OAI.BogomolovPop.main_graph_injective

The theorem states an injectivity result for a graph relation linking field isomorphisms to isomorphisms of Galois-theoretic data. Fix a prime ℓ and fields k ⊆ K and l ⊆ L, with k and l algebraically closed, K essentially of finite type over k and L over l, ℓ nonzero in k and in l, and transcendence degrees of K over k and of L over l both at least 2. IsomF(k,K,l,L) is the set of isomorphisms between the perfect closures of K and L inside their algebraic closures that carry the image of k onto the image of l, modulo the equivalence relation generated by Frobenius steps, where β is a step from α if β(x)=α(x)^p for all x for some prime p with p=0 in L. For a field F, PiA(ℓ,F) is the topological abelianization of the maximal pro-ℓ quotient of the absolute Galois group of F, and Delta is the kernel of the map from the quotient of that pro-ℓ group by the closure of the commutator of the closed commutator subgroup with the whole group, onto PiA. A map φ from PiA(L) to PiA(K) is bracket-compatible if the commutator pairing into Delta is carried by some topological isomorphism Delta(L) to Delta(K). IsomCUnits(ℓ,K,L) is the set of bracket-compatible topological isomorphisms PiA(L) to PiA(K), modulo the equivalence generated by φ ~ χ when some ℓ-adic unit u satisfies χ(x)=lim φ(x)^(appr_n(u)). PhiGraph(a,b) says a and b have representatives α and φ such that some extension β of α to the algebraic closures intertwines the Galois actions: every σ in the absolute Galois group of L has a τ in that of K with β∘τ=σ∘β on the algebraic closure of K and φ sends the image of σ in PiA(L) to the image of τ in PiA(K) The theorem concludes that for any a, a' in IsomF and any b in IsomCUnits, if both PhiGraph(a,b) and PhiGraph(a',b) hold, then a=a'. This is stated as an admitted theorem.