I know that $\mathsf{P}^A = \mathsf{EXP}$ for any $\mathsf{EXPTIME}$-complete language $A$.
Is it true that $\mathsf{DTIME}^A(n^k) = \mathsf{EXP}$
for any fixed $k$ and any $\mathsf{EXPTIME}$-complete oracle $A$?
If not, what do these complexity classes equal and why?
I am just confused because then it seems to me that we would then have $\mathsf{DTIME}^A(n^k) = \mathsf{DTIME}^A(n^j)$ for $k < j$ and this would contradict the fact that the Time Hierarchy Theorem holds under any oracle.