On the minimal free resolution of the universal ring for resolutions of length two

Hochster established the existence of a commutative noetherian ring \(\tilde C\) and a universal resolution \(U\) of the form \(0\to \tilde C^{e}\to \tilde C^{f}\to \tilde C^{g}\to 0\) such that for any commutative noetherian ring \(S\) and any resolution \(V\) equal to \(0\to S^{e}\to S^{f}\to S^{g}\to 0\), there exists a unique ring homomorphism \(\tilde C\to S\) with \(V=U\otimes_{\tilde C} S\). In the present paper we assume that \(f=e+g\) and we find the minimal resolution of \({\bf K}\otimes \tilde C\) by free \(B\)-modules, where \(\bf K\) is a field of characteristic zero and \(B\) is a polynomial ring over \(\bf K\). Our techniques are geometric. We use the Bott algorithm and the Representation Theory of the General Linear Group. As a by-product of our work, we resolve a family of maximal Cohen-Macaulay modules defined over a determinantal ring.