On unstable modules over the Dickson algebras, the Singer functors \(R_{s}\) and the functors \(Fix_{S}\)
This paper continues the author's work [J. Pure Appl. Algebra 214, No. 8, 1435--1449 (2010; Zbl 1186.55009)] on unstable modules and Singer's functor \(R_s\) from the category \(\mathcal{U}\) of unstable modules over the mod \(2\) Steenrod algebra to the category \(D_s-\mathcal{U}\) of unstable modules equipped with an action of the Dickson algebra \(D_s\) [\textit{W. M. Singer}, Math. Proc. Camb. Philos. Soc. 90, 279--292 (1981; Zbl 0479.55013)]. An approximation \(\tilde{R}_s\) of \(R_s\) is given, and used in proving that the natural transformation of functors on \(\mathcal{U}\), from \(\text{Fix}_s(H^*V_s \otimes_{D_s} R_s(-))\) to \(1_{\mathcal U}\), is an isomorphism for all natural numbers \(s\). This implies that \(R_s\) gives a fully-faithful embedding of \(\mathcal{U}\) in \(D_s-\mathcal{U}\).
scientific article; zbMATH DE number 6138568