LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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Algebra operations on smooth germs are representative independent

Statement

The addition, multiplication, and scalar-multiplication operations of The local algebra of smooth function germs do not depend on the chosen representatives.

Facts & Assumptions

Given: Germs $[f] = [f^{'}]$ and $[g] = [g^{'}]$ in $C_{p}^{\infty} (M)$ .

[F1]

Equality of germs means equality on some neighbourhood of $p$ (The germ of a smooth function at a point).

[F2]

The local algebra operations are defined by pointwise operations on representatives (The local algebra of smooth function germs).

Proof

technique · direct

1.1

By [F1], there are neighbourhoods of $p$ on which $f = f^{'}$ and $g = g^{'}$ . On the intersection neighbourhood, one also has $f + g = f^{'} + g^{'}$ , $f g = f^{'} g^{'}$ , and $c f = c f^{'}$ .

F1given

2.1

Therefore the germs determined by these sums, products, and scalar multiples are the same, so the operations in [F2] are representative independent.

F2step 1.1

3.1

Hence the algebra operations are well defined on germs.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The local algebra of smooth function germs.

Dependency tree · two levels

3 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)