Lemma 15.108.4. Let $R$ be a ring. Let

$K \subset L \subset M$

be $R$-modules such that $K$, $L/K$, and $M/L$ are finite projective $R$-modules. Then the diagram

$\xymatrix{ \det (K) \otimes \det (L/K) \otimes \det (M/L) \ar[r] \ar[d] & \det (L) \otimes \det (M/L) \ar[d] \\ \det (K) \otimes \det (M/K) \ar[r] & \det (M) }$

commutes where the maps are those of Lemma 15.108.2.

Proof. Omitted. Hint: after localizing at a prime of $R$ we can assume $K \subset L \subset M$ is isomorphic to $R^{\oplus a} \subset R^{\oplus a + b} \subset R^{\oplus a + b + c}$ and in this case the result is an evident computation. $\square$

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