Betti cones over fibre products

Let R be a fibre product of standard graded algebras over a field. We study the structure of syzygies of finitely generated graded R-modules. As an application of this, we show that the existence of an R-module of finite regularity and infinite projective dimension forces R to be Koszul. We also look at the extremal rays of the Betti cone of finitely generated graded R-modules, and show that when depth (R) =1, they are spanned by the Betti tables of pure R-modules if and only if R is Cohen-Macaulay with minimal multiplicity.