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.
The canonical map sends g to its coherent system of residue classes
Definition
For an abstract group , the canonical map to the profinite completion is
where the th coordinate is the coset of in . Compatibility of these coordinates is immediate from the quotient-transition maps in The profinite completion is the inverse limit of the finite quotients G over N.
Depends on
Used by
- The profinite completion of the integers is the inverse limit of the rings Z mod n Example
- A homomorphism induces a continuous homomorphism of profinite completions Theorem
- The canonical map to the profinite completion has kernel equal to the finite residual and has dense image Theorem
- The profinite completion is initial among continuous homomorphisms from G to profinite groups Theorem
Dependency tree · two levels
7 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
- Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
- H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)