On the structure of free resolutions of length 3 (Q917613)

Let \((R,F)\) be a pair where \(R\) is a commutative ring, and \[ F: 0\to F_3\overset{d_3} {\longrightarrow} F_2\overset{d_2} {\longrightarrow} F_1\overset{d_1} {\longrightarrow} F_0 \] a complex of length 3 and of type \((r_1,r_2,r_3),\) i.e. \(\mathrm{rank}(d_ i)=r_i\), \(\mathrm{rank}(F_i)=f_i=r_i+r_{i+1}\), \(i=1,2,3,\) \(r_4=0\). The aim of the paper is to construct such a pair \((R_{\text{gen}},F_{\text{gen}})\) which the author hopes to be ``generic'' for complexes of length 3, that is for any pair \((S,G)\) of the same type, there is a unique homomorphism \(h: R_{\text{gen}}\to S\) such that \(G=F_{\text{gen}}\otimes_R S\). It is shown that to prove the genericity it is sufficient to prove the vanishing of the homology of a certain family of complexes over the enveloping algebra of the defect Lie algebra. Finally, the complete classification of possible multiplications modulo the maximal ideal on the finite free resolution of length 3 over a local ring is given.