### Abstract

Let (R,m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set [Formula presented] is bounded below by 1/d!e(R‾) where R‾=R/Ann(M). Moreover, when Mˆ is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

Original language | English (US) |
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Pages (from-to) | 442-472 |

Number of pages | 31 |

Journal | Advances in Mathematics |

Volume | 347 |

DOIs | |

State | Published - Apr 30 2019 |

Externally published | Yes |

### Keywords

- Hilbert-Samuel multiplicities
- Koszul homology
- Lech's inequality
- Stückrad–Vogel conjecture

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## Cite this

Klein, P., Ma, L., Quy, P. H., Smirnov, I., & Yao, Y. (2019). Lech's inequality, the Stückrad–Vogel conjecture, and uniform behavior of Koszul homology.

*Advances in Mathematics*,*347*, 442-472. https://doi.org/10.1016/j.aim.2019.02.029