# Different algebra-structures on $\operatorname{THH}(\mathbb F_p)$?

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 $$\operatorname{THH}(\mathbb F_p)\simeq\tau_{\ge0}\mathbb Z_p^{tC_p}$$.