The resolution of the universal ring for finite length modules of projective dimension two

TL;DR

This paper constructs a universal resolution for certain finite length modules of projective dimension two, providing a characteristic-independent resolution and analyzing its properties and implications for Betti numbers.

Contribution

It explicitly constructs a non-minimal, coordinate-free universal resolution for modules of projective dimension two, applicable over polynomial rings and independent of characteristic.

Findings

The resolution allows computation of Tor groups with integers.

Betti numbers depend on the characteristic of the field when e,g ≥ 5.

The modules relate to resolutions over determinantal rings.

Abstract

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.