a) Let C ′ → u C → v C ′ ′ C' \xrightarrow{u} C \xrightarrow{v} C'' C ′ u C v C ′′ A A A P P P E E E
H o m g r A ( P , C ′ ) → H o m g r ( 1 , u ) H o m g r A ( P , C ) → H o m g r ( 1 , v ) H o m g r A ( P , C ′ ′ )
\mathrm{Homgr}_A(P, C') \xrightarrow{\mathrm{Homgr}(1,u)} \mathrm{Homgr}_A(P, C) \xrightarrow{\mathrm{Homgr}(1,v)} \mathrm{Homgr}_A(P, C'')
Homgr A ( P , C ′ ) Homgr ( 1 , u ) Homgr A ( P , C ) Homgr ( 1 , v ) Homgr A ( P , C ′′ ) and
H o m g r A ( C ′ ′ , E ) → H o m g r ( v , 1 ) H o m g r A ( C , E ) → H o m g r ( u , 1 ) H o m g r A ( C ′ , E )
\mathrm{Homgr}_A(C'', E) \xrightarrow{\mathrm{Homgr}(v,1)} \mathrm{Homgr}_A(C, E) \xrightarrow{\mathrm{Homgr}(u,1)} \mathrm{Homgr}_A(C', E)
Homgr A ( C ′′ , E ) Homgr ( v , 1 ) Homgr A ( C , E ) Homgr ( u , 1 ) Homgr A ( C ′ , E ) are exact sequences of complexes of k k k
b) Let 0 → C ′ → u C → v C ′ ′ → 0 0 \to C' \xrightarrow{u} C \xrightarrow{v} C'' \to 0 0 → C ′ u C v C ′′ → 0 A A A A A A C ′ C' C ′ C ′ ′ C'' C ′′ E E E
0 → H o m g r A ( E , C ′ ) → H o m g r ( 1 , u ) H o m g r A ( E , C ) → H o m g r ( 1 , v ) H o m g r A ( E , C ′ ′ ) → 0
0 \to \mathrm{Homgr}_A(E, C') \xrightarrow{\mathrm{Homgr}(1,u)} \mathrm{Homgr}_A(E, C) \xrightarrow{\mathrm{Homgr}(1,v)} \mathrm{Homgr}_A(E, C'') \to 0
0 → Homgr A ( E , C ′ ) Homgr ( 1 , u ) Homgr A ( E , C ) Homgr ( 1 , v ) Homgr A ( E , C ′′ ) → 0 0 → H o m g r A ( C ′ ′ , E ) → H o m g r ( v , 1 ) H o m g r A ( C , E ) → H o m g r ( u , 1 ) H o m g r A ( C ′ , E ) → 0
0 \to \mathrm{Homgr}_A(C'', E) \xrightarrow{\mathrm{Homgr}(v,1)} \mathrm{Homgr}_A(C, E) \xrightarrow{\mathrm{Homgr}(u,1)} \mathrm{Homgr}_A(C', E) \to 0
0 → Homgr A ( C ′′ , E ) Homgr ( v , 1 ) Homgr A ( C , E ) Homgr ( u , 1 ) Homgr A ( C ′ , E ) → 0 are exact sequences of complexes of k k k
In case a) , we remark that the sequences
H o m A ( P p , C q ′ ) → H o m A ( P p , C q ) → H o m A ( P p , C q ′ ′ )
\mathrm{Hom}_A(P_p, C'_q) \to \mathrm{Hom}_A(P_p, C_q) \to \mathrm{Hom}_A(P_p, C''_q)
Hom A ( P p , C q ′ ) → Hom A ( P p , C q ) → Hom A ( P p , C q ′′ ) and
H o m A ( C q ′ ′ , E p ) → H o m A ( C q , E p ) → H o m A ( C q ′ , E p )
\mathrm{Hom}_A(C''_q, E_p) \to \mathrm{Hom}_A(C_q, E_p) \to \mathrm{Hom}_A(C'_q, E_p)
Hom A ( C q ′′ , E p ) → Hom A ( C q , E p ) → Hom A ( C q ′ , E p ) are exact for all p , q ∈ Z p, q \in \mathbf{Z} p , q ∈ Z b) is analogous.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Nothing.
Nothing.