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In this paper, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the 𝔪-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.
-pure threshold, and the diagonal
-threshold are three important invariants of a graded
-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly
-regular rings. In this article, we prove that these relations hold only assuming that the algebra is
-pure. In addition, we present an interpretation of the
-pure Gorenstein graded
-algebras in terms of regular sequences that preserve
-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
. We also present analogous results and questions in characteristic zero.
Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. These generalized Lyubeznik numbers are defined in terms of D-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generated K-algebras. These new invariants are indicators of F-singularities in characteristic p > 0 and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.
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