Lemma 46.4.7. Let $A$ be a ring. For $F$ a module-valued functor on $\textit{Alg}_ A$ say $(*)$ holds if for all $B \in \mathop{\mathrm{Ob}}\nolimits (\textit{Alg}_ A)$ the functor $TF(B, -)$ on $B$-modules transforms a short exact sequence of $B$-modules into a right exact sequence. Let $0 \to F \to G \to H \to 0$ be a short exact sequence of module-valued functors on $\textit{Alg}_ A$.

1. If $(*)$ holds for $F, G$ then $(*)$ holds for $H$.

2. If $(*)$ holds for $F, H$ then $(*)$ holds for $G$.

3. If $H' \to H$ is morphism of module-valued functors on $\textit{Alg}_ A$ and $(*)$ holds for $F$, $G$, $H$, and $H'$, then $(*)$ holds for $G \times _ H H'$.

Proof. Let $B$ be given. Let $0 \to N_1 \to N_2 \to N_3 \to 0$ be a short exact sequence of $B$-modules. Part (1) follows from a diagram chase in the diagram

$\xymatrix{ 0 \ar[r] & TF(B, N_1) \ar[r] \ar[d] & TG(B, N_1) \ar[r] \ar[d] & TH(B, N_1) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & TF(B, N_2) \ar[r] \ar[d] & TG(B, N_2) \ar[r] \ar[d] & TH(B, N_2) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & TF(B, N_3) \ar[r] \ar[d] & TG(B, N_3) \ar[r] \ar[d] & TH(B, N_3) \ar[r] & 0 \\ & 0 & 0 }$

with exact horizontal rows and exact columns involving $TF$ and $TG$. To prove part (2) we do a diagram chase in the diagram

$\xymatrix{ 0 \ar[r] & TF(B, N_1) \ar[r] \ar[d] & TG(B, N_1) \ar[r] \ar[d] & TH(B, N_1) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & TF(B, N_2) \ar[r] \ar[d] & TG(B, N_2) \ar[r] \ar[d] & TH(B, N_2) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & TF(B, N_3) \ar[r] \ar[d] & TG(B, N_3) \ar[r] & TH(B, N_3) \ar[r] \ar[d] & 0 \\ & 0 & & 0 }$

with exact horizontal rows and exact columns involving $TF$ and $TH$. Part (3) follows from part (2) as $G \times _ H H'$ sits in the exact sequence $0 \to F \to G \times _ H H' \to H' \to 0$. $\square$

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