Commutative ring/Positive characteristic/Hilbert-Kunz multiplicity/Definition

Hilbert-Kunz multiplicity

Let $K$ denote a field of positive characteristic $p$, let $K\subseteq R$ be a noetherian ring and let $I\subseteq R$ be an ideal which is primary to some maximal ideal of height $d$. Then the Hilbert-Kunz multiplicity of $I$ is the limit (if it exists)

$\underset{e\to \mathrm{\infty }}{lim}\frac{\mathrm{length}(R/I^{[p^e]})}{p^{ed}}=\underset{e\to \mathrm{\infty }}{lim}\frac{{\phi }_I(e)}{p^{ed}}.$