Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(Objects are pairs $F,j$ consisting of a field $F$ and the embedding $j\colon K\to F$, such that transcendent degree is equal to $d$, morphisms between $(F_1,j_1)$ and $(F_2,j_2)$ are finite extensions $\psi\colon F_1\to F_2$ such that $\psi\circ j_1=j_2$). Consider the category of (covariant) 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 $(F^\times)\otimes \mathbb Q$. Is it true that all higher cohomology of this functor vanishes?
