Lemma 15.118.5. Let $R$ be a ring. Let $M'$ and $M''$ be two finite projective $R$-modules. Then the diagram

\[ \xymatrix{ \det (M') \otimes \det (M'') \ar[r] \ar[d]_{\epsilon \cdot (\text{switch tensors})} & \det (M' \oplus M'') \ar[d]^{\det (\text{swith summands})} \\ \det (M'') \otimes \det (M') \ar[r] & \det (M'' \oplus M') } \]

commutes where $\epsilon = \det ( -\text{id}_{M' \otimes M''}) \in R^*$ and the horizontal arrows are those of Lemma 15.118.2.

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