On the existence of $F$-thresholds and related limits

Abstract

The $F$-thresholds are important numerical invariants in prime characteristic, whose existence had been established only under certain assumptions. We show the existence of $F$-thresholds in full generality. We study properties of standard graded algebras over a field for which $F$-pure threshold and $F$-threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the $a$-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of $F$-thresholds and $F$-pure thresholds, obtaining existence of related limits in certain cases.

21 pages

References in corpus (5)

• F-thresholds and Bernstein-Sato polynomials
• On generalized Hilbert-Kunz function and multiplicity
• A note on the growth of regularity with respect to Frobenius
• F-thresholds of graded rings
• Globalizing F-invariants

Cited by in corpus (1)

• $D$-modules, Bernstein-Sato polynomials and $F$-invariants of direct summands