The \(BP \langle 2\rangle \)-cooperations algebra at odd primes (Q2301890)

In this paper, the author studies the algebra \(\pi_* (BP\langle 2 \rangle \wedge BP \langle 2 \rangle)\) for the Wilson spectrum \(BP\langle 2 \rangle \) at odd primes by using the mod \(p\) homology Adams spectral sequence. He follows his strategy for the prime \(2\) from [\textit{D. L. Culver}, Algebr. Geom. Topol. 19, No. 2, 807--862 (2019; Zbl 1420.55029)], to which he refers for some arguments. This reduces to studying \(A//E(2)_*\) as an \(E(2)_*\)-comodule and its cohomology \(Ext_{E(2)} (\mathbb{F}_p , A//E(2)_*)\). (Here, as usual, \(A\) is the Steenrod algebra, \(E(n)\) is the subalgebra generated by the Milnor primitives \(Q^t\), \(0 \leq t \leq n\), and \(*\) indicates the dual.) First the author establishes a splitting of \(E(2)\)-modules: \[ A//E(2)_* \cong S \oplus Q \] where \(S\) is a free \(E(2)\)-module and \(Q\) has cohomology that is \(v_2\)-torsion free. The proofs exploit Margolis homology. He then turns to analysing the cohomology of \(Q\). For this, he reduces to working with certain generalized Brown-Gitler comodules. (For \(i\in \mathbb{N}\), the \(i\)th family of Brown-Gitler comodules is obtained by considering a weight filtration of \(A//E(i)_*\).) In particular, one has the family \(\underline{\ell}_j\) of \(A\)-subcomodules of \(A//E(1)_*\). The author establishes the isomorphism of \(E(2)_*\)-comodules: \[ A//E(2)_* \cong \bigoplus_{k\geq 0} \Sigma^{2(p-1)k} \underline{\ell}_{\lfloor k/p \rfloor}. \] Thus he reduces to studying the cohomology of the \(\underline{\ell}_j\). He proposes an inductive strategy for calculating these cohomology groups, based upon his exact sequences of \(E(2)_*\)-comodules \[ 0 \rightarrow \Sigma^{2(p-1)pj} \underline{\ell}_j \otimes \underline{\ell}_i \rightarrow \underline{\ell}_{pj+i} \rightarrow R^{pj-1} A //E(1)_* \rightarrow \bigoplus_{k=i+1}^{p-1} \Sigma ^{\varphi (j,k)} \underline{\ell}_{j -1} \rightarrow 0, \] for \(j \geq 1\), \(0\leq i <p\), where \(R^{pj-1} A //E(1)_*\) is a quotient of \(A//E(1)_*\) arising from an explicit filtration. The strategy is illustrated at the prime \(p=3\) by calculations for small \(j\).