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.