Proposition 1.47
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The converse statement is incorrect if the polynomial system is not
homogeneous. A counter-example is (Y(X-1), Z(X-1), X): the system is not a
regular sequence, but the syzygy module is generated by the syzygies
(Z(X-1)*e3-X*e2, Y(X-1)*e3-X*e1, Y(X-1)*e2-Z(X-1)*e1).
I thank Vanessa Vitse for sending me this counter-example.