Terminal 3-folds that are not Cohen-Macaulay

An important local vanishing theorem for the minimal model program is the fact that klt singularities in characteristic zero are Cohen-Macaulay. In contrast, even in the narrow setting of terminal singularities of dimension 3, we show that Cohen-Macaulayness can fail in characteristic $p$ or mixed characteristic $(0,p)$ for $p$ equal to 2, 3, or 5. This is optimal, by work of Arvidsson-Bernasconi-Lacini. The examples are quotients of regular schemes by the cyclic group $G$ of order $p$. In characteristic $p$ or mixed characteristic, such quotients can exhibit a wide range of behavior. Our key technical tool is a sufficient condition for quotients by $G$ to have only toric singularities.