Algebras defined by powers of determinantal ideals

Abstract

Let X be an m x n matrix of indeterminates over a field K, A = K[X], and I= Z,(X) the ideal generated by the t-minors of X. The main objects of this article are the Rees algebra 2 = Y these characteristics will be called non-exceptional. We show how this result generalizes to an arbitrary integral domain of coefficients: The intersection of primary ideals which gives I’ in non-exceptional characteristics, always is the integral closure of I’. It follows immediately from the primary decomposition that the powers I’ are integrally closed in non-exceptional characteristics. Therefore &! is a normal domain, and the primary decomposition of I.!% turns out easy, giving some insight into the structure of Y. An interesting observation: The primary decomposition of the ideals I’ can be computed very quickly if one knows in advance that all these ideals are integrally closed. The best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group GL(m, K) x GL(n, K) on K[X] under which Z is stable, cf. [DEP] or [BV, Sect. 111. Applying the theory of U-invariants (Kraft [Kr]) one shows that ,!G$! has rational singularities, in particular &? and, hence, 9 are Cohen-Macaulay rings. We have no doubt that %! and 3 are Cohen-Macaulay in arbitrary non-exceptional characteristic. It seems however that in exceptional characteristic they are as far as possible from this property: The case t = 2