The Stacks project

Lemma 12.5.18. Let $\mathcal{A}$ be an abelian category. Let

\[ \xymatrix{ & & & x\ar[ld]\ar[rr]\ar[dd]^(.4)\alpha & & y\ar[ld]\ar[rr]\ar[dd]^(.4)\beta & & z\ar[ld]\ar[rr]\ar[dd]^(.4)\gamma & & 0\\ & & x'\ar[rr]\ar[dd]^(.4){\alpha '} & & y'\ar[rr]\ar[dd]^(.4){\beta '} & & z'\ar[rr]\ar[dd]^(.4){\gamma '} & & 0 & \\ & 0\ar[rr] & & u\ar[ld]\ar[rr] & & v\ar[ld]\ar[rr] & & w\ar[ld] & & \\ 0\ar[rr] & & u'\ar[rr] & & v'\ar[rr] & & w' & & & } \]

be a commutative diagram with exact rows. Then, the induced diagram

\[ \xymatrix@C=15pt{ \mathop{\mathrm{Ker}}(\alpha ) \ar[r] \ar[d] & \mathop{\mathrm{Ker}}(\beta ) \ar[r] \ar[d] & \mathop{\mathrm{Ker}}(\gamma ) \ar[r]^(.45){\delta } \ar[d] & \mathop{\mathrm{Coker}}(\alpha ) \ar[r] \ar[d] & \mathop{\mathrm{Coker}}(\beta ) \ar[r] \ar[d] & \mathop{\mathrm{Coker}}(\gamma ) \ar[d] \\ \mathop{\mathrm{Ker}}(\alpha ') \ar[r] & \mathop{\mathrm{Ker}}(\beta ') \ar[r] & \mathop{\mathrm{Ker}}(\gamma ') \ar[r]^(.45){\delta '} & \mathop{\mathrm{Coker}}(\alpha ') \ar[r] & \mathop{\mathrm{Coker}}(\beta ') \ar[r] & \mathop{\mathrm{Coker}}(\gamma ') } \]

commutes.

Proof. Omitted. $\square$

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