Regular bi-interpretability of Chevalley groups over local rings

Elena Bunina

doi:10.1007/s40879-023-00659-4

Regular bi-interpretability of Chevalley groups over local rings

Elena Bunina

Department of Mathematics

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

3 Scopus citations

Abstract

We prove that if G(R) = G_π(Φ , R) (E(R) = E_π(Φ , R)) is an (elementary) Chevalley group of rank > 1 , R is a local ring (with 12 for the root systems A₂, B_l, C_l, F₄, G₂ and with 13 for G₂) , then the group G(R) (or (E(R)) is regularly bi-interpretable with the ring R. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementarily definable, i.e., if for an arbitrary group H we have H≡ G_π(Φ , R) , then there exists a ring R^′≡ R such that H≅ G_π(Φ , R^′) .

Original language	English
Article number	64
Journal	European Journal of Mathematics
Volume	9
Issue number	3
DOIs	https://doi.org/10.1007/s40879-023-00659-4
State	Published - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Funding

Our sincere thanks go to Eugene Plotkin for very useful discussions regarding various aspects of this work and permanent attention to it.

Keywords

Chevalley groups
Elementary definability
Local rings
Regular bi-interpretability

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10.1007/s40879-023-00659-4

Cite this

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title = "Regular bi-interpretability of Chevalley groups over local rings",

abstract = "We prove that if G(R) = Gπ(Φ , R) (E(R) = Eπ(Φ , R)) is an (elementary) Chevalley group of rank > 1 , R is a local ring (with 12 for the root systems A2, Bl, Cl, F4, G2 and with 13 for G2) , then the group G(R) (or (E(R)) is regularly bi-interpretable with the ring R. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementarily definable, i.e., if for an arbitrary group H we have H≡ Gπ(Φ , R) , then there exists a ring R′≡ R such that H≅ Gπ(Φ , R′) .",

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author = "Elena Bunina",

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doi = "10.1007/s40879-023-00659-4",

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N2 - We prove that if G(R) = Gπ(Φ , R) (E(R) = Eπ(Φ , R)) is an (elementary) Chevalley group of rank > 1 , R is a local ring (with 12 for the root systems A2, Bl, Cl, F4, G2 and with 13 for G2) , then the group G(R) (or (E(R)) is regularly bi-interpretable with the ring R. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementarily definable, i.e., if for an arbitrary group H we have H≡ Gπ(Φ , R) , then there exists a ring R′≡ R such that H≅ Gπ(Φ , R′) .

AB - We prove that if G(R) = Gπ(Φ , R) (E(R) = Eπ(Φ , R)) is an (elementary) Chevalley group of rank > 1 , R is a local ring (with 12 for the root systems A2, Bl, Cl, F4, G2 and with 13 for G2) , then the group G(R) (or (E(R)) is regularly bi-interpretable with the ring R. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementarily definable, i.e., if for an arbitrary group H we have H≡ Gπ(Φ , R) , then there exists a ring R′≡ R such that H≅ Gπ(Φ , R′) .

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KW - Elementary definability

KW - Local rings

KW - Regular bi-interpretability

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Regular bi-interpretability of Chevalley groups over local rings

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Keywords

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