In terms of the valuative criterion of properness, for \(f:X\to Y\) a morphism of finite type of Noetherian schemes, given a regular curve \(C\) on \(Y\) corresponding to \(\operatorname{Spec}R\to Y\) (for \(R\) a DVR) and a lift of the generic point of \(C\) to \(X\), there is exactly one way to complete the curve with lifts of closed points.

Let \(X\) be a noetherian integral separated scheme which is regular in codimension one and let \(Y\) be a prime divisor with generic point \(\eta\). Then \({\mathcal{O}}_{X, \eta}\) is a DVR with residue field \(K\). There is a map \begin{align*} K^{\times}\to \operatorname{Div}(X) \\ f \mapsto (f) \coloneqq\sum_{y\in \operatorname{Div}(X)} v_y(f)\, y ,\end{align*} which is well-defined since infinitely many \(v_y(f) = 0\) since the non-regular locus of \(f\) is a proper closed subset of a Noetherian scheme, and thus contains only finitely many prime divisors. Any divisor in the image is called prinicipal.

In mixed characteristic algebraic geometry, the basic geometric objects are smoot p-adic formal schemes over the ring of integers \(\mathcal{O}_{K}\) of a complete algebraically closed nonarchimedean field \(K / { {\mathbf{Q}}_p }\).These rings are often non-Noetherian, e.g. the value group of \({\mathcal{O}}_K\) is a divisible group. Replacing \({\mathcal{O}}_K\) with a DVR like \({ {\mathbf{Z}}_{\widehat{p}} }\) is not ideal. Applications of these non-Noetherian rings: perfectoid MOC geometry, descent for fine topologies( pro-etale, quasi-syntomic, v topology, arc topology), and the theory of delta rings.

Tags: #todo #NT/algebraic Refs: DVR

\(v\) is a discrete valuation if \(v(K) \cong {\mathbf{Z}}\leq {\mathbf{R}}\). See also DVR.