Corollary

Tag 0E3E, permanent. Cite it as [Bourbaki, Tag 0E3E].

Algebra, chapter X, § 9. Complexes de Koszul

Suppose moreover that the modules $M/(x_1 M + \cdots + x_{i-1} M)$ are separated for the (x)-adic topology, and let $(a_{ij}) \in \mathrm{GL}_n(A)$ . Put

y_i = \sum_j a_{ij} x_j \quad \text{and} \quad y = (y_1, \ldots, y_n) .

Then the sequence y is regular for M, and one has $\theta_y = \det(a_{ij})^{-1} \theta_x$ . Indeed, the sequence y is regular for M by Prop. 6 (X, p. 158) and Theorem 1 (X, p. 160); the last assertion follows from Prop. 8, and from Prop. 4 of X, p. 119.

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

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Cites

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