  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.

Commutative Algebra