The notion of residual intersections was introduced by Artin and Nagata. Let I be an ideal in a local Cohen-Macaulay ring R, and A = (a_1, \ldots, a_s) \subsetneq I. Then J = A:I is called an s-residual intersection of I if ht(J) \geq s \geq ht(I). Residual intersections provide a generalization of linkage. Indeed, if J = A:I and I = A:J for A a regular sequence, I and J are said to be linked.

I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard determinantal ideals and Pfaffian ideals arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with X. Ni, J. Torres and J. Weyman.