Kommutativer Ring/Positive Charakteristik/Hilbert-Kunz Multiplizität/en/Definition

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)

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