Some extensions of theorems of Kn\"orrer and Herzog-Popescu

A construction due to Kn\"orrer shows that if $N$ is a maximal Cohen-Macaulay module over a hypersurface defined by $f+y^2$, then the first syzygy of $N/yN$ decomposes as the direct sum of $N$ and its own first syzygy. This was extended by Herzog-Popescu to hypersurfaces $f+y^n$, replacing $N/yN$ by $N/y^{n-1}N$. We show, in the same setting as Herzog-Popescu, that the first syzygy of $N/y^{k}N$ is always an extension of $N$ by its first syzygy, and moreover that this extension has useful approximation properties. We give two applications. First, we construct a ring $\Lambda^\#$ over which every finitely generated module has an eventually $2$-periodic projective resolution, prompting us to call it a "non-commutative hypersurface ring". Second, we give upper bounds on the dimension of the stable module category (a.k.a. the singularity category) of a hypersurface defined by a polynomial of the form $x_1^{a_1} + \dots + x_d^{a_d}$.