Example 3

Tag 001T, permanent. Cite it as [Bourbaki, Tag 001T].

Algebra, VIII, p. 26, § 2. The Structure of Modules of Finite Length

Let X be a $C^r$ manifold (VAR, R, 5.1.5) and $x$ a point of X. Let $\mathscr{O}_x$ be the ring of germs at $x$ of $C^r$ functions with values in the field of scalars K. Then $\mathscr{O}_x$ is a commutative local ring, and its maximal ideal consists of the

germs of the functions that are zero at $x.*$

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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