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 a hypersurface defined by f+y², 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+yⁿ, replacing N/yN by N/yⁿ⁻¹N. We show, in the same setting as Herzog–Popescu, that the first syzygy of N/y^kN 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 Λ 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 x₁^a₁+…+x_d^a_d.