On some properties of birational derived splinters

A Noetherian reduced ring $A$ is called a birational derived splinter if for all proper birational maps $X\to\operatorname{Spec}(A)$, the canonical map $A\to Rf_*\mathcal{O}_X$ splits. In equal characteristic zero this property characterizes rational singularities, but much less can be said in positive or mixed characteristics. In this paper, we prove some fundamental properties of this notion, including the behavior under localization, taking a pure subring, taking direct limit, and along an \'etale extension. In particular, direct limit of rational singularities in characteristic zero has rational singularities. Then, we study residue extensions (in arbitrary characteristic), and openness and regular extensions in positive characteristic, parallel to Datta-Tucker and the author's previous works on splinters.