THE RESOLUTION OF THE UNIVERSAL RING FOR MODULES OF RANK ZERO AND PROJECTIVE DIMENSION TWO

R S. In the present paper we assume that f = e + g and we nd a resolution F of R by free P-modules, where P is a polynomial ring over the ring of integers. The resolution F is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use F to calculate TorP (R; Z). If e and g both at least 5, then TorP (R; Z) is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of K Z R by free K Z P-modules depend on the characteristic of the eld K. We record the modules in the minimal K Z P resolution of K ZR in terms of the modules which appear when one resolves divisors over the determinantal ring dened by the 2 2 minors of an e g matrix. Introduction