A minimal free resolution of a finitely generated graded module 
over a commutative Noetherian 
-graded ring 
in which all maps are homogeneous module homomorphisms, i.e.,
they map every homogeneous element to a homogeneous element of the same degree. It
is usually written in the form
 |
(1)
|
where 
indicates the ring 
with the shifted graduation such that, for all 
,
 |
(2)
|
For all nonnegative integers 
and all integers 
, 
is the number of copies of 
appearing in the 
th module of the resolution, and is called graded Betti
number. The ordinary 
th Betti number is 
.
For example, if 
is the polynomial ring ![K[X_1,X_2,X_3]](/images/equations/GradedFreeResolution/Inline15.svg)
over a field 
, with the usual graduation, the graded free resolution of
is
 |
(3)
|
In 
,
the constant polynomials have degree 2. It follows that 
has degree 5. Similarly, 
has degree 5 in 
.
The graded free resolution can be used to compute the Hilbert function.
