# 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 Jmultiplicity 2013-03-22 18:15:26 2013-03-22 18:15:26 yshen (21076) yshen (21076) 6 yshen (21076) Definition msc 13H15