# j-multiplicity

Let $(R,\U0001d52a)$ be a Noetherian^{} local ring^{} with proper ideal^{} $I$. Define

$$j(I)=\underset{n\to \mathrm{\infty}}{lim}\frac{(d-1)!}{{n}^{d-1}}{\mathrm{length}}_{R}({H}_{\U0001d52a}^{0}({I}^{n}/{I}^{n+1}))$$ |

and call it the j-multiplicity of $I$. Here ${H}_{\U0001d52a}^{0}(\bullet )$ is the 0-th local cohomology functor. When $I$ is $\U0001d52a$-primary, it is same as the Hilbert-Samuel multiplicity ${e}_{I}(R)$.

Title | j-multiplicity |
---|---|

Canonical name | Jmultiplicity |

Date of creation | 2013-03-22 18:15:26 |

Last modified on | 2013-03-22 18:15:26 |

Owner | yshen (21076) |

Last modified by | yshen (21076) |

Numerical id | 6 |

Author | yshen (21076) |

Entry type | Definition |

Classification | msc 13H15 |