  A resolution-free definition of rational singularities is introduced, and it
is proved that for a variety admitting a resolution of singularities, so in
particular in characteristic zero, this is equivalent to the usual definition.
It is also demonstrated that pseudo-rational singularities are rational. The
main theorem, a Kempf-type criterion for rational singularities, is new even in
characteristic zero contrasting an earlier result of Cutkosky that a similar
result with slightly different assumptions could not hold.
  Several applications are included. In particular, an Elkik-type theorem,
proving that Cohen-Macaulay dlt singularities are rational in arbitrary
characteristic.
