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External relative Jacobian criterion and openness of smooth maps
PDF p. 15: Theorem6.7 second Bertini application
Statement
Import the ordinary relative Jacobian criterion in standard smooth presentations and openness of smooth morphisms: adding equations whose differentials are independent in a smooth relative -fold yields a smooth relative -fold near that point. In particular its image contains an open neighbourhood of the parameter. The criterion is valid in every characteristic and survives base change.
Source contract
Given: A smooth finite-presentation presentation over the base, and an invertible maximal minor for the original equations together with the additional equations.
Goal: Supply the exact scheme-level nonempty-parameter argument.
[F1] Stacks Algebra Definition10.137.5 and Lemma10.137.6 give the standard-smooth Jacobian presentation and its smoothness and relative differential rank; Morphisms Lemma29.35.11 gives such presentations locally for smooth maps.
[F2] Stacks Morphisms Lemma29.35.10 says smooth maps are universally open.
External input
Import [F1]–[F2]. The augmented minor condition verifies the displayed standard smooth definition after localization. No characteristic-zero generic-smoothness assertion or dominance assumption is imported.
Depends on
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
An invertible defining Jacobian minor gives a standard smooth algebra with the expected free differential rank; smooth morphisms are locally standard smooth and universally open.
Hypotheses: Arbitrary rings/base schemes with the actual finite-presentation and invertible-minor hypotheses; no characteristic restriction. Apply to a smooth chart times the coefficient affine space.
Reading limits: Read actual full Definition10.137.5 and Lemma10.137.6 with its displayed proof, and the full statements and printed proof references of Morphisms29.35.10–29.35.11. The omitted localization details and earlier complete commutative-algebra foundations are imported, not newly reconstructed.
Sources
- The Stacks Project, standard smooth presentations and openness · Algebra Definition10.137.5 and Lemma10.137.6, complete proof; Morphisms Lemmas29.35.10–29.35.11 at https://stacks.math.columbia.edu/tag/01V4.