Asymptotic Invariants and Flatness of Local Endomorphisms

Author

Nikita Miasnikov, Graduate Center, City University of New York

Date of Degree

6-2014

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Lucien Szpiro

Subject Categories

Mathematics

Keywords

Pure sciences, African, Asymptotic, Endomorphism, Length, Local, Ring

Abstract

For a local endomorphism of a noetherian local ring we introduce 3 asymptotic invariants one of which we call entropy. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.

Recommended Citation

Miasnikov, Nikita, "Asymptotic Invariants and Flatness of Local Endomorphisms" (2014). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/511