We will present joint work with Luchezar L. Avramov and Srikanth B. Iyengar. In the following (R,m) will denote a commutative local Noetherian ring, and L, M finitely generated R-modules. It is well-known that if R is a Gorenstein ring and L an R-module, then Ext_R^i(L,R) = 0 for i > dim R. The natural question that we wonder is: what if R is not Gorenstein? We'll prove that if there exists an M such that L is contained in M, mL is not equal to mM and mM is contained in L, then Ext_R^i(L,R) does not vanish for infinitely many i's if and only if R is not Gorenstein. The main tool used is stable cohomology, in particular a new decomposition theorem of stable cohomology.