The resolution of the universal ring for modules of rank zero and projective dimension two

Abstract

Hochster established the existence of a commutative noetherian ring R and a universal resolution U of the form 0 → R e → R f → R g → 0 such that for any commutative noetherian ring S and any resolution V equal to 0 → S e → S f → S g → 0, there exists a unique ring homomorphism R → S with V = U ⊗ R S. In the present paper we assume that f = e + g and we find 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 Tor P • (R, Z). If e and g both are at least 5, then Tor P • (R, Z) is not a free abelian group; and therefore, the graded Betti numbers in the minimal resolution of K K K ⊗ Z R by free K K K ⊗ Z P-modules depend on the characteristic of the field K K K. We record the modules in the minimal K K K ⊗ Z P resolution of K K K ⊗ Z R in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the 2 × 2 minors of an e × g matrix.

An Explicit, Characteristic-Free, Equivariant Homology Equivalence Between Koszul Complexes

“…there exists a unique ring homomorphism → S with = ⊗ S. In Kustin (2007), we found a free resolution of the universal ring over an integral polynomial ring, , in the border case f = e + g. The resolution is not minimal; indeed, if e and g are both at least 5; then Hashimoto's Theorem can be used to prove that does not posses a generic minimal resolution over the ring of integers. Nonetheless, for each field K, we are able to use the resolution to express the modules in the minimal homogeneous resolution of ⊗ K by free ⊗ K modules in terms of the homology of (0.1).…”

“…The homology of (0.1) is known when K is a field of characteristic zero. With the answer in hand, but without appealing to Kustin (2007), the geometric method of calculating syzygies was directly applied in Kustin and Weyman (2007) to find the modules in the minimal resolution of ⊗ K, when K is a field of characteristic zero. Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E * ⊗ G induces the Koszul complex (0.1) and its dual…”