A RIGIDITY THEOREM FOR EXT

The goal of this paper is to show that if R is an unramified hypersurface, if M and N are finitely generated R-modules, and if Ext(R)(n)(M, N) = 0 for some n <= grade(R)M, then Ext(R)(i)(M, N) = 0 for i <= n. A corollary to this says that if M not equal 0, then Ext(R)(i)(M, M) not equal 0 for 0 <= i <= grade(R)M. We also give an extension of this result to complete intersections. These results are related to a question of Jorgensen and results of Dao.