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External valuative dimension comparison on locally Noetherian schemes

PDF p. 4: Motivic graded spectral sequence; PDF p. 9: Eventual filtration vanishing

Statement

The Jaffard valuative dimension of a domain is the supremum of the Krull dimensions of valuation overrings inside its fraction field. For an arbitrary ring it is the supremum of these dimensions over prime quotients; for a scheme take the supremum over affine opens. It is local on open covers. For every locally Noetherian scheme X it equals its Krull dimension.

Source contract

Given: Schemes and commutative rings with identity, in fixed universes.

Goal: Import the precise convention and Noetherian comparison required by the source filtration theorem.

[F1] EHIK §2.3 definitions and Proposition2.3.2(1),(9) supply the convention and comparison.

External input

Import [F1] with the exact source record and reading limits. Its “rank of a valuation ring” is its Krull dimension; the equivalent chain-of-overrings definition is given on p9. The Noetherian comparison includes reducible and nonreduced schemes. It is not the false assertion that all valuation rings in arbitrary extension fields mapping to X have bounded dimension.

Depends on

Used by · 2

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Elden Elmanto, Marc Hoyois, Ryomei Iwasa and Shane Kelly, Cdh descent, cdarc descent, and Milnor excision · Math.Ann.379 (2021),1011–1045; §2.3 definitions and Lemma2.3.1 p.9; Proposition2.3.2(1),(8),(9) pp.9–10

Jaffard valuative dimension of a domain is the supremum of ranks, equivalently Krull dimensions, of valuation overrings in its fraction field; for a general ring take prime quotients, and for a scheme take affine opens. This is local on open covers; Krull dimension≤valuative dimension; equality holds for locally Noetherian schemes.

Hypotheses: Arbitrary schemes for the definitions and open-local property; locally Noetherian for equality. Nilpotents and reducible schemes are included through prime quotients.

Reading limits: Read §2.3 definitions, Lemma2.3.1 and entire Proposition2.3.2 with proof pp.9–10, plus the rank/scheme descriptions in Corollary2.3.4. Jaffard’s original results cited in the proof are imported, not reconstructed. The definitions coincide with the completed local supplier, not an alternative notion requiring every valuation morphism to have bounded rank.

Sources