Given a regular scheme$X$ and subschemes $Y,Z$ with defining ideal sheaves $\mathcal{I},\mathcal{J}$, respectively, the intersection multiplicity $m(x,Y,Z)$ at a generic point $x$ of (an irreducible component of) the intersection $Y\cap Z$ is given by the Serre intersection (or multiplicity) formula in terms of torsion groups: