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

Abstract

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 $\mathbf{K}\otimes \tilde{C}$ 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.

20 pages