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 . 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 ps, s>0, that have finite projective dimension over a hypersurface ring.