Alphabeta MathHow statement and proof provenance work
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Localisation And Faithfully Flat Base Change Of Regular Sequences
Statement
An -regular sequence remains regular after localization whenever the localized terminal quotient is nonzero, and remains regular after faithfully flat base change.
Facts & Assumptions
Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Regular Sequence On A Module, Localisation of modules is exact, Flat and faithfully flat modules and ring homomorphisms.
Proof
Localization preserves injectivity, so each regularity condition survives; the stated nonzero localized terminal quotient prevents the convention from failing at the end.
Faithfully flat tensoring preserves and reflects injectivity and nonzero modules. Apply this successively to the quotient stages to obtain the base-change assertion.
Depends on
Used by
- Polynomial extension preserves Cohen--Macaulayness Corollary
- A flat local map splits regular sequences into base and fibre parts Lemma
- Expected ranks and determinantal regular sequences force a free complex to be exact Lemma
- Order and simultaneous normal crossings are preserved by smooth morphisms Lemma
- Regularity ascends and descends along a flat local homomorphism Lemma
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)