Remark

Tag 0E3B, permanent. Cite it as [Bourbaki, Tag 0E3B].

Algebra, chapter X, § 9. Complexes de Koszul

Let $\mathfrak{x}=(x_1,\ldots,x_n)$ be a sequence of elements of $A$ such that $H_1(\mathfrak{x},A)=0$ ; then the kernel of the surjective homomorphism $u:A^n\to\mathfrak{x}$ such that

u\left(\sum a_i e_i\right)=\sum a_i x_i

is generated by the elements

X_j e_i-X_i e_j

; consequently, the

A

-algebra

A[X_1,\ldots,X_n]/\mathfrak{d}

is isomorphic to the symmetric algebra

S_A(\mathfrak{x})

(III, p. 69, prop. 4). It therefore follows from Theorem 1 that the algebra homomorphism

S_A(\mathfrak{x})\longrightarrow\bigoplus_n \mathfrak{x}^n

induced by the canonical injection of

\mathfrak{x}

into

\bigoplus_n\mathfrak{x}^n

is an isomorphism. The same holds for the homomorphism

S_A(\mathfrak{x}/\mathfrak{x}^2)\longrightarrow\bigoplus_n\mathfrak{x}^n/\mathfrak{x}^{n+1}.

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

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