Situation 15.7.1 (08KK)—The Stacks project

Situation 15.7.1. Let $A, A', B, B', I$ be as in Situation 15.6.1. Let $B' \to D'$ be a ring map. Set $D = D' \otimes _{B'} B$, $C' = D' \otimes _{B'} A'$, and $C = D' \otimes _{B'} A$. This leads to a big commutative diagram

\[ \xymatrix{ C & & & C' \ar[lll] \\ & A \ar[ul] & A' \ar[l] \ar[ru] \\ & B \ar[u] \ar[ld] & B' \ar[l] \ar[u] \ar[rd] \\ D \ar[uuu] & & & D' \ar[lll] \ar[uuu] } \]

of rings. Observe that we do not assume that the map $D' \to D \times _ C C'$ is an isomorphism ¹. In this situation we have the functor

15.7.1.1

\begin{equation} \label{more-algebra-equation-relative-functor} \text{Mod}_{D'} \longrightarrow \text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'},\quad L' \longmapsto (L' \otimes _{D'} D, L' \otimes _{D'} C', can) \end{equation}

analogous to (15.6.3.1). Note that $L' \otimes _{D'} D = L \otimes _{D'} (D' \otimes _{B'} B) = L \otimes _{B'} B$ and similarly $L' \otimes _{D'} C' = L \otimes _{D'} (D' \otimes _{B'} A') = L \otimes _{B'} A'$ hence the diagram

\[ \xymatrix{ \text{Mod}_{D'} \ar[r] \ar[d] & \text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'} \ar[d] \\ \text{Mod}_{B'} \ar[r] & \text{Mod}_ B \times _{\text{Mod}_ A} \text{Mod}_{A'} } \]

is commutative. In the following we will write $(N, M', \varphi )$ for an object of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$, i.e., $N$ is a $D$-module, $M'$ is an $C'$-module and $\varphi : N \otimes _ B A \to M' \otimes _{A'} A$ is an isomorphism of $C$-modules. However, it is often more convenient think of $\varphi $ as a $D$-linear map $\varphi : N \to M'/IM'$ which induces an isomorphism $N \otimes _ B A \to M' \otimes _{A'} A = M'/IM'$.

[1] But $D' \to D \times _ C C'$ is surjective by Lemma 15.6.5.

The Stacks project

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