The Lascar groups and the first homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over B=\acl^{\eq}(B) in any theory $T$. Depending on a ternary relation \indo^* satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of \indo^*, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772

The Lascar groups and the 1st homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over B=\acl^{\eq}(B) in any theory $T$. Depending on a ternary relation \indo^* satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of \indo^*, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772

The relativized Lascar groups, type-amalgamation, and algebraicity

We apply compact group theory to obtain some model-theoretic results about the relativized Lascar Galois group of a strong type

SHELAH-STRONG TYPE AND ALGEBRAIC CLOSURE OVER A HYPERIMAGINARY (Model theoretic aspects of the notion of independence and dimension)

We characterize Shelah-strong type over a hyperimagianary with the algebraic closure of a hyperimaginary. Also, we present and take a careful look at an example that witnesses acl[eq](ℯ) is not interdefinable with acl(ℯ) where ℯ is a hyperimaginary

Second cohomology groups and finite covers

For D an infinite set, k>1 and W the set of k-sets from D, there is a natural closed permutation group G_k which is a non-split extension of \mathbb{Z}_2^W by \Sym(D). We classify the closed subgroups of G_k which project onto \Sym(D)$. The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms.Comment: Typos corrected; change of title to 'Second cohomology groups and finite covers of infinite symmetric groups' in published versio

Model Theory: Around Valued Fields and Dependent Theories

The general topic of the meeting was “Valued fields and related structures”. It included both applications of model theory, as well as so-called “pure” model theory: the classification of first order structures using new techniques extending those developed in stable theories

Stable domination and independence in algebraically closed valued fields

We seek to create tools for a model-theoretic analysis of types in algebraically closed valued fields (ACVF). We give evidence to show that a notion of 'domination by stable part' plays a key role. In Part A, we develop a general theory of stably dominated types, showing they enjoy an excellent independence theory, as well as a theory of definable types and germs of definable functions. In Part B, we show that the general theory applies to ACVF. Over a sufficiently rich base, we show that every type is stably dominated over its image in the value group. For invariant types over any base, stable domination coincides with a natural notion of `orthogonality to the value group'. We also investigate other notions of independence, and show that they all agree, and are well-behaved, for stably dominated types. One of these is used to show that every type extends to an invariant type; definable types are dense. Much of this work requires the use of imaginary elements. We also show existence of prime models over reasonable bases, possibly including imaginaries