What is the relation between Gröbner basis and syzygies?

Question

Let $I=(f_1,f_2,f_3,f_4,f_5) $ an ideal in $S=\mathbb {C}[w,x,y,z ]$.

$f_1=w^2-xz $

$f_2=wx-yz$

$f_3=x^2-wy$

$f_4=xy-z^2$

$f_5=y^2-wz $

I want to compute the syzygies of $S/I $ using Gröbner basis.

I know the definition of syzygies and I need to know the $S$-graded minimal free resolution for $S/I $ to compute the syzygies.

What is the relation between Gröbner basis and syzygies? I noticed that $G $={$f_1,f_2,f_3,f_4,f_5$} is a Gröbner basis of $I $.

Answer 1

Let $F = (f_1, \dots, f_5)\in S^5$ then $F$ is a Gröbner basis of the ideal $I = \langle f_1, \dots, f_5 \rangle$ if and only if every syzygy $(g_1, \dots, g_5)$ of the leading monomial $\mathop{LM}(I)$ can be lifted to a syzygy of $F$.

You can find details about the connection between Gröbner basis and syzygies for example in Computational Commutative Algebra 1 by Kreuzer and Robbiano and in Using Algebraic Geometry by Cox, Little and O'Shea.