Next we want to prove the other direction: ifk is a field whose ultrapower∗kis in the class
Let Yk →P1k be aG-HKG-cover of curves. We want to construct a lifting of this cover
over the ring R(k). First, form the base change of this cover under the diagonal embedding
k ,→∗kto get a G-coverY∗
k→P1∗k. This is also a HKG-cover, as the base change preserves
the ramification behaviour of the cover. (We also see this because the base change is really the ultrapower of copies of our original cover.)
Now, by the assumption that∗k∈CG,g,π, we know that the cover Y∗k →P1∗k has a lifting
overR(∗k), denote it by
YR(∗k)→P1R(∗k).
Recall that the diagonal map k ,→∗kis an elementary embedding. We know by AKE-
principle thatW(k) andW(∗k) are elementarily equivalent, which implies that also the rings
R(k) and R(∗k) are elementarily equivalent. This means that R(∗k) can be elementarily embedded into an ultrapower ofR(k), denote this byR(∗k),→ ∗R(k). Note that the two
“stars” might be different: we don’t necessanily have a mapR(∗k),→∗R(k).
We can also do the same with the elementary diagram of R(k). The models of this
theory are elementary extensions ofR(k), and again by the AKE-principle the ringR(∗k) is such an extension. This way we can make sure that the embedding R(∗k),→ ∗R(k) is an
R(k)-homomorphism.
Now we can form the base change of the cover YR(∗k)→P1
R(∗k) to get a new cover
Y∗R(k)→P1∗R(k).
What does the curveY∗R(k) look like? Since the cover is defined over the ring∗R(k), we can
consider it as an ultraproduct of covers overR(k). Taking the special fiber of Y∗R(k) → P1
∗R(k) then gives a cover Y∗k → P 1
“big” ultrapower of k. Because we constructed the embedding R(∗k) → ∗R(k) to be an
R(k)-homomorphism, the residue extension∗k→ ∗kis also ak-embedding. The functoriality
of base change (and special fiber) then implies that this special fiber
Y∗k →P1∗k
is the base change of our original coverYk→P1k in the diagonal embedding k ,→ ∗k.
Since the base change is liftable over∗R(k), the lifting gives a lifting overR(k) in almost
all coordinates of the ultrapower. In particular, this means the set of liftings must be nonempty.
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