Regular ring/Characterization with Hilbert-Kunz multiplicity/Fakt

Let ${}R$ be a local noetherian ring of positive characteristic. Then the following hold.

(1) The Hilbert-Kunz multiplicity of ${}R$ is  ${}e_{HK}(R)\geq 1$.  (2) If ${}R$ is unmixed, then  ${}e_{HK}(R)=1$  if and only if ${}R$ is regular.

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