integral scheme

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integral scheme

Reduced and irreducible.

A scheme is integral if the ring \({\mathcal{O}}_X (U )\) is an integral domain for every open set \(U\).

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Weil divisor

A prime divisor on \(X\in {\mathsf{Sch}}\) is a closed integral subscheme of codimension one (idea: irreducible subvarietes).

- 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

For Weil divisors, only consider schemes which are noetherian, integral, separated, and regular in codimension one.

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