1968/01/01 by K. W. Gruenberg · 1 citation
Mathematics · #Finite Group Theory Research #Geometric and Algebraic Topology #Advanced Algebra and Geometry
paper · doi:10.1112/jlms/s1-43.1.239
openalex publication_date 1968/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
The object of this note is to present a very direct and completely unsophisticated proof of the universal coefficient theorem in the cohomology theory of groups. It makes use of formulae (see (4), below) that express the integral homology groups of a group G in terms of the ideal structure of ZF, the integral group ring of a free group F having G as a homomorphic image. The formula in dimension 2 is essentially the same as the well-known one of Schur and Hopf (see (5) below). If G is our given group, let 1-> R-> F-> G-> 1 be any free presentation of G. This was shown in [2] to determine the following G-free resolution of Z: l0t2-> &1&91- » ZG-+ Z-> 0, (1) where J5 " is the augmentation ideal of F (i.e., the kernel of the natural augmentation ZF-> Z) and 0t is the kernel of ZF-> ZG induced by F-> G. Consequently, for each i>0 and any G-module M, Ht(G, M)cz Ker (Y®M-+X®M), (2) ZG ZG I where X = ^ " / ^ " + i, Y = &W/0F+i if i = 2/i +1, v n if i=2«. Since 1/®Z ^ U/U & for any G-module L/, where ^ is the augmentation ideal of G, ZG we have, putting M = Z in (2), tf2n+! (G, Z) ~ H2n(G, Z) ^ Ker (^"/^" Thus for all H2n(G, L)1^ ^ n These formulae can be read off from the diagram below. Mnemonic for constructing the diagram: the left-hand vertices are the numerators of the resolution; so are the right-hand vertices but transposed down one step and multiplied on the right by &. If we recall [2; top of p. 488] that 0t 0t % R/R', as G-modules, then