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

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe

Combinable Extensions of Abelian Groups

The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method à la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification