We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.
AbstractWe prove a functorial correspondence between a category of logarithmic $$\mathfrak {sl}_2$$
sl
2
-connections on a curve $${\mathsf {X}}$$
X
with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover "Equation missing". The proof is by constructing a pair of inverse functors $$\pi ^\text {ab}, \pi _\text {ab}$$
π
ab
,
π
ab
, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $$\pi _*$$
π
∗
.
A new proof of the Grauert direct image theorem
