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

Abstract

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 $Re{p}_{\bar{K}}(Aut(W))$, and we use it to construct a generic form $\tilde{W}$ over a commutative $K_0$ algebra $B_W$ (so that forms of $W$ are exactly the specializations of $\tilde{W}$). 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.

47 pages. A more detailed description of the kernel completion was added