j-multiplicity

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

$$j(I)=\underset{n\to \mathrm{\infty }}{lim}\frac{(d-1)!}{n^{d-1}}{\mathrm{length}}_R(H_{𝔪}^0(I^n/I^{n+1}))$$

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