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)

${}\lim _{e\rightarrow \infty }{\frac {\operatorname {length} \,(R/I^{[p^{e}]})}{p^{ed}}}=\lim _{e\rightarrow \infty }{\frac {\varphi _{I}(e)}{p^{ed}}}\,.$