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-dim _R  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-dim _R  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-dim _R  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.

Key Words:

• Buchsbaum ring
• Gorenstein dimension
• u.s.d-sequences

2000 Mathematics Subject Classification:

• Primary 13C140, 13D05, 13H10
• Secondary 13D45

Acknowledgment

The authors are deeply grateful to the referee for his or her useful pointed comments on the paper. This research was in part supported by a grant from IPM (No. 81130030 and No. 82130118).

Notes

^#Communicated by I. Swanson.