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Seminar

Fellowship of the Ring, National Seminar: The quest for F-rational signature January 28, 2021 (01:00 PM PST - 02:30 PM PST)
Parent Program: --
Location: MSRI: Online/Virtual
Speaker(s) Ilya Smirnov (Stockholm University)
Description

To attend this seminar, you must register in advance, by clicking HERE.

Title: 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. 

Speaker: Ilya Smirnov, Stockholm University
 

Video

The Quest For F-Rational Signature

Abstract/Media

To attend this seminar, you must register in advance, by clicking HERE.

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. 

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The Quest For F-Rational Signature