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Let $X$ be a scheme over an algebraically closed field $k$ of positive characteristic $p$. Recall that the absolute Frobenius morphism $F : X \to X$ is the map which is the identity on points and the $p^{th}$ power morphism on functions. Recall also that we say that $X$ is Frobenius split if there is an $\mathcal O_X$-linear morphism splitting the $p^{th}$ power morphism $\mathcal O_X \to F_* \mathcal O_X$.

Now, whenever one sees the definition of Frobenius splitting, it is always stated for an algebraically closed field $k$. However, the definitions above make perfectly good sense for any scheme over $\mathbb F_p$, and in fact many Frobenius-split schemes, eg flag varieties, are "split over $\mathbb F_p$" in the sense that the Frobenius splitting is the appropriate base-change of a morphism $F_* \mathcal O_X \to \mathcal O_X$ for a scheme $X$ over $\mathbb F_p$. (Although I defined the Frobenius morphism above for schemes over $k$ it also makes sense for any scheme over $\mathbb F_p$). My question is: Why is the definition of Frobenius splitting always stated for schemes over an algebraically closed field, when one can state it more generally for schemes over any field containing $\mathbb F_p$? Is this just convention or is there a deeper reason?

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# Frobenius splitting over non-algebraically closed fields

Let $X$ be a scheme over an algebraically closed field $k$ of positive characteristic $p$. Recall that the absolute Frobenius morphism $F : X \to X$ is the map which is the identity on points and the $p^{th}$ power morphism on functions. Recall also that we say that $X$ is Frobenius split if there is an $\mathcal O_X$-linear morphism splitting the $p^{th}$ power morphism $\mathcal O_X \to F_* \mathcal O_X$.

Now, whenever one sees the definition of Frobenius splitting, it is always stated for an algebraically closed field $k$. However, the definitions above make perfectly good sense for any scheme over $\mathbb F_p$, and in fact many Frobenius-split schemes, eg flag varieties, are "split over $\mathbb F_p$" in the sense that the Frobenius splitting is the appropriate base-change of a morphism $F_* \mathcal O_X \to \mathcal O_X$ for a scheme $X$ over $\mathbb F_p$. (Although I defined the Frobenius morphism above for schemes over $k$ it also makes sense for any scheme over $\mathbb F_p$). My question is: Why is the definition of Frobenius splitting always stated for schemes over an algebraically closed field, when one can state it more generally for schemes over $\mathbb F_p$? Is this just convention or is there a deeper reason?