Initial Algebras of Determinantal Rings, Cohen–Macaulay and Ulrich Ideals

Let K be a field and X an m×n matrix of indeterminates over K. Let K[X] denote the polynomial ring generated by all the indeterminates Xij . For a given positive integer r ≤ min{m, n}, we consider the determinantal ideal Ir+1 = Ir+1(X) generated by all r + 1 minors of X if r < min{m, n} and Ir+1 = (0) otherwise. Let Rr+1 = Rr+1(X) be the determinantal ring K[X]/Ir+1. Determinantal ideals and rings are well-known objects, and the study of these objects has many connections with algebraic geometry, invariant theory, representation theory, and combinatorics. See Bruns and Vetter [BrV] for a detailed discussion. In the first part of this paper we develop an approach to determinantal rings via initial algebras. We cannot prove new structural results on the rings Rr+1 in this way, but the combinatorial arguments involved are extremely simple. They yield quickly that Rr+1, with respect to its classical generic point, has a normal semigroup algebra as its initial algebra. Using general results about toric deformations and the properties of normal semigroup rings, one obtains immediately that Rr+1 is normal and Cohen–Macaulay, has rational singularities in characteristic 0, and is F -rational in characteristic p. Toric deformations of determinantal rings have been constructed by Sturmfels [St] for the coordinate rings of Grassmannians (via initial algebras) and by Gonciulea and Lakshmibai [GoL] for the class of rings considered by us. The advantage of our approach, compared to that of [GoL], is its simplicity. Moreover, it allows us to determine the Cohen–Macaulay and Ulrich ideals of Rr+1. Suppose that 1 ≤ r < min{m, n} and let p (resp., q) be the ideal of Rr+1 generated by the r-minors of the first r rows (resp. the first r columns) of the matrix X. The ideals p and q are prime ideals of height 1 and hence they are divisorial, because Rr+1 is a normal domain. The divisor class group Cl(Rr+1) is isomorphic to Z and is generated by the class [p] = −[q] (see [BrH, Sec. 7.3; BrV, Sec. 8]). The symbolic powers of p and q coincide with the ordinary ones. Therefore, the ideals p and q represent all reflexive rank-1 modules. The goal of Section 4 is to show that p (resp., q ) is a Cohen–Macaulay ideal if and only if k ≤ m − r (resp., k ≤ n− r). In addition, we prove that the powers pm−r and qn−r are even Ulrich ideals.