Complexity class separation in the presence of relativization barriers Give an example of complexity classes $M$ and $N$ and oracles $A$ and $B$ such that 
 1. $M^A=N^A$ and 
 2. $M^B\neq N^B$ and
 3. $M \neq N$.
H. Buhrman, L. Fortnow, and T. Thierauf. Nonrelativizing separations. In Proceedings of the 13th IEEE Conference on Computational Complexity, pages 8-12. IEEE, New York, 1998. (also from Lance's homepage).  They show unconditionally that $MA_{EXP} \not\subseteq P/poly$, and also build an oracle relative to which $MA_{EXP} \subseteq P/poly$.