The projective dimension 
of a unital R-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, p. 91). For a nonzero module, projective dimension is 0 iff
the module is projective.
For example, the module 
has projective dimension 1 over 
for every integer 
, 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.
