  In this short note we introduce a notion of extremality for Betti numbers of
a minimal free resolution, which can be seen as a refinement of the notion of
Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an
arbitrary submodule of a free S-module are preserved when taking the generic
initial module. We relate extremal multigraded Betti numbers in the minimal
resolution of a square free monomial ideal with those of the monomial ideal
corresponding to the Alexander dual simplicial complex and generalize theorems
of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs
of classical criteria due to Hochster, Reisner, and Stanley.
