  In this paper, we introduce a new homological invariant called
quasi-projective dimension, which is a generalization of projective dimension.
We discuss various properties of quasi-projective dimension. Among other
things, we prove the following. (1) Over a quotient of a regular local ring by
a regular sequence, every finitely generated module has finite quasi-projective
dimension. (2) The Auslander--Buchsbaum formula and the depth formula for
modules of finite projective dimension remain valid for modules of finite
quasi-projective dimension. (3) Several results on vanishing of Tor and Ext
hold for modules of finite quasi-projective dimension.
