Today I am amused by a ingenious generalization of Principal Ideal Theorem.
Recall that the Principal Ideal Theorem says over a noetherian ring, any
prime ideal minimal over a ideal generated by r elements has height less
than or equal to r. Eagon generalized this theorem to determinantal ideals
in his thesis. The statement is as follows.
Let M be a r*s matrix over a northerian ring, let I(t) denote the ideals
generated by the t*t minors of M, then any prime ideal minimal over I(t)
has height less than or equal to (r-t+1)(s-t+1).
The Principal Ideal Theorem is the special case when t=1.
The original proof was given by Eagon in “Ideals defined by matrices and a
certain complex associated with them” using genelized Koszul complex.
There is also a short proof in the appendix of Matsumura’ s Commutative
Ring Theory”.
Shuchao
04/23/08
