On the homology of the universal Steenrod algebra at odd primes (Q480368)

Let \(p\) be a prime number, \(\mathbb{F}_{p}\) the field with \(p\) elements and \(\mathcal{Q}\) the universal Steenrod algebra over \(\mathbb{F}_{p}\) (see [\textit{J. P. May}, Lect. Notes Math. 168, 153--231 (1970; Zbl 0242.55023)], [\textit{M. A. Mandell}, Topology 40, No. 1, 43--94 (2001; Zbl 0974.55004)], [\textit{D. Chataur} and \textit{M. Livernet}, Commun. Algebra 33, No. 11, 4337--4360 (2005; Zbl 1095.55010)]). We know that the homology \(H_{s,t}(\mathcal{Q}) = Tor_{s,t}^{\mathcal{Q}}(\mathbb{F}_{p}, \; \mathbb{F}_{p}) = 0\) if \(s \neq t\) see [\textit{M. Brunetti} and \textit{A. Ciampella}, Colloq. Math. 109, No. 2, 179--192 (2007; Zbl 1151.16027)]. In this paper, the author gives, for \(p\) an odd prime, an explicit description of the diagonal homology of \(\mathcal{Q}\): \(D_{*}(\mathcal{Q}) = \oplus_{k \geq 0}Tor_{k,k}^{\mathcal{Q}}(\mathbb{F}_{p}, \; \mathbb{F}_{p})\) extending the case \(p = 2\) (see [\textit{A. Ciampella} and \textit{L. A. Lomonaco}, Fundam. Math. 183, No. 3, 245--252 (2004; Zbl 1069.55014)]). In particular, he shows that \(D_{*}(\mathcal{Q})\) is isomorphic as a coalgebra, to \(\oplus_{n \geq 0}\Gamma_{n}\) where \(\Gamma_{n} = \Big( H^{*}(B(\mathbb{Z}/p\mathbb{Z})^{n}; \;\mathbb{F}_{p})[L_{n}^{-1}] \Big )^{GL_{n}(\mathbb{F}_{p})}\) is the ring of invariants of the localisation out of the Euler class \(L_{n}\) of \(H^{*}(B(\mathbb{Z}/p\mathbb{Z})^{n};\;\mathbb{F}_{p})\).