F-Rationality of Determinantal Rings and Their Rees Rings

Let X be an m × n matrix (m ≤ n) of indeterminates over a field K of positive characteristic, and denote the ideal generated by its t-minors by It . We show that the Rees ringR(It ) ofK[X], as well as the algebra At generated by the t-minors, are F-rational if charK > min(t, m − t). Without a restriction on characteristic this holds for K[X]/Ir+1 and the symbolic Rees ring R(It ). The determinantal ring K[X]/Ir+1 is actually F-regular, as was previously proved by Hochster and Huneke [13] and Conca and Herzog [5] through different approaches. Our main tool is the filtration induced by the straightening law. The associated graded ring with respect to this filtration is typically given by a Segre productK[H ] #Nm F(X), whereH is a normal semigroup representing the weights of the standard bitableaux present in the object under consideration, N represents all the possible weights, and F(X) parameterizes the set of standard bitableaux of K[X]. The ring F(X) itself is the Segre product F1(X) #Nm F2(X), where F1(X) (resp. F2(X)) are the coordinate rings of the flag varieties associated withX (resp. the transpose of X). We prove that F(X) is F-regular. Normal semigroup rings are also F-regular since they are direct summands of polynomial rings. Furthermore, F-regularity is inherited by Segre products, and F-rationality is preserved under deformations. Hence a ring with an associated graded ring of type K[H ] #Nm F(X) is (at least) F-rational. This applies especially to K[X]/Ir+1,R(It ), and At . The results and the method of this paper are a variant of the method applied by Bruns [1] in characteristic 0, where F-rationality is to be replaced by the property of having rational singularities. By a theorem of Smith [15], our results in positive characteristic actually imply those previously obtained in characteristic 0.