On a Consequence of the Order Ideal Conjecture

Given a minimal set of generators x of an ideal I of height d in a regular local ring (R,m, k) we prove several cases for which the map Kd(x;R) ⊗ k → Tord (R/I, k) is the 0-map. As a consequence of the order ideal conjecture we derive several cases for which Kd+i(x;R)⊗ k → Tord+i(R/I, k) are 0-maps for i ≥ 0. In order to solve the syzygy problem in the equicharacteristic case Evans and Griffith proved the following: Theorem A ([E-G1]; Th. 2.4, [E-G2]). Let R be a local ring containing a field. Let M be a finitely generated k-th syzygy of finite projective dimension and let x be a minimal generator of M . Then the order ideal OM (x) = {f(x) | f ∈ HomR(M,R)} has grade at least k. We drop M from the notation for order ideals when there is no scope for confusion. Afterwards Bruns and Herzog extended the above theorem for finite complexes of finitely generated free modules in the following way. Theorem B (Th. 9.5.2, [B-H]). Let (R,m) be a local ring containing a field and let F• : 0 → Fs φs −→ Fs−1 → · · · → F1 φ1 −→ F0 → 0 be a complex of finitely generated free R-modules. Then for every j, 1 ≤ j ≤ s and for every e ∈ Fj with e ̸∈ mFj + Imφj+1, codimension O (φj(e)) ≥ codimF• + j, where codimF• = inf {codimension Iri(φi)− i | ri = s ∑ j=i (−i)j−i rankFj}. The existence of big Cohen-Macaulay modules, due to Hochster ([H2]), played an important role in the proofs of these two theorems. Bruns and Herzog observed the following as a consequence of the above Theorems. AMS Subject Classification: Primary 13D02, 13D22, Secondary 13C15, 13D25, 13H05