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TheoremStatement: Literature-sourcedProof: Literature-sourced
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Dense-open isomorphism for proper birational maps

PDF p. 10: Lemma 5.1, weight-zero restriction; PDF p. 5: Lemma 2.4, finite-type cdh topology

Statement

Let k be a field and let f:P→Q be a proper birational morphism between integral finite-type k-schemes. Birational here means that the dominant map induces an isomorphism of function fields. There is a dense open subscheme V⊆Q such that the entire inverse image f−1(V)→V is an isomorphism.

External input

Import this specialization of Stacks Project, tag 0BAJ, using its reduced-target branch. The function-field convention is consistent with tag 0552. These established inputs are cited rather than reconstructed.

Depends on

Used by · 1

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Stacks Project, Generically finite morphisms · Tag 0BAJ, displayed Lemma 29.52.6; online text retrieved 2026-11-09

A proper birational morphism between integral finite-type schemes over a field is an isomorphism over a dense open of its target.

Hypotheses: Use the quasi-compact, locally finite type, reduced-target branch; both schemes have one irreducible component.

Reading limits: Read the complete statement and displayed proof of tag 0BAJ. The cited finite-morphism and birational-open lemmas were not reconstructed. The history endpoint was unavailable; no commit-specific version is claimed.

Stacks Project, Birational morphisms · Tag 0552, displayed Lemma 29.51.7; online text retrieved 2026-11-09

For integral schemes locally of finite type over a base, birationality is equivalent to an isomorphism of generic residue fields over their common base point.

Hypotheses: Here both schemes are integral and finite type over the field k; the isomorphism is induced by the given dominant morphism.

Reading limits: Read the complete statement and displayed proof to check the function-field convention. Its cited rational-map foundations were not reconstructed.

Sources