Unstable algebras over an operad (Q2214744)

This paper defines and studies a notion of unstable algebra over an operad in characteristic \(2\). This generalizes the classical notion of an unstable algebra over the (mod \(2\)) Steenrod algebra \(\mathcal{A}\) and is shown to recover other notions of unstable modules studied in the literature by Carlsson, Brown-Gitler, and Campbell-Selick. For the whole article, the base field is \(\mathbb{F}:=\mathbb{F}_2\). An unstable module over the Steenrod algebra \(\mathcal{A}\) is a graded \(\mathcal{A}\)-module satisfying the \emph{instability relation} \(\operatorname{Sq}^ix = 0\) for all \(i>|x|\). The category of unstable modules over the Steenrod algebra is denoted by \(\mathcal{U}\). An unstable algebra is a commutative, associative algebra satisfying the \emph{Cartan formula} \[ \operatorname{Sq}^i(x\cdot y) = \sum_{i_1 + i_2 = i}\operatorname{Sq}^{i_1}x\cdot\operatorname{Sq}^{i_2}y \] and the \emph{instability relation} \(\operatorname{Sq}_0x = x\cdot x\), where \(\operatorname{Sq}_0x:=\operatorname{Sq}^{|x|}x\). Given a symmetric operad \(\mathcal{P}\) concentrated in degree \(0\), one can consider \(\mathcal{P}\)-algebras in \(\mathcal{U}\). These algebras are shown to satisfy a generalization of the Cartan formula and -- as a corollary -- that \(\operatorname{Sq}_0\) is automatically compatible with the algebraic structure. Next, a generalization of the notion of unstable algebra is considered. Given a symmetric operation \(\star\in\mathcal{P}(2)\), a \emph{\(\star\)-unstable \(\mathcal{P}\)-algebra} is defined to be a \(\mathcal{P}\)-algebra in \(\mathcal{U}\) such that \(\operatorname{Sq}_0x = \star(x, x)\). For example, the classical notion of unstable algebras is recovered by considering \(\mathcal{P}=\mathrm{uCom}\) the operad encoding unital commutative algebras and by taking \(\star\) to be the product. Given an operad \(\mathcal{P}\) and a symmetric operation \(\star\), the free \(\star\)-unstable \(\mathcal{P}\)-algebra \(K^\star_\mathcal{P}(M)\) generated by an unstable module \(M\) is explicitly described as a quotient of the free \(\mathcal{P}\)-algebra on \(M\). The main result of the article relates \(K^\star_\mathcal{P}(M)\) with the free \(\mathcal{P}\)-algebra on \(\Sigma\Omega M\) provided that \(\star\) and \(M\) satisfy certain conditions. Here, \(\Sigma\) denotes the suspension, and \(\Omega\) is its left adjoint. \textbf{Theorem 6.11.} If \(\star\) is \(\mathcal{P}\)-central, then for all connected reduced unstable modules \(M\) there exists an isomorphism of graded \(\mathcal{P}\)-algebras between the \(\star\)-unstable \(\mathcal{P}\)-algebra \(K^\star_\mathcal{P}(M)\) and the free \(\mathcal{P}\)-algebra generated by \(\Sigma\Omega M\). The isomorphism is not natural and depends on the choice of a section \(s:\Sigma\Omega M\to M\). Finally, it is shown that the Brown-Gitler algebra, the Carlsson algebra, and the Campbell-Selick algebra are all \(\star\)-unstable \(\mathcal{P}\)-algebras for appropriate \(\mathcal{P}\) and \(\star\), and Theorem 6.11 is applied to show that in fact they are all \emph{free} \(\star\)-unstable \(\mathcal{P}\)-algebras generated by some unstable modules.