The Stacks project

Lemma 13.4.3. Let $\mathcal{D}$ be a pre-triangulated category. Let

\[ (a, b, c) : (X, Y, Z, f, g, h) \to (X', Y', Z', f', g', h') \]

be a morphism of distinguished triangles. If two among $a, b, c$ are isomorphisms so is the third.

Proof. Assume that $a$ and $c$ are isomorphisms. For any object $W$ of $\mathcal{D}$ write $H_ W( - ) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(W, -)$. Then we get a commutative diagram of abelian groups

\[ \xymatrix{ H_ W(Z[-1]) \ar[r] \ar[d] & H_ W(X) \ar[r] \ar[d] & H_ W(Y) \ar[r] \ar[d] & H_ W(Z) \ar[r] \ar[d] & H_ W(X[1]) \ar[d] \\ H_ W(Z'[-1]) \ar[r] & H_ W(X') \ar[r] & H_ W(Y') \ar[r] & H_ W(Z') \ar[r] & H_ W(X'[1]) } \]

By assumption the right two and left two vertical arrows are bijective. As $H_ W$ is homological by Lemma 13.4.2 and the five lemma (Homology, Lemma 12.5.20) it follows that the middle vertical arrow is an isomorphism. Hence by Yoneda's lemma, see Categories, Lemma 4.3.5 we see that $b$ is an isomorphism. This implies the other cases by rotating (using TR2). $\square$


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