecture.

Known results

  • Confluence is characterized completely: all aia_i are idempotent and a1a3=a1a4=a2a3=a2a4=0a_1a_3=a_1a_4=a_2a_3=a_2a_4=0, yielding seven parameter choices.
  • φa\varphi_a is surjective in weight 33 and injective in several biarities, including (1,4)(1,4), (4,1)(4,1), (1,2)(1,2), and (2,1)(2,1).
  • For parameters a=(a1,0,a3,0)a=(a_1,0,a_3,0), Nédélec proves a partial Koszulness classification.

3 April 2025 partial results

Nédélec’s article reformulates the remaining question for the confluent parameter set as equivalence between weight-33 bijectivity of φa\varphi_a and Koszulness of Pa\mathcal{P}_a, but presents this only as a conjecture; no proof, counterexample, or verification is reported.

Current status (as of September 2026): Confluence is classified and several partial cases are known, but the equivalence with weight-33 bijectivity and Koszulness remains open.

Sources

Solutions 0

No solutions have been posted yet.