This paper is an intrusion of Model Theory into Algebraic Geometry, and its author attaches a price to the methods used for the obtention of its results which is more important than the value of the results themselves. This is reflected in the nature of the questions disseminated in the text. It studies the correlations between the group of automorphisms of an algebraically closed field K and the group of automorphisms of a structure S definable in it. The parameters involved in the definitions will play an important rôle, as well as the very peculiar properties of the Model Theory of Algebraically Closed Fields. It begins with a commentary of a result of A.V. Borovik, which was the source of its inspiration, making explicit its dependence on the famous Theorem of Borel and Tits on abstract isomorphisms between algebraic simple groups, considered from a model-theoretic point of view. It finally leads to a description of the automorphisms of finite order of a simple algebraic group (over an algebraically closed field), and of its superstable groups of automorphisms, based on general arguments from Model Theory, demanding only a minimal inspection of the structure of the group; to derivate some consequences of it, we have to complete a somehow elliptic argument of Altinel, Borovik and Cherlin, in a proof which is nevertheless crucial in their inductive context. In fact, our version of the Theorem of Borel and Tits does not depend of the presence of a law of group; it is valid more generally in what we call autonomous constructible structures, which are the infinite structures S definable in an algebraically closed field K, for which anything which is definable on S in the language of the field K is definable (with parameters) in the language of S. Significant examples of such structures are given by the multifields, whose plain definition hides a sophisticated Galois theory in characteristic p; in any autonomous constructible structure S, a multifield is defined without parameters, and controls the automorphisms of S in the following sense: the automorphisms of S whose action on the multifield has a finite order form a group denoted by Autmax(S), which is the largest definable group of automorphisms of S, and also its largest superstable group of automorphisms. In characteristic zero, the same result is easy to handle because then a unifield, that is a copy L of the base field K, is definable in S without parameters. When S is a simple algebraic group G, in the ordinary cases such a copy L exists even in characteristic p, but there are special cases, in the presence of an exceptional endogeny, where only a bifield (L1,L2) is definable without parameters; Autmax(G) is the kernel of the action of the automorphisms of G on the field L in the general case, on the bifield (L1,L2) in the special case; in fact, it is the group of geometric automorphisms of G. The existence of Autmax(G) is obtained by straightforward model-theoretic methods. But it is of a sovereign importance to realize that its connected component is formed by the inner automorphisms of G; since Autmax(G) is definable, that is definably isomorphic to an algebraic group, this is a purely geometric result, which is in fact well-known; its proof necessitates a minutious description of the structure of algebraic simple groups, but it is not necessary to reproduce it to obtain its model-theoretic corollary: in a context of finite Morley rank, a definable connected group of automorphisms of G is composed of inner automorphisms.

Paramètres dans les corps algébriquement clos

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发表日期：

2025年10月29日

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Bruno Poizat

摘要

This paper is an intrusion of Model Theory into Algebraic Geometry, and its author attaches a price to the methods used for the obtention of its results which is more important than the value of the results themselves. This is reflected in the nature of the questions disseminated in the text. It studies the correlations between the group of automorphisms of an algebraically closed field K and the group of automorphisms of a structure S definable in it. The parameters involved in the definitions will play an important rôle, as well as the very peculiar properties of the Model Theory of Algebraically Closed Fields. It begins with a commentary of a result of A.V. Borovik, which was the source of its inspiration, making explicit its dependence on the famous Theorem of Borel and Tits on abstract isomorphisms between algebraic simple groups, considered from a model-theoretic point of view. It finally leads to a description of the automorphisms of finite order of a simple algebraic group (over an algebraically closed field), and of its superstable groups of automorphisms, based on general arguments from Model Theory, demanding only a minimal inspection of the structure of the group; to derivate some consequences of it, we have to complete a somehow elliptic argument of Altinel, Borovik and Cherlin, in a proof which is nevertheless crucial in their inductive context. In fact, our version of the Theorem of Borel and Tits does not depend of the presence of a law of group; it is valid more generally in what we call autonomous constructible structures, which are the infinite structures S definable in an algebraically closed field K, for which anything which is definable on S in the language of the field K is definable (with parameters) in the language of S. Significant examples of such structures are given by the multifields, whose plain definition hides a sophisticated Galois theory in characteristic p; in any autonomous constructible structure S, a multifield is defined without parameters, and controls the automorphisms of S in the following sense: the automorphisms of S whose action on the multifield has a finite order form a group denoted by Aut_max(S), which is the largest definable group of automorphisms of S, and also its largest superstable group of automorphisms. In characteristic zero, the same result is easy to handle because then a unifield, that is a copy L of the base field K, is definable in S without parameters. When S is a simple algebraic group G, in the ordinary cases such a copy L exists even in characteristic p, but there are special cases, in the presence of an exceptional endogeny, where only a bifield (L₁,L₂) is definable without parameters; Aut_max(G) is the kernel of the action of the automorphisms of G on the field L in the general case, on the bifield (L₁,L₂) in the special case; in fact, it is the group of geometric automorphisms of G. The existence of Aut_max(G) is obtained by straightforward model-theoretic methods. But it is of a sovereign importance to realize that its connected component is formed by the inner automorphisms of G; since Aut_max(G) is definable, that is definably isomorphic to an algebraic group, this is a purely geometric result, which is in fact well-known; its proof necessitates a minutious description of the structure of algebraic simple groups, but it is not necessary to reproduce it to obtain its model-theoretic corollary: in a context of finite Morley rank, a definable connected group of automorphisms of G is composed of inner automorphisms.