Localizations of quasi-algebraic extensions

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Localizations of quasi-algebraic extensions. / Степанов, Алексей Владимирович ; Мордосевич, Андрей Вениаминович.

In: European Journal of Mathematics, Vol. 10, No. 4, 70, 22.11.2024.

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

BibTeX

@article{c656ecd29c654d4b918bbb2999073dfa,

title = "Localizations of quasi-algebraic extensions",

abstract = "The article is part of a project aimed at describing the subgroup lattice of a Chevalley group over a ring. Namely, we consider the subgroups of a Chevalley group G(Φ,A), containing the elementary subgroup E(Φ,R) over a subring R of A. Conjecturally, if Φ is a simply laced root system, then the lattice is standard if and only if the ring extension R⊆A is quasi-algebraic. Roughly speaking, a ring extension R⊆A is quasi-algebraic if it cannot be reduced to the extension F⊆F[t], where F is a field and t is an independent variable, by taking subrings of A and quotients. To prove the conjecture we plan to use a localization method. The first step is to consider the case when R is semilocal. We show that in this case all finitely generated R-subalgebras of A are semilocal as well. In fact, the latter condition is even equivalent, thus, we obtain a new characterization of quasi-algebraic extensions.",

keywords = "13B02, Commutative ring, Localization, Quasi-algebraic extension, Semilocal ring",

author = "Степанов, {Алексей Владимирович} and Мордосевич, {Андрей Вениаминович}",

year = "2024",

month = nov,

day = "22",

doi = "10.1007/s40879-024-00786-6",

language = "English",

volume = "10",

journal = "European Journal of Mathematics",

issn = "2199-675X",

publisher = "Springer Nature",

number = "4",

}

RIS

TY - JOUR

T1 - Localizations of quasi-algebraic extensions

AU - Степанов, Алексей Владимирович

AU - Мордосевич, Андрей Вениаминович

PY - 2024/11/22

Y1 - 2024/11/22

N2 - The article is part of a project aimed at describing the subgroup lattice of a Chevalley group over a ring. Namely, we consider the subgroups of a Chevalley group G(Φ,A), containing the elementary subgroup E(Φ,R) over a subring R of A. Conjecturally, if Φ is a simply laced root system, then the lattice is standard if and only if the ring extension R⊆A is quasi-algebraic. Roughly speaking, a ring extension R⊆A is quasi-algebraic if it cannot be reduced to the extension F⊆F[t], where F is a field and t is an independent variable, by taking subrings of A and quotients. To prove the conjecture we plan to use a localization method. The first step is to consider the case when R is semilocal. We show that in this case all finitely generated R-subalgebras of A are semilocal as well. In fact, the latter condition is even equivalent, thus, we obtain a new characterization of quasi-algebraic extensions.

AB - The article is part of a project aimed at describing the subgroup lattice of a Chevalley group over a ring. Namely, we consider the subgroups of a Chevalley group G(Φ,A), containing the elementary subgroup E(Φ,R) over a subring R of A. Conjecturally, if Φ is a simply laced root system, then the lattice is standard if and only if the ring extension R⊆A is quasi-algebraic. Roughly speaking, a ring extension R⊆A is quasi-algebraic if it cannot be reduced to the extension F⊆F[t], where F is a field and t is an independent variable, by taking subrings of A and quotients. To prove the conjecture we plan to use a localization method. The first step is to consider the case when R is semilocal. We show that in this case all finitely generated R-subalgebras of A are semilocal as well. In fact, the latter condition is even equivalent, thus, we obtain a new characterization of quasi-algebraic extensions.

KW - 13B02

KW - Commutative ring

KW - Localization

KW - Quasi-algebraic extension

KW - Semilocal ring

UR - https://www.mendeley.com/catalogue/03fe57e1-c780-383f-95e7-b6c6ef8042d6/

U2 - 10.1007/s40879-024-00786-6

DO - 10.1007/s40879-024-00786-6

M3 - Article

VL - 10

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 4

M1 - 70

ER -

ID: 127604904