Remark

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

Algebra, chapter X, § 1. Compléments d’algèbre linéaire

Considering in (ii) a finite presentation A_s^J \xrightarrow{c} A_s^I \to P \to 0, one obtains the condition (ii’) still equivalent to the preceding ones:

(ii’) For every finite matrix (c_{ij})_{i \in I, j \in J} of elements of A, every solution

e = (e_i)_{i \in I} \in E^I

of the system of linear homogeneous equations

\sum_{i \in I} c_{ij} e_i = 0,\quad j \in J,

can be written b_1 z_1 + \cdots + b_n z_n, where b_1, ..., b_n \in E and where, for r = 1, ..., n, z_r = (z_{r,i})_{i \in I} is a solution in A^I of the system of equations

\sum_{i \in I} z_{r,i} c_{ij} = 0,\quad j \in J.

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

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