  We investigate homological properties of perfect algebras of prime
characteristic. The principle is as follows: perfect algebras resolve the
singularities. For example, we show any module over the ring of absolute
integral closure has finite flat dimension. Under some mild conditions, we show
any module over that ring has finite projective dimension. We compute weak
dimension and global dimension of perfect rings in a series of nontrivial
cases. Some interesting applications are given. In particular, we answer some
questions asked by Shimomoto.
