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Cartier modules on toric varieties

Assume that $X$ is an affine toric variety of characteristic $p > 0$. Let $Δ$ be an effective toric $Q$-divisor such that $K_X+Δ$ is $Q$-Cartier with index not divisible by $p$ and let $ϕ_Δ:F^e_* O_X \to O_X$ be the toric map corresponding to $Δ$. We identify all ideals $I$ of $O_X$ with $ϕ_Δ(F^e_* I)=I$ combinatorially and also in terms of a log resolution (giving us a version of these ideals which can be defined in characteristic zero). Moreover, given a toric ideal $\ba$, we identify all ideals $I$ fixed by the Cartier algebra generated by $ϕ_Δ$ and $\ba$; this answers a question by Manuel Blickle in the toric setting.