Résumé. Wedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable (Cherlin, Shelah) and supersimple (Pillay, Scanlon, Wagner) ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed that a small field is algebraically closed, and asked whether a small field should be commutative. We shall answer this question positively in characteristic p>0.

Résumé. A stable field of charactersitic p>0 has finite dimension over his centre. This also holds for a simple field of charcacteristic p>0.

Résumé. We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a 0-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, rings, categories, groupoids and preorders which are 0-type-definable in a small structure. For an A-type-definable group G_A (where the set A may be infinite) in a small and simple structure, we deduce that:
    (1) For any finite tuple g in G_A, there is a definable group containing g.
    (2) If G_A is included in some definable set X such that boundedly many translates
of G_A cover X, then G_A is the conjunction of definable groups.

Résumé. A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n.

Résumé. According to O. Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic 2 is a field.

Résumé. We investigate some common points between stable structures and weakly small structures and define a structure M to be fine if the Cantor-Bendixson rank of the topological space S_f(dcl^eq(A)) is an ordinal for every formula f(x,y) where x is of arity 1 and every finite subset A of M. By definition, a theory is fine if all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subset A of a fine group G, the traces on the algebraic closure acl(A) of A of definable subgroups of G over acl(A) which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroup of finite index, nor Artin-Schreier extension.

Résumé. Let G be a group having a simple theory. For any nilpotent subgroup N of class n, there is a definable nilpotent group E of G which is virtually 'nilpotent of class at most 2n' and finitely many translates of which cover N. The group E is definable using parameters in N and normalised by N_G(N). If S is a soluble subgroup of G of derived length ℓ, there is a definable soluble group F whih is virtually 'soluble of derived length at most 2ℓ' and contains S. The group F is definable with parameters in S and normalised by N_G(S). Analogous results are shown in the more general setting where the ambient group G is defined by the conjunction of infinitely many formulas in a struture having a simple theory. In that case, the envelopes E and F are defined by the conjunction of infinitely many formulas.

Résumé. If G is a group with supersimple theory having finite SU-rank, the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If H is any group with supersimple theory, the subgroup of H generated by all of its normal solvable subgroups is definable and solvable.

Résumé. An infinite group having a supersimple theory has a finite series of definable subgroups whose factors are infinite and either virtually-FC or virtually-simple modulo a finite FC-centre. We deduce that a group which is type-definable in a supersimple theory has a finite series of relatively definable subgroups whose factors are either abelian or simple groups. In this decomposition, the non-abelian simple factors are unique up to isomorphism.

Résumé. We consider a group G that does not have the independence property and study the definability of certain subgroups of G using parameters from a fixed elementary extention G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A⊂G and any external subgroup H⊂G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X∩G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

Résumé. We give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable and whose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013.

Résumé. On considère un analogue de la topologie de Zariski sur un corps gauche K muni d’une transformation pseudo-linéaire T, et l’on y définit une géométrie algébrique élémentaire : ensembles T-affines, T-morphismes, et une notion de comorphisme qui témoigne d’une dualité entre la catégorie des ensembles T-affines et celle des K[t;σ,δ]-modules. En s’appuyant sur des résultats de P. Cohn, on montre, lorsque σ et δ commutent, que K a une extension K sur laquelle chaque fonction de K[T] est surjective. Sur K, la projection d’un constructible est constructible, et un théorème des zéros est valide. Dans un prochain article, on applique ces résultats aux corps gauches NIP.

Résumé. Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain NIP valued fields of characteristic p with Dickson's construction of cyclic algebras, we provide examples of noncommutative NIP division ring of characteristic p and show that an NIP division ring of characteristic p has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite NIP fields have no Artin-Schreier extension. The result extends to NTP₂ division rings of characteristic p, using results of Chernikov, Kaplan and Simon. We also highlight consequences that concern NIP or simple difference fields.

Titre. Quelques propriétés algébriques des structures minces et menues, rang de Cantor Bendixson, espaces topologiques généralisés.

Soutenue le 10 décembre 2009, sous la direction de Frank O. Wagner (Lyon 1)

Rapporteurs. Françoise Point (Mons-Hainaut, Paris 7), Enrique Casanovas (Barcelona)

Jury. Elisabeth Bouscaren (Orsay), Enrique Casanovas, Abderezak Ould Houcine (Lyon 1), Françoise Point, Frank Wagner

Abstract. Small theories appear in the '60s together with Vaught's conjecture. They include all the possible counter examples to this conjecture. Weakly small structures are introduced by Belegradek. They include both minimal and small structures. It is well known that the definable sets of a weakly small structure are ranked by the Cantor-Bendixson rank, when one fixes a finite paramater set. The difficulty of studying weakly small structures comes from the fact that when one increases this parameter set, the rank also increases, and one does not know how to control its growth. The corner stone of the dissertation is the simple remark that the Cantor-Bendixson rank does not increase when one adds finitely many algebraic elements in the parameter set. We deduce a local descending chain condition on definable subgroups (by formulas using parameters in the algebraic closure of a fixed finite set), and introduce a notion of local almost stabiliser. We deduce algebraic properties of weakly small structures. Among them,

Theorem. A weakly small division ring of characteristic p>0 is locally finite dimensional over its centre. A small division ring of characteristic p>0 is a field.

(A partial answer to Problem 6.1.15 of Wagner's book Simple Theories), and

Theorem. An infinite weakly small group has an infinite abelian subgroup.

(This answers Question 2.8 of Wagner's article Groups in simple theories). We then turn to structures that are type-definable in a small theory, showing

Theorem. A group that is finitary type-definable (using parameters in a finite set) in a small theory is the intersection of definable groups.

(An answer to Problem 6.1.14 of Wagner's book Simple Theories). We extend the result to monoids, rings, fields, categories and groupoids that are finitary small type-definable (still over finitely many parameters) in a small theory. We give local definability results concerning groups and fields which are type-definable over an arbitrary set of parameters in theories that are both small and simple. Finally, we reintroduce the Cantor-Bendixson rank in its topological context, and show that the Cantor derivative can be seen as a derivation in a semi-ring of topological spaces. In an attempt to find a global Cantor rank for stable structures, we try to eliminate the word denumerable, omnipresent when one does topology, by replacing it by a regular cardinal κ. We develop the notions of κ-metrisable space, κ-topology, κ-compactness etc. and show an analogue of Urysohn's metrisability Lemma and Cantor-Bendixson Theorem.