Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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.