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 $R=\Bbbk [x_{1},...,x_{m}]$ is the polynomial ring over a field $\Bbbk$ 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. 
Amanda Beecher