The quest for F-rational signature

Abstract: Strongly F-regular singularities are one of the fundamental classes of singularities defined by the properties of Frobenius endomorphism. This class of mild singularities can be detected using F-signature, an invariant of a local ring with many good properties. Through this connection we obtain a powerful tool for studying strongly F-regular singularities, for example, several results on "mildness" of F-regular can be quantified using F-signature.

Another fundamental class of singularities in positive characteristic are F-rational singularities. While generally more severe, this class of singularities is in many aspects analogous to strongly F-regular singularities and can be even understood by "moving" the definition of F-regularity to take place in the dualizing module. Naturally, there has been interest in adapting the definition of F-signature to work with F-rational singularities.

While there is no complete solution yet, I am convinced that such a theory should exist. As my evidence, I will present results of a joint work with Kevin Tucker, and prior works of Hochster and Yao, and Sannai.

My talk will be self-contained. I will discuss all necessary background, such as definitions, properties, and relations between these notions, in the first half and then proceed to more technical results in the second half.

commutative algebra

Audience: researchers in the topic

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Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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