Proposition 13

Tag 053V, permanent. Cite it as [Bourbaki, Tag 053V].

Let M be a lattice of V and u a K-automorphism of V. Then : (6)

-\chi(M, u(M)) = \operatorname{div}(\det(u)).

For all

p \in P

, then

\bigwedge^n (u(M)_p) = \bigwedge^n (u(M_p))

; if

(e_i)_{1 \leq i \leq n}

is a basis of

M_p

, then

\bigwedge^n (M_p) = A_p \cdot e_1 \text{ } A \text{ } e_2 \text{ } A \ldots A \text{ } e_n,

and $\bigwedge^n (u(M_p)) = A_p \cdot \det(u) e_1 \wedge e_2 \mathbf{A} \ldots \mathbf{A} e_n$ , whence the proposition by virtue of formula (5).

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

Cited by

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