Let R {\displaystyle {}R} be a local noetherian ring of positive characteristic. Then the following hold.
(1) The Hilbert-Kunz multiplicity of R {\displaystyle {}R} is e H K ( R ) ≥ 1 {\displaystyle {}e_{HK}(R)\geq 1} . (2) If R {\displaystyle {}R} is unmixed, then e H K ( R ) = 1 {\displaystyle {}e_{HK}(R)=1} if and only if R {\displaystyle {}R} is regular.
