Lemma 15.108.3. Let $R$ be a ring. Let

$\xymatrix{ 0 \ar[r] & M' \ar[r] \ar[d]^ u & M \ar[r] \ar[d]^ v & M'' \ar[r] \ar[d]^ w & 0 \\ 0 \ar[r] & K' \ar[r] & K \ar[r] & K'' \ar[r] & 0 }$

be a commutative diagram of finite projective $R$-modules whose vertical arrows are isomorphisms. Then we get a commutative diagram of isomorphisms

$\xymatrix{ \det (M') \otimes \det (M'') \ar[r]_-\gamma \ar[d]_{\det (u) \otimes \det (w)} & \det (M) \ar[d]^{\det (v)} \\ \det (K') \otimes \det (K'') \ar[r]^-\gamma & \det (K) }$

where the horizontal arrows are the ones constructed in Lemma 15.108.2.

Proof. Omitted. Hint: use the second construction of the maps $\gamma$ in Lemma 15.108.2. $\square$

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