Nearly commuting matrices

Zhibek Kadyrsizova

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)199-218
Number of pages20
JournalJournal of Algebra
Volume497
DOIs
Publication statusPublished - Mar 1 2018

Fingerprint

Algebraic Set
Irreducible Components
Intersection
Commutator
Singularity
Ring

Keywords

  • Commuting matrices
  • F-purity
  • Frobenius
  • Singularities

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Nearly commuting matrices. / Kadyrsizova, Zhibek.

In: Journal of Algebra, Vol. 497, 01.03.2018, p. 199-218.

Research output: Contribution to journalArticle

Kadyrsizova, Zhibek. / Nearly commuting matrices. In: Journal of Algebra. 2018 ; Vol. 497. pp. 199-218.
@article{6fd01d3f454f4d388047c06f48e4fb74,
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",
year = "2018",
month = "3",
day = "1",
doi = "10.1016/j.jalgebra.2017.10.019",
language = "English",
volume = "497",
pages = "199--218",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Nearly commuting matrices

AU - Kadyrsizova, Zhibek

PY - 2018/3/1

Y1 - 2018/3/1

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.

KW - Commuting matrices

KW - F-purity

KW - Frobenius

KW - Singularities

UR - http://www.scopus.com/inward/record.url?scp=85035115221&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035115221&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2017.10.019

DO - 10.1016/j.jalgebra.2017.10.019

M3 - Article

VL - 497

SP - 199

EP - 218

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -