# Is the Cox ring of a $\mathbb{Q}$-factorial, $F$-regular, Mori dream space $F$-regular?

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?

• 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. Jan 25 at 17:06
• Thank you @KarlSchwede! Jan 26 at 4:01