 
Summary: How to make free resolutions with Macaulay2
Chris Peterson and Hirotachi Abo
1. What are syzygies?
Let k be a field, let R = k[x0, . . . , xn] be the homogeneous coordinate ring
of Pn
and let X be a projective variety in Pn
. Consider the ideal I(X) of X.
Assume that {f0, . . . , ft} is a generating set of I(X) and that each polynomial
fi has degree di. We can express this by saying that we have a surjective
homogenous map of graded Smodules:
t
i=0
R(di) I(X),
where R(di) is a graded Rmodule with grading shifted by di, that is,
R(di)k = Rkdi
.
In other words, we have an exact sequence of graded Rmodules:
t
i=0 R(di)
&&MMMMMMMMMM
