2025/12/17 by Murayama, Takumi
#13B10 (Primary) 13A35 #13D22 #14B05 (Secondary) #14B25 #14D06 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.15563
We prove that a Noetherian ring R is a splinter if and only if for every equidimensional surjective morphism Spec(S) → Spec(R), the map R → S is pure. This yields a large, nontrivial class of ring maps that are automatically pure. More generally, we prove that a locally Noetherian scheme Y is locally a splinter if and only if every locally equidimensional morphism X → Y is strongly pure. Special cases of our results show that equidimensional fibrations over normal Q-schemes or regular schemes of arbitrary characteristic are strongly pure. The main ingredient is a new factorization result for locally equidimensional morphisms of schemes, which is of independent interest. Additionally, we prove a weak Boutot-type theorem for F-rationality, which says that F-rationality descends under pure ring maps that are locally equidimensional under universally catenary assumptions. This statement is false without the locally equidimensional hypothesis.