Upper Bounds for Betti Numbers of Multigraded Modules

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded R-module, where R = k[x1, . . . , xm] is the polynomial ring over a field k in m variables. The bound is given in terms of the rank and the first two Betti numbers of the module. An example is given which achieves these bounds simultaneously in each homological degree. Using Alexander duality, a bound is established for the total multigraded Bass numbers of a finite multigraded module in terms of the first two total multigraded Bass numbers. Introduction Let R = k[x1, . . . , xm] be the polynomial ring over a field k in m variables with the standard Z grading and L a finite Z (multigraded) R-module. Much work has been done on establishing lower bounds for Betti numbers of L, initially motivated by the Buchsbaum-Eisenbud-Horrocks conjecture for finite modules over regular local rings. This conjecture was shown to hold for R/I when I is a monomial ideal by Evans and Griffith [3] and generally for all multigraded modules by Charalambous [1] and Santoni [6]. On the other hand, little is known about upper bounds for the Betti numbers. The main result of this paper gives sharp upper bounds for the total Betti number of L in each homological degree in terms of the rank and the first two Betti numbers of the module L. Main Theorem. For i ≥ 2, we have