1. By definition, we have a ring map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$. Post-compose with the canonical map $\mathbb Z_p\to\mathbb F_p$, we get a ring map $\mathbb Z_p\to\operatorname{THH}(\mathbb F_p)$;
  2. On Nikolaus-Scholze, Chapter Examples, section Rings of characteristic $p$, they computed $\operatorname{TC}_*(\mathbb F_p)=\mathbb Z_p$ when $*=-1,0$, and $\operatorname{TC}_*(\mathbb F_p)=0$ when $*\neq-1,0$. Consequently, there is an $S^1$-equivariant ring map $\mathbb Z_p\to\operatorname{THH}(\mathbb F_p)$ where $\mathbb Z_p$ is endowed with the trivial $S^1$-action. Moreover, this map preserves the cyclotomic structures;
  3. As an upshot of the preceding, they succeed to identify $\operatorname{THH}(\mathbb F_p)$ with $\tau_{\ge0}\mathbb Z_p^{tC_p}$ as rings with $S^1$-actions, hence the Tate-valued Frobenius $\mathbb Z_p\to\mathbb Z_p^{tC_p}$ induces a ring map $\mathbb Z_p\to\operatorname{THH}(\mathbb F_p)$.

On Nikolaus-Scholze, they claim that these three structures are different. I wonder how to show this. I find it difficult to characterize the map $\mathbb F_p\to\operatorname{THH}(\mathbb F_p)$ along the identification $\operatorname{THH}(\mathbb F_p)\simeq\tau_{\ge0}\mathbb Z_p^{tC_p}$.


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