Splinter-type characterizations of singularities

The Direct Summand Theorem states that if $R$ is a commutative Noetherian ring, then any finite extension $R\to S$ splits as a map of $R$-modules. This suggests the notion of a splinter as a class of singularities, where we say a ring $R$ is a splinter if any finite extension of rings $R\to S$ splits as a map of $R$-modules. In this talk, I'll present examples and discuss the history of using splinter-type conditions to classify singularities, including work of Bhatt and Kov\’acs on rational singularities, with the goal of introducing a recent result giving a splinter-type characterization of klt singularities.