Hochster established the existence of a commutative noetherian ring $\Cal R$ and a universal resolution $\Bbb U$ of the form $0\to \Cal R^{ e}\to \Cal R^{ f}\to \Cal R^{ g}\to 0$ such that for any commutative noetherian ring $S$ and any resolution $\Bbb V$ equal to $0\to S^{ e}\to S^{ f}\to S^{ g}\to 0$, there exists a unique ring homomorphism $\Cal R\to S$ with $\Bbb V=\Bbb U\otimes_{ \Cal R} S$. In the present paper we assume that $f=e+g$ and we find a resolution $\Bbb F$ of $\Cal R$ by free $\Cal P$-modules, where $\Cal P$ is a polynomial ring over the ring of integers. The resolution $\Bbb F$ is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use $\Bbb F$ to calculate $\operatorname{ Tor}^{ \Cal P}_{ \bullet}(\Cal R, \Bbb Z)$. If $e$ and $g$ both at least 5, then $\operatorname{ Tor}^{ \Cal P}_{ \bullet}(\Cal R, \Bbb Z)$ is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of $\pmb K\otimes_{ \Bbb Z} \Cal R$ by free $\pmb K\otimes_{ \Bbb Z} \Cal P$-modules depend on the characteristic of the field $\pmb K$. We record the modules in the minimal $\pmb K\otimes_{ \Bbb Z} \Cal P$ resolution of $\pmb K\otimes_{ \Bbb Z} \Cal R$ in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the $2\times 2$ minors of an $e\times g$ matrix.
Categories: math.AC