Algebra and Combinatorics Seminar

Date: March 21, 2025

Time: 3:00PM - 3:50PM

Location: BLOC 302

Speaker: Suchitra Pande, University of Utah

Title: A Gorenstein criterion in positive characteristics via the F-pure threshold

Abstract: This talk concerns singularities of local rings and numerical invariants used to detect mild singularities. Some classical notions of mild singularities include normal, Cohen-Macaulay and Gorenstein. However, when the ring has prime characteristic, the Frobenius map provides powerful new ways to detect mild singularities via the notions of F-purity and F-regularity. For instance, Hochster and Huneke proved that strongly F-regular rings are automatically normal and Cohen-Macaulay, but not always Gorenstein. In this talk, we will discuss a criterion for a strongly F-regular standard graded ring to be Gorenstein. This criterion was conjectured by Hirose, Watanabe and Yoshida and relies on the relationship between the F-pure threshold and another classical invariant of graded rings called the a-invariant. In this talk, we will discuss a proof of this conjecture, extending previous partial results of Singh, Takagi and Varbaro, and De Stefani and Núñez-Betancourt. We will review the relevant preliminary notions before presenting the proof.