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

Hochster established the existence of a commutative noetherian ring $\tilde C$andauniversalresolution$U$oftheform$0\to \tilde C^{e}\to \tilde C^{f}\to \tilde C^{g}\to 0$suchthatforanycommutativenoetherianring$S$andanyresolution$V$equalto$0\to S^{e}\to S^{f}\to S^{g}\to 0$, there exists a unique ring homomorphism $\tilde{C}\to S$ with $V=U\otimes_{\tilde C} S$\mathrm{.}Inthepresentpaperweassumethat$f=e+g$ and we find the minimal resolution of $\mathbf{K}\otimes \tilde{C}$ by free $B$-modules, where $\mathbf{K}$ is a field of characteristic zero and $B$ is a polynomial ring over $\mathbf{K}$. 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.