A graded Betti number 
counts the copies of 
in homological position 
of a minimal graded free
resolution of a finitely generated module 
over the standard graded polynomial
ring ![S=k[x_1,...,x_n]](/images/equations/GradedBettiNumber/Inline5.svg)
,
where 
is a field. The shift is 
, and the maps in the
minimal resolution have entries in 
(Bruns and Herzog 1998). Equivalently, using
Tor,
 |
The total Betti number in position 
is 
.
For example, 
has graded Betti numbers 
, with all others
0. They record the two generators of the ideal and their
relation in total degree 5. The lex-plus-powers
conjecture proposes simultaneous upper bounds
for graded Betti numbers.
