  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.
