On the upper semi-continuity of HSL numbers

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Let B be an affine Cohen-Macaulay algebra over a field of characteristic p. For every prime ideal p⊂ B, let Hₚ denote H Bₚₚ Bₚ(Bₚ). Each such Hₚ is an Artinian module endowed with a natural Frobenius map Θ and if Nil(Hₚ) denotes the set of all elements in Hₚ killed by some power of Θ then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an e≥ 0 such that Θᵉ Nil(Hₚ)=0. The smallest such e is the HSL-number of Hₚ which we denote HSL(Hₚ). The main theorem in this paper shows that for all e>0, the sets { p (B) | HSL(Hₚ) < e } are Zariski open, hence HSL is upper semi-continuous. An application of this result gives a global test exponent for the calculation of Frobenius closures of parameter ideals in Cohen-Macaulay rings.