Determinantal ideal

Summary

A determinantal ideal is the ideal generated by all minors of a fixed size of a matrix with entries in a commutative ring. Take the generic matrix, whose entries are independent variables. Then the zero set of the ideal is the determinantal variety of matrices of rank less than the size of the minors. These ideals are among the best understood classes of ideals. For the generic matrix they are prime, and their quotient rings are Cohen–Macaulay with a known height. They appear in the study of resolutions, of rank conditions, and of secant varieties.