  For a Noetherian local domain $R$ let $R^+$ be the absolute integral closure
of $R$ and let $R_{\infty}$ be the perfect closure of $R$, when $R$ has prime
characteristic. In this paper we investigate the projective dimension of
residue rings of certain ideals of $R^+$ and $R_{\infty}$. In particular, we
show that any prime ideal of $R_{\infty}$ has a bounded free resolution of
countably generated free $R_{\infty}$-modules. Also, we show that the analogue
of this result is true for the maximal ideals of $R^+$, when $R$ has residue
prime characteristic. We compute global dimensions of $R^+$ and $R_{\infty}$ in
some cases. Some applications of these results are given.
