j-multiplicity

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

j(I)=\lim_{n\to\infty}\frac{(d-1)!}{n^{d-1}}\operatorname{length}_{R}(H_{% \mathfrak{m}}^{0}(I^{n}/I^{n+1}))

and call it the j-multiplicity of $I$ . Here $H_{\mathfrak{m}}^{0}(\bullet)$ is the 0-th local cohomology functor. When $I$ is $\mathfrak{m}$ -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