The Cohomology of Modules over a Complete Intersection Ring

We investigate the cohomology of modules over commutative complete intersection rings. The first main result is that if M is an arbitrary module over a complete intersection ring R, and if Ext 2n R (M, M) = 0 for some n ≥ 1 then M has finite projective dimension. The second main result gives a new proof of the fact that the support variety of a Cohen-Macaulay module whose completion is indecomposable is projectively connected. iii ACKNOWLEDGMENTS First and foremost, I would like to thank my thesis advisor Srikanth B. Iyengar for his guidance and support throughout my graduate career. I would also like to thank the members of my committee, Luchezar Avramov and Mark Walker, whose help has been invaluable.