Unstable algebras over an operad

Sacha Ikonicoff

Homology, homotopy, and applications, Tome 23 (2021) no. 1, pp. 119-141.

Résumé

The aim of this article is to define and study a notion of unstable algebra over an operad that generalises the classical notion of unstable algebra over the Steenrod algebra. For this study we focus on the case of characteristic $2$. We define $\star$-unstable $\mathcal{P}$-algebras, where $\mathcal{P}$ is an operad and $\star$ is a commutative binary operation in $\mathcal{P}$. We then build a functor that takes an unstable module $M$ to the free $\star$-unstable $\mathcal{P}$-algebra generated by $M$. Under some hypotheses on $\star$ and on $M$, we identify this unstable algebra as a free $\mathcal{P}$-algebra. Finally, we give some examples of this result, and we show how to use our main theorem to obtain a new construction of the unstable modules studied by Carlsson, Brown–Gitler, and Campbell–Selick, that takes into account their internal product.

Détail

DOI : 10.4310/HHA.2021.v23.n1.a8

Classification : 17A30, 18D50, 55S10
Keywords: Steenrod algebra, operad, unstable module, unstable algebra