Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(J,jects are field extensions, morphisms are finite extensions).  Consider the category of functors $Fun(\mathcal F_d, \mathcal{Vect})$ from this category to the category of vector spaces over $\mathbb Q$. What is known about its cohomological dimension?

Especially we have a functor which associate to a field extension $K\to F$ the multiplicative group $F^\times$. Is it true that all higher cohomology of this functor vanishes?