The main theorem of this paper gives a general technique for constructing an oracle which separates a deterministic and nondeterministic complexity class while the classes at a higher level are equal. As a corollary of the main theorem we obtain a result in [Book, Wilson, Xu, "Relativizing Time, Space, and Time-Space," SIAM J. Comput., 11(3), 1982, 571-581) that there is an oracle A such that P^A ≠ NP^A yet DEXT^A = NEXT^A. Another corollary is that for some oracle A, DTIME^A(O(n)) ≠ NTIME^A(O(n)) and DTIME^A(n²) = NTIME^A(n²). Some connections with sparse sets are then discussed.