DERIVATION MODULE AND THE HILBERT–KUNZ MULTIPLICITY OF THE COORDINATE RING OF A PROJECTIVE MONOMIAL CURVE

Let n<inf>0</inf>, n<inf>1</inf>, . . ., n<inf>p</inf> be a sequence of positive integers such that n<inf>0</inf> <n<inf>1</inf> <· · ·<n<inf>p</inf>, gcd(n<inf>0</inf>, n<inf>1</inf>, . . ., n<inf>p</inf>)=1. Let S = (0, n<inf>p</inf>), (n<inf>0</inf>, n<inf>p</inf> − n<inf>0</inf>), . . ., (n<inf>p</inf><inf>−1</inf>, n<inf>p</inf> − n<inf>p</inf><inf>−1</inf>), (n<inf>p</inf>, 0) be an affine semigroup in N². The semigroup ring k[S] is the coordinate ring of the projective monomial curve in the projective space P<inf>k</inf>^p⁺¹, which is defined parametrically by x<inf>0</inf> = v^np, x<inf>1</inf> = uⁿ⁰ v^np⁻ⁿ⁰, . . ., x<inf>p</inf> = u^np⁻¹ v^np^−np−1, x<inf>p</inf><inf>+1</inf> = u^np. In this article, we consider the case when n<inf>0</inf>, n<inf>1</inf>, . . ., n<inf>p</inf> forms an arithmetic sequence, and give an explicit set of minimal generators for the derivation module Der<inf>k</inf>(k[S]). Further, we give an explicit formula for the Hilbert–Kunz multiplicity of the coordinate ring of a projective monomial curve.