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f-purity

Nagoya Mathematical Journal (1)

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F-THRESHOLDS OF GRADED RINGS
Part of
- Local theory
- General commutative ring theory
ALESSANDRO DE STEFANI, LUIS NÚÑEZ-BETANCOURT
Journal:

Nagoya Mathematical Journal / Volume 229 / March 2018

Published online by Cambridge University Press:

07 December 2016, pp. 141-168

Print publication:

March 2018
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The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.