## Abstract

A construction due to Knörrer 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 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 x_{1}^{a1}+…+x_{d}^{ad}.

Original language | English (US) |
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Pages (from-to) | 94-120 |

Number of pages | 27 |

Journal | Journal of Algebra |

Volume | 571 |

DOIs | |

State | Published - Apr 1 2021 |

## Keywords

- Huneke issue
- Hypersurface rings
- Maximal Cohen–Macaulay modules
- Stable module category
- Syzygies

## ASJC Scopus subject areas

- Algebra and Number Theory