Cohomology of the sub-Hopf algebra \(D\) of the Steenrod algebra
Let \(A\) be the mod\,\(2\) Steenrod algebra, \(D\) be the sub-Hopf-algebra of \(A\) generated by \(\{P_t^s, s< t\}\) where \(P_t^s=Sq(0,\dots,0, 2^s, 0,\dots)\), \(2^s\) at the \(t\)-th place and \(\{Sq(r_1,\dots ,r_n), r_i\geq 0, n\geq 1\}\) denotes the Milnor basis of \(A\). The aim of this paper is the computation (at least the understanding) of the mod\,\(2\) cohomology of \(D\): \(H^{*}(D; \mathbb{F}_{2})= \text{Ext}_D^{*}(\mathbb{F}_2; \mathbb{F}_2)\). One of the motivation of this question is the result of \textit{J. H. Palmieri} [Ann. Math. (2) 149, 421--449 (1999; Zbl 0932.55021)] which states that the inclusion \(D \hookrightarrow A\) induces an \(F\)-isomorphism: \(H^{*}(A; \mathbb{F}_2) \rightarrow [H^{*}(D; \mathbb{F}_2)]^{A/IA}\) where \(I\) is the augmentation ideal of \(D\) and an \(F\)-isomorphism is an isomorphism up to nilpotence. Using technical methods of the May spectral sequence, the author computes \(H^{*}(D; \mathbb{F}_2)\), for \(* \leq 2\).
scientific article; zbMATH DE number 5146862
scientific article