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

Hilbert-Kunz function

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. Then the Hilbert-Kunz function is the function

$\varphi _{I}\colon \mathbb {N} \longrightarrow \mathbb {N} ,e\longmapsto \varphi _{I}(e)=\operatorname {length} \,(R/I^{[p^{e}]}),$

where ${}I^{[p^{e}]}$ is the extended ideal under the ${}e$-th iteration of the Frobenius homomorphism

$R\longrightarrow R,f\longmapsto f^{p^{e}}.$