Let a,b be two ideals of a noetherian local ring (R,m,k) such that the ideal a+b is m-primary. It can happen that dimR/a+dimR/b > dimR, but it has been conjectured that this is not the case when there is a finitely generated module M of finite projective dimension, annihilated by the ideal a, with dimM = dimR/a.
We strengthen the condition on R/a and prove the formula dimR/a+dimR/b £ dimR when the local ring R contains a field and when the ring R/a has a canonical module of finite injective dimension over R (thus forcing the ring R to be Cohen-Macaulay).
Not surprisingly, one of the key points in this proof is the improved new intersection theorem, valid in this case.
[Back to Anne-Marie Simon homepage]