On Singer's invariant-theoretic description of the lambda algebra: A mod \(p\) analogue (Q1906929)

The lambda algebra \(\Lambda_p\) at the prime \(p\) is known as the \(E_1\)-term of the Adams spectral sequence for computing the homotopy groups of spheres at a prime \(p\). In his paper [Trans. Am. Math. Soc. 280, 673-693 (1983; Zbl 0533.55013)] \textit{W. M. Singer} describes the lambda algebra \(\Lambda_2\) in terms of modular invariant theory at the prime 2. As the authors write, this paper gives a \(\text{mod } p\) analogue of Singer's description of the lambda algebra for an odd prime \(p\). Actually, they construct a differential coalgebra \(\Gamma^+\) which is dual to the lambda algebra \(\Lambda_p\) using modular invariants of \(\text{GL} (n, \mathbb{Z}/p)\). Associating to this, they also construct a complex \(\Gamma^+ M\) for an \(A(p)\)-module \(M\) whose homology is \(\text{Tor}^{A(p)} (\mathbb{Z}/ p,M)\) as the image of the total power. This complex has advantages: simpler differentials and the diagonal \(A(p)\)-action.