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A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{O}_{X}} $. Such a map $ \phi $ is called a splitting. A closed sub-scheme $ Y $ of $ X $ is compatibly split if there exists a splitting $ \phi $ such that $ \phi(F_{\ast}(\mathcal{I}_{Y})) \subseteq \mathcal{I}_{Y} $. Note that $ \mathcal{I}_{Y} \subseteq \phi(F_{\ast}(\mathcal{I}_{Y})) $ already.

If $ X $ is a normal variety, then $ X $ if $ F $-regular if for all effective Weil divisors $ D $ there is an $ e \in \mathbb{N} $ such that $ \mathcal{O}_{X} \to F^{e}_{\ast}(\mathcal{O}_{X}) \to F^{e}_{\ast}(\mathcal{O}_{X}(D)) $ splits.

It is a theorem that the following statements are equivalent for a projective variety $ Z $ (I am using Hartshorne's definition of variety as an integral, separated, scheme of finite type, over an algebraically closed field $ k $) : a) $ Z $ is a projective, $ F $-regular (respectively $ F $-split), variety b) for all invertible sheaves $ \mathcal{L} \in \operatorname{Pic}(Z) $, the section ring $ R(Z,\mathcal{L}) \cong \oplus_{m \in \mathbb{N}_{0}} H^{0}(Z, \mathcal{L}^{\otimes m}) $ is $ F $-regular (respectively $ F $-split), c) the ring $ R(Z, \mathcal{L}) $ is $ F $-regular (respectively $ F $-split) for some ample $ \mathcal{L} \in \operatorname{Pic}(Z) $.

If $ Z $ is a $ \mathbb{Q} $-factorial, $ F $-regular, Mori dream space (so $ \operatorname{Cox}(Z) $ is finitely generated) then is $ \operatorname{Cox}(Z) $ an $ F $-regular ring?

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    $\begingroup$ Yes. To check if a (multi-)graded ring is F-regular one has to check only homogeneous things. Then use the global F-splittings along divisors on $X$ and see what they do to various sections on the Cox ring. $\endgroup$ Jan 25 at 17:06
  • $\begingroup$ Thank you @KarlSchwede! $\endgroup$
    – Schemer1
    Jan 26 at 4:01

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