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 $e_{I}(R)$ .

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