We will discuss versions of the Frobenius homomorphism for a ring spectrum $R$: the *Tate-valued Frobenius* $R \to R^{tC_{p}}$ and the *Frobenius on topological Hochschild homology* $\mathrm{THH} (R) \to\mathrm{THH} (R)^{tC_{p}}$. Similar to ordinary algebra, these morphisms play an important role in higher algebra and are related to various concepts in stable homotopy theory and algebraic $K$-theory. We discuss the notion of *perfectness*, which is to say that these morphisms are equivalences, and relate this notion to the Segal conjecture, the red-shift conjecture, and the classification of spaces by stable data.

Frobenius homomorphisms in higher algebra

Thomas Nikolaus

Abstract