On the upper semi-continuity of HSL numbers

Abstract

Let $B$ be an affine Cohen-Macaulay algebra over a field of characteristic $p$. For every prime ideal $\mathfrak{p}\subset B$, let ${\text{H}}_{\mathfrak{p}}$ denote $H_{\mathfrak{p}{B}_{\mathfrak{p}}}^{\dim B_{\mathfrak{p}}}\left(\widehat{B_{\mathfrak{p}}}\right)$. Each such ${\text{H}}_{\mathfrak{p}}$ is an Artinian module endowed with a natural Frobenius map $\Theta$ and if $\text{Nil}({\text{H}}_{\mathfrak{p}})$ denotes the set of all elements in ${\text{H}}_{\mathfrak{p}}$ killed by some power of $\Theta$ then a theorem by Hartshorne-Speiser and Lyubeznik shows that there exists an $e\ge 0$ such that ${\Theta }^e\text{Nil}({\text{H}}_{\mathfrak{p}})=0$. The smallest such $e$ is the HSL-number of ${\text{H}}_{\mathfrak{p}}$ which we denote $\text{HSL}({\text{H}}_{\mathfrak{p}})$. The main theorem in this paper shows that for all $e>0$, the sets $\{\mathfrak{p}\in \text{Spec}(B)|\text{HSL}({\text{H}}_{\mathfrak{p}})<e\}$ 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.

References in corpus (1)

• Finiteness properties of local cohomology for F-pure local rings

Cited by in corpus (2)

• Global parameter test ideals
• The support of local cohomology modules