The projective dimension 
of a module 
over a unit ring 
is the smallest nonnegative
integer 
for which there is an exact sequence
 |
with every 
a projective module. It is infinite
if no such finite projective resolution
exists (Weibel 1994). For a nonzero module,
projective dimension is 0 iff the module
is projective.
For example, the module 
has projective dimension 1 over 
when 
, as shown by the exact
sequence
 |
It is not a projective module, since every projective module over 
is a free module. The finitistic
dimension takes the supremum of projective dimensions
that are finite.
