Descent, fields of invariants and generic forms via symmetric monoidal categories

Let $W$ be a finite dimensional algebraic structure (e.g. an algebra) over a field $K$ of characteristic zero. We study forms of $W$ by using Deligne's Theory of symmetric monoidal categories. We construct a category $\mathcal{C}_W$, which gives rise to a subfield $K_0\subseteq K$, which we call the field of invariants of $W$. This field will be contained in any subfield of $K$ over which $W$ has a form. The category $\mathcal{C}_W$ is a $K_0$-form of $Rep_{\bar{K}}(Aut(W))$, and we use it to construct a generic form $\widetilde{W}$ over a commutative $K_0$ algebra $B_W$ (so that forms of $W$ are exactly the specializations of $\widetilde{W}$). This generalizes some generic constructions for central simple algebras and for $H$-comodule algebras. We give some concrete examples arising from associative algebras and $H$-comodule algebras. As an application, we also explain how can one use the construction to classify two-cocycles on some finite dimensional Hopf algebras.Comment: 47 pages. A more detailed description of the kernel completion was adde

Groupoids, imaginaries and internal covers

Let $T$ be a first-order theory. A correspondence is established between internal covers of models of $T$ and definable groupoids within $T$. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between: covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.Comment: Local improvements; thanks to referee of Turkish Mathematical Journal. First appeared in the proceedings of the Paris VII seminar: structures alg\'ebriques ordonn\'ee (2004/5

On finite imaginaries

We study finite imaginaries in certain valued fields, and prove a conjecture of Cluckers and Denef.Comment: 15p

Imaginaries and definable types in algebraically closed valued fields

The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point