Integral topological Hochschild homology of connective complex K-theory

Abstract We compute the homotopy groups of normal upper T normal upper H normal upper H left parenthesis normal k normal u right parenthesis T H H ( k u ) $\mathrm{THH}(\mathrm{ku})$ as a normal k normal u Subscript asterisk k u ∗ $\mathrm{ku}_\ast$ -module using the descent spectral sequence for the map normal upper T normal upper H normal upper H left parenthesis normal k normal u right parenthesis right arrow normal upper T normal upper H normal upper H left parenthesis normal k normal u divided by normal upper M normal upper U right parenthesis T H H ( k u ) → T H H ( k u / M U ) $\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$ , which is the motivic spectral sequence for normal upper T normal upper H normal upper H left parenthesis normal k normal u right parenthesis T H H ( k u ) $\mathrm{THH}(\mathrm{ku})$ in the sense of Hahn–Raksit–Wilson. We reduce the computation of homotopy groups to the algebra of the universal formal group law, providing a systematic way to compute THH of quotients of MU. We compute the upper E 2 E 2 $E_2$ -page of the motivic spectral sequence computing upper T upper H upper H left parenthesis normal k normal u right parenthesis