Let $Y,Z$ be $\mathbb{F}_p$-schemes, they are both models of scheme $X$ over $\overline{\mathbb{F}_p}$ . Let $F$ be the absolute Frobenius of $\overline{\mathbb{F}_p}$, If $1_Y\times F$ and $1_Z\times F$, two endomorphisms of X are equal, can we deduce that $Y\cong Z$?
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2$\begingroup$ Yes, that is part of descent for coherent sheaves: consider the pushforward of the structure sheaf of the graph of an isomorphism in $(Y\times_{\text{Spec}\ \mathbb{F}_p} Z)\times_{\text{Spec}\ \mathbb{F}_p} \text{Spec}\ \overline{\mathbb{F}}_p$. $\endgroup$– Jason StarrCommented Jul 3, 2022 at 23:14
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