In this paper we use tight closure and Gro bner basis theory to prove that ladder determinantal rings have rational singularities.
We show that the ladder determinantal rings of a certain class of ladders, which we call wide ladders, are F-rational. Though F-rationality is only defined in positive characteristic, recent results of Smith [26] imply that these ladder determinantal rings are pseudorational in the sense of Lipman and Tessier [20], and in characteristic 0 are of F-rational type which in turn implies that they have rational singularities.
We will further show that an arbitrary ladder determinantal ring is an algebra retract of the determinantal ring of a wide ladder. Thus we may apply Boutot's theorem [3] to conclude that all ladder determinantal rings defined over an algebraically closed field of characteristic 0 have rational singularities. With some more effort one probably could avoid Boutot's theorem and prove instead that all ladder determinantal rings are F-rational; see Remark 4.5. For this we have to check that certain simplicial complexes which arise from the ladder are shellable. Assuming the ladders are wide simplifies the arguments considerably, and, as we hoped, makes the proof more readable.
Ladder determinantal rings were introduced by Abhyankar [1] in his studies of singularities of Schubert varieties of flag manifolds. An important subclass of general ladders are the one-sided ladders. In [22] Mulay showed that one-sided ladder determinantal rings occur as coordinate rings of certain affine sets in Schubert varieties, and Ramanathan [24] showed that all Schubert varieties have rational singularities. So their results cover a special case of our Theorem 1.7. Another special case, namely that of ladder determinantal rings which are complete intersections, has been treated by Glassbrenner and Smith [12].