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- Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links
A scheme \(X\) is locally noetherian if it can be covered by affine open subschemes Spec \(A_{i}\) where each \(A_{i}\) is noetherian.
\(X\) is noetherian if it is locally noetherian and only finitely many Spec \(A_{i}\) is needed to cover \(X\).
- Tags: - #AG/basics - Refs: - #todo/add-references - Links: - smooth scheme - closed point - Cohen-Macaulay - Noetherian scheme
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.
- Tags: - #AG/basics - Refs: - See Bhatt-Lurie 2022 for Cartier-Witt divisors, generalized Cartier divisors, and their relation to prisms: https://arxiv.org/pdf/2201.06120.pdf#page=1 #resources/papers/2022 - Links: - divisor - Ring theory: - Krull dimension - regular ring - Scheme theory: - Noetherian scheme - integral scheme - separated scheme - closed subscheme