Let $R$ be a local Noetherian ring and $I$ an $R$-ideal. What can we say about the ideal $I$ if the projective dimension of $R/I^i$ for $i \ge 1$ is a finite number which is independent of $i$, i.e., $\operatorname{pd}_R R/I^i = c$ for all $i$?
Of course, in the obvious case when $R$ is regular and $I$ is primary to the maximal ideal, one cannot say much. What about the case where $R$ is not regular, so having a finite projective dimension is not automatic? I'm fine with assuming $R$ to be a complete intersection.