Pure ring maps arise naturally in algebraic geometry and commutative algebra. For example, inclusions of rings of invariants by linearly reductive groups, split maps, and faithfully flat maps are all pure. It is useful to know when a map is pure because pure maps satisfy effective descent properties and many classes of singularities are preserved under the operation of taking pure subrings. For example, a theorem of Boutot (for rings of finite type over a field) and myself (in general) says that for Noetherian \(\mathbf Q\)-algebras, pure subrings of rings with (at worst) rational singularities have (at worst) rational singularities. Such results are often called Boutot-type theorems.

In this talk, I will discuss a new class of pure ring maps which arise geometrically from families of varieties of the same dimension. In fact, this yields a characterization of the splinter property: A Noetherian ring \(R\) is a splinter (i.e., all module-finite extensions \(R \to S\) split) if and only if every locally equidimensional surjective morphism \(Spec(S) \to Spec(R)\) is pure. Since not all pure maps are locally equidimensional and Boutot-type theorems fail for some classes of singularities like \(F\)-rationality, this raises the question: Are there "weak" Boutot-type theorems for pure ring maps that are also equidimensional? I will discuss my affirmative solution to this question for \(F\)-rationality.