Some extensions of theorems of Knörrer and Herzog–Popescu

A construction due to Knörrer shows that if N is a maximal Cohen–Macaulay module over hypersurface defined by f+y2, then the first syzygy of N/yN decomposes as direct sum and its own syzygy. This was extended Herzog–Popescu hypersurfaces f+yn, replacing N/yn−1N. We show, in same setting Herzog–Popescu, N/ykN always an extension syzygy, moreover this has useful approximation properties. give two applications. First, we construct ring Λ which every finitely generated eventually 2-periodic projective resolution, prompting us call it “non-commutative ring”. Second, upper bounds on dimension stable category (a.k.a. singularity category) polynomial form x1a1+…+xdad.