Session S07 - Differential operators in algebraic geometry and commutative algebra

Monday, July 19, 17:00 ~ 17:30 UTC-3

Bernstein-Sato theory for singular rings in positive characteristic

Eamon Quinlan-Gallego

University of Michigan, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Given an ideal $\mathfrak{a}$ in a smooth $\mathbb{C}$-algebra $R$, its Bernstein-Sato polynomial $b_{\mathfrak{a}}(s)$ is an invariant with origins in complex analysis that measures the singularities of the zero locus of $\mathfrak{a}$.

There are two recent generalizations of this construction. In one direction, Àlvarez-Montaner, Huneke, and Núñez-Betancourt have shown that ideals in certain non-smooth $\mathbb{C}$-algebras (direct summands) still admit Bernstein-Sato polynomials. In the other direction, work of Mustaţă, Bitoun and myself shows that we can also define Bernstein-Sato invariants in smooth algebras over fields of positive characteristic.

In this talk I present joint work with J. Jeffries and L. Núñez-Betancourt in which we that the combined generalization is possible. Namely, we show that Bernstein-Sato invariants exist in positive characteristic when the ambient ring has mild singularities (direct summand or graded F-finite representation type).

Joint work with Jack Jeffries (University of Nebraska-Lincoln) and Luis Núñez-Betancourt (Centro de Investigación en Matemáticas).

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