On the generic free resolutions (Q1263618)

With the aid of structure theorems by Buchsbaum and Eisenbud for finite free resolutions, Hochster has given a universal pair for resolutions of \(length\quad 2.\) This means a ring R and a resolution \({\mathbb{F}}\) over R such that for each resolution \({\mathbb{G}}\) of the same size over a ring S there is a unique map \(R\to S\) so that \({\mathbb{G}}={\mathbb{F}}\otimes_ RS.\) Hochster proved that R is a noetherian normal CM-ring. - In this paper the authors consider the same problem for resolutions of \(length\quad 3.\) For resolutions of type (1,n,n,1) a generic ring is found, i.e. Hochster's result is generalized but for unicity. The authors conjecture that the ring is noetherian. The paper is highly technical by necessity. The methods used come from combinatorics and algebraic geometry.