Determinantal ideal theorem

From Commalg
Jump to: navigation, search

History

The result was proved for polynomial rings by Macaulay and for arbitrary Noetherian rings by Eagon.

Statement

Let M be a p \times q matrix with entries over a Noetherian ring R. Denote by I_k(M) the ideal generated by the k \times k minors of M. Then, the codimension of any prime ideal minimal over I_k(M) is at most (p - k + 1)(q - k + 1).

The case k = p = 1 yields Krull's height theorem and the case p = q = k = 1 yields Krull's principal ideal theorem.