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I'm looking for a reference for the following result (all rings commutative with unit) :

Let $A$ be a ring, $B$ be an $A$-algebra which is free of finite rank as an $A$-module, $M$ be a free $B$-module of finite rank and $u$ an endormorphism of $M$. Then we have

$\det_A(u) = N_{B/A}(\det_B(u))$

Zorba le Grec
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  • In the general case, do you mean when we don't assume that $B$ is a separable $A$-algebra? – Alex Youcis Oct 22 '13 at 21:57
  • I guess not. Note that it might just be wishfull thinking on might part to expect it to be true. If it's false I would be interested in a counter example and any kind of statement which gives some necessary (or sufficient) conditions for it to be true. – Zorba le Grec Oct 23 '13 at 05:16

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