Remark

Tag 0CU4, permanent. Cite it as [Bourbaki, Tag 0CU4].

Commutative Algebra, chapter IX, § 1. Vecteurs de Witt

Let $J$ be the set of integers $j \geq 1$ . For every element $j$ of $J$ , define the polynomial $\varphi_j$ of $\mathbf{Z}[(X_j)_{j \in J}]$ by the formula

\varphi_j = \sum_d d X_d^{j/d},

where the sum is taken over the elements of $J$ which divide $j$ . For every integer $n \geq 0$ , we have

\varphi_{p^n} = \Phi_n(X_{p^0}, ..., X_{p^n}).

For every ring $A$ and every element $m$ of $J$ , we denote by $\varphi_m$ the mapping from $A^J$ into $A$ which associates to $(a_j)_{j \in J}$ $\varphi_m((a_j)_{j \in J})$ ; we denote by $\varphi_A$ , or simply $\varphi$ , the mapping from $A^J$ into itself which associates to $a = (a_j)_{j \in J}$ $(\varphi_m(a))_{m \in J}$ .

Let $\mathscr{A} = \mathbf{Z}[(X_j)_{j \in J}, (Y_j)_{j \in J}]$ be the ring of polynomials with integer coefficients in the two families of indeterminates $X = (X_j)_{j \in J}$ and $Y = (Y_j)_{j \in J}$ . One can show (p. 51, exerc. 34) that there exist in $\mathscr{A}$ elements

s = (s_j)_{j \in J}, \quad p = (p_j)_{j \in J} \quad \text{and} \quad i = (i_j)_{j \in J},

characterized by the following equalities:

\varphi_{\mathscr{A}}(s) = \varphi_{\mathscr{A}}(\mathbf{X}) + \varphi_{\mathscr{A}}(\mathbf{Y})

\varphi_{\mathscr{A}}(p) = \varphi_{\mathscr{A}}(\mathbf{X}) \varphi_{\mathscr{A}}(\mathbf{Y})

\varphi_{\mathscr{A}}(i) = - \varphi_{\mathscr{A}}(\mathbf{X}) .

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

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