The little finitistic dimension of a unital associative algebra 
is the supremum of the finite projective dimensions of its finitely
generated modules. For left modules, it is
 |
The big finitistic dimension 
uses all left modules of
finite projective
dimension, without the finite-generation requirement. Consequently, 
. For a field
both are 0, since every module is a vector
space and hence a free module.
The little finitistic dimension conjecture asserts that 
is finite for every algebra of finite vector
space dimension over a field.
