Syzygy Theorems via Comparison of Order Ideals on a Hypersurface

We introduce a weak order ideal property that suffices for establishing the Evans-Griffith Syzygy Theorem. We study this weak order ideal property in settings that allow for comparison between homological algebra over a local ring \(R\) versus a hypersurface ring \(R/(x^n)\). Consequently we solve some relevant cases of the Evans-Griffith syzygy conjecture over local rings of unramified mixed characteristic \(p\), with the case of syzygies of prime ideals of Cohen-Macaulay local rings of unramified mixed characteristic being noted. We reduce the remaining considerations to modules annihilated by \(p^s\), \(s>0\), that have finite projective dimension over a hypersurface ring.