Direct summands and klt-type singularities

For a ring $R$ with an action of a reductive group $G$, let $R^G$ be the subring of invariant elements. Under some assumptions on the base field the inclusion $R^G \to R$ is a direct summand as an $R^G$-module. This algebraic fact is quite useful for studying the geometry of quotients by the reductive group actions. In 1987 J.-F. Boutot used this fact to prove that a quotient of a variety with rational singularities by the action of a reductive group still has rational singularities. While this was known before, Boutot's proof was shorter and simpler. A similar thing happened recently: in 2021 Braun, Greb, Langlois, and Moraga proved that a quotient of a klt-type singularity by the action of a reductive group is of klt-type. Their proof was geometric, but just a year after them Z. Zhuang found a shorter proof of that result, using the algebraic fact about direct summands as well as some other tricks from commutative algebra. I'll discuss Zhuang's argument.