Remark

Tag 0A1Y, permanent. Cite it as [Bourbaki, Tag 0A1Y].

Lie Groups and Lie Algebras, chapter II, § 1. Enveloping bigebra of a Lie algebra

(1) Let $p$ be a prime number such that $p \cdot 1 = 0$ in $K$ . The binomial formula and the congruences $\binom{p}{i} \equiv 0 \pmod{p}$ for $1 \leq i \leq p - 1$ imply that

P(E) is stable under the mapping $x \mapsto x^p$ .

(2) By definition, the diagram

0 \longrightarrow P(E) \longrightarrow E^+ \xrightarrow{c^+} E^+ \otimes E^+

is an exact sequence. If $K'$ is a commutative ring and $\rho : K \to K'$ a ring homomorphism, $\rho^*(E) = E \otimes_K K'$ is a $K'$ -bigebra and the inclusion $P(E) \to E$ defines a homomorphism of Lie $K'$ -algebras

\alpha : P(E) \otimes_K K' \to P(E \otimes_K K').

If $K'$ is flat over $K$ (Commutative Algebra, Chapter I, § 2, no. 3, Definition 2), it follows from loc. cit. that the diagram

0 \longrightarrow P(E) \otimes_K K' \longrightarrow E^+ \otimes_K K' \xrightarrow{c^+ \otimes_K \mathrm{Id}_{K'}} (E^+ \otimes_K K') \otimes_{K'} (E^+ \otimes_K K')

is an exact sequence, which implies that $\alpha$ is an isomorphism.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Commutative Algebra, Chapter I, § 2, no. 3, Definition 2 not yet on this site: AC I