Valued fields, metastable groups

• Ehud Hrushovski ^[1] ; Silvain Rideau-Kikuchi ^[2]
  1. [1] Hebrew University of Jerusalem

     Hebrew University of Jerusalem

     Israel

     
  2. [2] Université Denis Diderot

     Université Denis Diderot

     París, Francia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 3, 2019
• Idioma: inglés
• DOI: 10.1007/s00029-019-0491-x
• Enlaces
  • Texto completo (pdf)
• Resumen

  • We introduce a class of theories called metastable, including the theory of algebraically closed valued fields ( ACVF ) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort Γ ) if every type over a sufficiently rich base structure can be viewed as part of a Γ -parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from Γ , and a definable direct limit system of groups with stably dominated generic. In the case of ACVF , among definable subgroups of affine algebraic groups, we characterize the groups with stably dominated generics in terms of group schemes over the valuation ring. Finally, we classify all fields definable in ACVF .