Nearly commuting matrices

Zhibek Kadyrsizova

doi:10.1016/j.jalgebra.2017.10.019

Nearly commuting matrices

Zhibek Kadyrsizova

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F-pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings.

Original language	English
Pages (from-to)	199-218
Number of pages	20
Journal	Journal of Algebra
Volume	497
DOIs	https://doi.org/10.1016/j.jalgebra.2017.10.019
Publication status	Published - Mar 1 2018

Keywords

Commuting matrices
F-purity
Frobenius
Singularities

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2017.10.019

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title = "Nearly commuting matrices",

abstract = "We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F-pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings.",

keywords = "Commuting matrices, F-purity, Frobenius, Singularities",

author = "Zhibek Kadyrsizova",

note = "Funding Information: This paper is the part of the thesis written by the author during her doctorate study at the University of Michigan. The author would like to express an enormous gratitude to Mel Hochster for the support and guidance received while working on the paper. This work was supported in part by NSF grant DMS-1401384 and the Barbour Scholarship at the University of Michigan. The author also thanks the referees for their helpful comments and suggestions.",

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N1 - Funding Information: This paper is the part of the thesis written by the author during her doctorate study at the University of Michigan. The author would like to express an enormous gratitude to Mel Hochster for the support and guidance received while working on the paper. This work was supported in part by NSF grant DMS-1401384 and the Barbour Scholarship at the University of Michigan. The author also thanks the referees for their helpful comments and suggestions.

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N2 - We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F-pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings.

AB - We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F-pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings.

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KW - F-purity

KW - Frobenius

KW - Singularities

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