An invariant of free central extensions

If G is a free abelian finitely generated group, the 'most-general' 2-cocycle on G with trivial action is the map G × G → f Λ²G, defined as follows. Let e₁,...,e_n be a basis of G and f the bilinear map satisfying f(e_i, e_j) = e_i ∧ e_j if i < j and =0 if i ≥ j. We show that K^αG has global dimension 1 where K is the field of fractions of the group ring C[Λ²G] and α ε{lunate} H²(G, K^*) is represented by the map above. More generally for every finitely generated group we define an invariant ξ(G) and gl.dim(K^αG) = 1 is equivalent to ξ(G) = 1 for G free abelian. We also show that if G₁,..., G_n are non-commutative free groups, then ξ(G₁ × {divides} × G_n) = n. In general, if G is not a torsion group, 1 ≤ ξ(G) ≤ cd_C(G).