On products of beta and gamma elements in the homotopy of the first Smith-Toda spectrum (Q6657457)

\(\bullet\) Objective:\N\N-- The paper determines the first cohomology of the monochromatic comodule \(M^1_2\) at odd primes.\N\N-- It establishes the non-triviality of certain products of beta (\(\beta\)) and gamma (\(\gamma\)) elements in the homotopy groups of the Smith-Toda spectrum \(V(1)\).\N\N\(\bullet\) Results:\N\N-- The cohomology at primes greater than three was verified by the first author in previous work. This paper extends the analysis by determining the cohomology at the prime \(p=3\) through elementary calculations.\N\N-- Explicit generators for the cohomology are provided to facilitate further computations in monochromatic spectral sequences.\N\N\(\bullet\) Applications:\N\NThe computed cohomology serves as a foundational step for studying the cohomology of other modules, such as \(M^3_0\), which remains a long-standing objective in algebraic topology.\N\N\(\bullet\) Methodology:\N\N-- The Adams-Novikov spectral sequence is employed to connect computations in the cohomology of comodules to the homotopy of the Smith-Toda spectrum.\N\N-- Explicit formulas for differential maps and connecting homomorphisms in cohomology groups are derived.\N\N\(\bullet\) Key Theorems:\N\N-- Theorem 1.8: Provides conditions under which products of gamma elements with beta elements are non-trivial in the \(E_3\)-page of the spectral sequence.\N\N-- Theorem 2.9: Details the structure of the first cohomology \(H^1 M^1_2\), including explicit cyclic modules and their relations.\N\N\(\bullet\) Future Work: The results of this paper are expected to aid in the computation of the elusive cohomology \(H^\ast M^3_0\), a challenging problem in the field.\N\N\(\bullet\) Conclusions: This work deepens the understanding of the Smith-Toda spectrum and its connections to the algebraic and geometric topology of \(p\)-local spectra, particularly through the behavior of beta and gamma elements in stable homotopy theory. The explicit cohomology computations and techniques presented are likely to have implications for related studies in higher chromatic homotopy theory.