Lemma 50.18.2 (0H9F)—The Stacks project

Lemma 50.18.2. Let $R_1 \to R_2$ be a ring homomorphism. For $i = 1, 2$ consider commutative diagrams

\[ \xymatrix{ 0 \ar[r] & K_ i^0 \ar[r] & L_ i^0 \ar[r] & M_ i^0 \ar[r] & 0 \\ & & L_ i^{-1} \ar[u]_{\partial _ i} \ar@{=}[r] & M_ i^{-1} \ar[u] } \]

of $R_ i$-modules as in Lemma 50.18.1. Assume we have maps $K_1^0 \to K_2^0$, $L_1^ j \to L_2^ j$, $M_1^ j \to M_2^ j$ compatible with the given ring map $R_1 \to R_2$ and compatible with the maps in the displayed diagrams. If the maps $M_1^ j \otimes _{R_1} R_2 \to M_2^ j$ are isomorphisms, then the diagrams

\[ \xymatrix{ \wedge ^ p(H^0(L_1^\bullet )) \ar[d] \ar[r]_-{c^ p} & \wedge ^ p(K_1^0) \otimes _{R_1} \det (M_1^\bullet ) \ar[d] \\ \wedge ^ p(H^0(L_2^\bullet )) \ar[r]^-{c^ p} & \wedge ^ p(K_2^0) \otimes _{R_2} \det (M_2^\bullet ) } \]

commute.

Proof. This follows from the explicit description of the map given in the proof of Lemma 50.18.1. Note that we need the assumption that $M_1^ j \otimes _{R_1} R_2 \to M_2^ j$ are isomorphisms to see that the basis we use for $M_1^{-1}$ maps to a basis for $M_2^{-1}$ (and hence in checking the compatibility in both constructions we can use the “same” basis). $\square$

The Stacks project

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