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