Determinantal ideal theorem

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.