The algebraic structure of twisted topological Hochschild homology

Topological Hochschild homology (THH) is an invariant of ring spectra developed by B\"okstedt. In recent years many equivariant analogues to THH have emerged. One example is twisted THH which is an invariant of $C_n$-equivariant ring spectra developed by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell. In this paper, we study the algebraic structure of twisted THH, and perform some computations. Specifically, we compute $C_2$-twisted THH of the Real bordism spectrum and show that the $C_p$-twisted THH of geometric ring $C_p$-spectra reduces to a computation of classical THH. We extend the algebraic structure of twisted THH to the twisted B\"okstedt spectral sequence of Adamyk, Gerhardt, Hess, Klang, and Kong. We show that, under appropriate flatness conditions and for $R$ a commutative ring $C_p$-spectrum, the $C_p$-twisted B\"okstedt spectral sequence is a spectral sequence of commutative $\underline{E}_\star(R)$-algebras.