1 6 O ct 1 99 3 PROJECTIVE RESOLUTIONS FOR GRAPH PRODUCTS

Let Γ be a finite graph together with a group Gv at each vertex v. The graph product G(Γ) is obtained from the free product of all Gv by factoring out by the normal subgroup generated by {ghgh; g ∈ Gv, h ∈ Gw} for all adjacent v, w. In this note we construct a projective resolution for G(Γ) given projective resolutions for each Gv, and obtain some applications. Let Γ be a finite graph together with a group Gv at each vertex v. The graph product G(Γ) is obtained from the free product of all Gv by factoring out by the normal subgroup generated by {ghgh; g ∈ Gv, h ∈ Gw} for all adjacent v, w. In this note we construct a projective resolution for G(Γ) given projective resolutions for each Gv, and obtain some applications. This is quite easy to do, since G(Γ) is built up from the vertex groups by direct products and amalgamated free products. Let A be an arbitrary group, and let R be a commutative ring. A projective resolution for A is an exact sequence P : . . . → Pn dn → Pn−1 → . . . P1 → P0 → R → 0 of projective (right) RA-modules. We shall always assume that P0 = RA. Let G = A∗C B and let Q and N be projective resolutions of B and C. Suppose that N ⊗RC RA and N ⊗RC RB are summands of P and Q respectively. Let Sk be the quotient of (Pk ⊗RA RG) ⊕ (Qk ⊗RB RG) by the submodule generated by all (n⊗RC 1,−n⊗RC 1), with the obvious map from Sk to Sk−1. Lemma 1. In the above situation, . . . Sk → Sk−1 → . . . → S1 → RG → R → 0 is a projective resolution for G. Proof. It is easy to check that each Sk is projective. Let IA be the augmentation ideal of A, with similar notation for B,C,G. We can regard the resolution P as finishing with P2 → P1 → IA → 0, and this sequence