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

Abstract

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 \(x_1^{a_1} + \dots + x_d^{a_d}\).