Let $X$ be a smooth projective $k$-scheme, where $k=\mathbb{F}_p$ and $p$ is prime. We have an identification of the de Rham cohomology of $X$ with $H^*_{crys}(X/k)$: $H^*_{DR}(X/k)\cong H^*_{crys}(X/k)$ (note that I am not considering crystalline cohomology $H^*_{crys}(X/W)=\lim H^*_{crys}(X/W_n)$, where $W=\mathbb{Z}_p$ is the Witt ring of $k$, and $W_n=W/p^n$). My understanding is that $H^*_{crys}(X/k)$ is the cohomology of a sheaf of $k$-algebras $\mathcal{O}$ on a site. Thus, we may consider the Frobenius endomorphism of $\mathcal{O}$: $F:\mathcal{O}\to \mathcal{O}$, $t\mapsto t^p$ for every section $t$. Clearly, $F$ induces a linear (because $k=\mathbb{F}_p$) endomorphism of any injective resolution of $\mathcal{O}$, and so we get an induced endomorphism $F^*$ of $H^*_{DR}(X/k)$.

Questions: is $F^*$ always the identity? If not, how to compute $F^*$? For example, what is $F^*$ if $X$ is a projective space (other examples would be welcome too)?

I tried to find this information in standard references, but they always consider the case of $X/\mathbb{Z}_p$, and they consider $H^*_{crys}[1/p]$.

Let $X$ be a smooth projective $k$-scheme, where $k=\mathbb{F}_p$ and $p$ is prime. We have an identification of the de Rham cohomology of $X$ with $H^*_{crys}(X/k)$: $H^*_{DR}(X/k)\cong H^*_{crys}(X/k)$ (note that I am not considering crystalline cohomology $H^*_{crys}(X/W)=\lim H^*_{crys}(X/W_n)$, where $W=\mathbb{Z}_p$ is the Witt ring of $k$, and $W_n=W/p^n$). My understanding is that $H^*_{crys}(X/k)$ is the cohomology of a sheaf of $k$-algebras $\mathcal{O}$ on a site. Thus, we may consider the Frobenius endomorphism of $\mathcal{O}$: $F:\mathcal{O}\to \mathcal{O}$, $t\mapsto t^p$ for every section $t$. Clearly, $F$ induces a linear (because $k=\mathbb{F}_p$) endomorphism of any injective resolution of $\mathcal{O}$, and so we get an induced endomorphism $F^*$ of $H^*_{DR}(X/k)$.

Questions: is $F^*$ always the identity? If not, how to compute $F^*$? For example, what is $F^*$ if $X$ is a projective space (other examples would be welcome too)?

I tried to find this information in standard references, but they always consider the case of $X/\mathbb{Z}_p$, and they consider $H^*_{crys}(X/W)[1/p]$.