Consider $g_1=x^2, g_2=y^2, g_3=xy+yz\in k[x,y,z]$ with a field $k$. We consider the reverse lexicographic order, and put $x>y>z$. I want to find the generators of the syzygies.
Eisenbud CA book, p739, exercise 15.27, says that it is $$(y^2,-x^2,0),(0,x+z,-y),((x+z)y,0,-x^2).$$
However my computation yields $$(y^2,-x^2,0),(0,x+z,-y),(y,0,-x+z).$$
Which is the correct generators?