Parametrizing Ω ₁(Z):G=C _∞×Φ

Abstract

Both first and second minimality criteria fail in a case of particular interest, namely when G is a direct product G=F×Φ in which F is a free group and Φ is finite. As we observed in Proposition 13.2, when Φ is the trivial group, the conclusion also fails. Nevertheless, using a rather more intricate argument, we are still able to show that the conclusion is sustained when the finite factor Φ is nontrivial; that is we shall show:

Third minimality criterion::

$\mathcal{I}$ lies at the minimal level of Ω ₁(Z) when G is a direct product F _m ×Φ where F _m is a free group of rank m≥1 and Φ is finite and nontrivial.

The results of this section first appeared in Johnson (J. Algebra 337:181–194, 2011). The proof requires a knowledge of all the syzygies Ω _r (Z) over Z[F _n ×C _m ] so we begin by giving a complete resolution of Z in this case.