A Class of Grade Three Determinantal Ideals

Abstract

Abstract. Let k be a field containing the field Q of rational numbers and let R = k[xij|1 ≤ i ≤ m, 1 ≤ j ≤ n] be the polynomial ring over a field k with indeterminates xij. Let It(X) be the determinantal ideal generated by the t-minors of an m × n matrix X = (xij). Eagon and Hochster proved that It(X) is a perfect ideal of grade (m − t + 1)(n − t + 1). We give a structure theorem for a class of determinantal ideals of grade 3. This gives us a characterization that It(X) has grade 3 if and only if n = m + 2 and It(X) has the minimal free resolution F such that the second differential map of F is a matrix defined by complete matrices of grade n + 2.