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.
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Ananthnarayan et al. (2024) studied this question.
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