Kan extensions in Enriched Category Theory by Eduardo J. Dubuc

By Eduardo J. Dubuc

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Extra resources for Kan extensions in Enriched Category Theory

Example text

Since ν is strongly G-invariant, ν(z) = ν(z1 ) = · · · = ν(zd ) by Theorem 17. So, as in the proof of Theorem 17, ν(Δ(z)) ν(z). Remark 41. This property has the following geometric interpretation. e. there exists a variety Y of the same type over C such that X Y ⊗C K and the function ﬁeld of X is F . Each Δ of Lie(G), deﬁnes a tangent vector ﬁeld on Y hence on X. Since X is complete, each invariant valuation ν possesses a center Z ⊂ X. Proposition 40 asserts that this center remains invariant under the vector ﬁeld Δ (see [24], [7], [1], and [3] for comments).

The following assertions are equivalent 1. The vector ﬁeld X is tangent to the formal curve γ. 2. t. ν. 3. t. ν. Proof To say that X is tangent to the formal curve γ means that the two vectors γ (t) = are collinear in (19) s x1 (t) xs (t) and X(γ(t)) = P1 ◦ γ(t) , Ps ◦ γ(t) . It also means that exists a nonzero λ(t) ∈ C((t)) such that x1 (t) = λ(t)P1 ◦ γ(t) = λP1∗ xs (t) = λ(t)Ps ◦ γ(t) = λPs∗ . This also means that for the substitution morphism ϕ : F → C((t)), f → f ∗ we have d d ∂f ∂f (ϕ(f )) = (f ∗ ) = (γ(t)) · x1 (t) + · · · + (γ(t)) · xs (t) dt dt ∂x1 ∂xs ∂f ∂f = λ(t) P1 + · · · + Ps ◦ γ(t) ∂x1 ∂xs = λ(t)(∂f )∗ = λ(t)ϕ(∂f ).

Therefore we must have ν(ϕ) 0. Since this holds for all valuations, K ⊂ T (F/K) ⊂ Rν = K. ν∈S ∗ (F/K) This concludes the proof. 4. 6. Invariant valuations and the Lie algebra of G. Let (F/K, ∂) be a Picard-Vessiot extension with constants C and diﬀerential Galois group G. t. any G-invariant valuation ν of F/C, which is going to be the main purpose of the next section, we are going to focus on other derivations of F . 27, p. 20), the Lie algebra of G can be identiﬁed with the C-vector space of all K-derivations Δ of F commuting with ∂: Lie(G) = {Δ ∈ DerK (F )|[Δ, ∂] = 0}.