Continuity of a vertical generization

Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 7.11(2).

A vertical generization v/H of a continuous valuation on a Huber ring is again continuous. The cofinality half of the argument is topology-free and lives in TauCeti.RingTheory.Valuation.Coarsen as Valuation.cofinalValue_coarsenByUnits_restrict. Wedhorn states the hypothesis as H ⊊ Γ_v: the convex subgroup is proper in the value group, not in the ambient codomain. That is why the coarsening here is applied to v.restrict, which is the presentation of v on its own value group; H ≠ ⊤ is then literally Wedhorn's properness.

Properness is not decoration. If H were all of Γ_v the coarsening would take only the values 0 and 1, so {a | w a < w b} would be the support of v, and continuity would force that support to be open — which it need not be.

Main results

• Valuation.IsContinuous.coarsenByUnits_restrict: the coarsening of a continuous valuation by a proper convex subgroup of its value group is continuous.

References

• T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 7.11 and Corollary 1.21.

theorem Valuation.IsContinuous.coarsenByUnits_restrict {A : Type u_1} [CommRing A] [TopologicalSpace A] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] [IsTopologicalRing A] [TauCeti.Huber.IsHuberRing A] {v : Valuation A Γ₀} (hv : v.IsContinuous) {H : TauCeti.ConvexSubgroup (↑v).ValueGroup₀ˣ} (hH : H ≠ ⊤) : (v.restrict.coarsenByUnits H).IsContinuous

Wedhorn Remark 7.11(2). A vertical generization of a continuous valuation by a proper convex subgroup of its value group is again continuous.