Abstract:
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.
