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{switch 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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