Buchsbaum and monomial conjecture dimension

Abstract

We define two new homological invariants for a finitely generated module M over a commutative Noetherian local ring R, its Buchsbaum dimension B-dim R M, and its Monomial conjecture dimension MC-dimR M. It will be shown that these new invariants have certain nice properties we have come to expect from homological dimensions. Over a Buchsbaum ring R, every finite module M has B-dimR M < ∞; conversely, if the residue field has finite B-dimension, then the ring R is Buchsbaum. Similarly R satisfies the Hochster Monomial Conjecture if only if MC-dimR k is finite, where k is the residue field of R. MC-dimension fits between the B-dimension and restricted flat dimension Rfd of Christensen et al. [Christensen, L. W., Foxby, H.-B., Frankild, A. (2002). Restricted homological dimensions and Cohen-Macaulayness. J. Algebra 251(1):479-502]. B-dimension itself is finer than CM-dimension of Gerko [Gerko, A. A. (2001). On homological dimensions. Sb. Math. 192(7-8): 1165-1179] and we have equality if CM-dimension is finite. It also satisfies an analog of the Auslander-Buchsbaum formula. Copyright © 2004 by Marcel Dekker, Inc.