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.


Toric varieties are $ F $-split. Does anyone know of an example of an affine, toric, variety $ U\_{\sigma} $ and a point $ x \in U\_{\sigma} $ such that $ x $ is not compatibly split? Here I am using Hartshorne's definition of a "variety" as an integral, separated, scheme of finite type over an algebraically closed field.



Does there exist a point $ x $ of an affine toric variety $ U_{\sigma} $ such that $ x $ is not compatibly split?
Actually, on an affine variety, if it is $F$-split, then it is compatibly $F$-split with every point (possibly changing the splitting). This is not true in the projective case (an ordinary projective elliptic curve is $F$-split, but not compatibly split with any point). It is also not true for non-closed points (for instance the generic point of a cusp in $\mathbb{A}^2$ cannot be compatibly split, since that would force the cusp itself to be $F$-split).


**Lemma.** *If $R$ is a $F$-split $F$-finite ring, then for any maximal ideal $Q \subseteq R$, there is a Frobenius splitting compatibly split with $m$.*


*Proof.* Suppose $R$ is an $F$-split ring and $Q \in Spec R$ is any closed point. Let $\phi : F\_\* R \to R$ be an $F$-splitting. If $\phi$ is compatible with $Q$ we are done, so suppose its not compatible with $Q$. Thus $\phi(F\_\* Q) \not\subseteq Q$. Since outside of $Q$, $\phi$ is surjective, we actually see that $\phi(F\_\* Q) = R$. Thus there exists $x\_1 \in Q$ with $\phi(F\_\* x\_1) = 1$. Let $\phi\_1 = \phi \circ (\cdot F\_\* x)$. In other words, $\phi\_1(F\_\* y) = \phi(F\_\* xy)$. Note $\phi\_1$ is also a Frobenius splitting (since it also sends $1$ to $1$).


If $\phi\_1$ is compatible with $Q$, we are also done, so we repeat the process. Continuing in in this way, we eventually come to a Frobenius splitting $\phi\_{n}$ where $n > p d + 1$ where $d$ is the number of generators of the ideal $Q = (f\_1, \dots, f\_d)$. Now,
$$1 = \phi\_n(F\_\* 1) = \phi(F\_\* x\_1 x\_2 \dots x\_n).$$
By construction, $x\_1 x\_2 \dots x\_n \in Q^{pd + 1} \subseteq (f\_1^p, \dots, f\_d^p) =: Q^{[p]}$, that containment is basically the pigeonhole principal. On the other hand, an easy computation shows that $\phi(F\_\* Q^{[p]}) \subseteq Q$ and so we see that $1 = \phi\_n(F\_\* 1) \in Q$, a contradiction. Thus at some earlier point we must have had a Frobenius splitting $\phi\_i$ that was compatible with $Q$.


