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 CWCW, which gives rise to a subfield K0⊆KK0⊆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 CWCW is a K0K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0K0-algebra BWBW (so that forms of W   are exactly the specializations of 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 one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.