Regular bi-interpretability of Chevalley groups over local rings

2022/08/29 by E. I. Bunina, Bunina, Elena
Medicine · #Autoimmune Neurological Disorders and Treatments #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2208.13623

openalex publication_date 2022/08/29 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28

Abstract

In this paper 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 (1)/(2) for the root systems \mathbf A₂, \mathbf B_l, \mathbf C_l, \mathbf F₄, \mathbf G₂ and with (1)/(3) for \mathbf 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 elementary definable, i. e., if for an arbitrary group~H we have H≡ G_π(Φ, R), than there exists a ring R'≡ R such that H≅ G_π(Φ,R').

Related

• Regular bi-interpretability and finite axiomatizability of Chevalley groups
• Defining R and G(R)
• Automorphisms of Chevalley groups over commutative rings
• Defining \mathbbA in G(\mathbbA)
• Some Properties of Twisted Chevalley Groups