The Stacks project

Lemma 22.20.4. In Situation 22.20.2 given any object $x$ of $\mathcal{A}$, and the cone $C(1_ x)$ of the identity morphism $1_ x : x \to x$, the identity morphism on $C(1_ x)$ is homotopic to zero.

Proof. Consider the admissible short exact sequence given by axiom (C).

\[ \xymatrix{ x \ar@<0.5ex>[r]^ a & C(1_ x) \ar@<0.5ex>[l]^{\pi } \ar@<0.5ex>[r]^ b & x[1]\ar@<0.5ex>[l]^ s } \]

Then by Lemma 22.20.1, identifying hom-sets under shifting, we have $1_ x=\pi d(s)=-d(\pi )s$ where $s$ is regarded as a morphism in $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{A}}^{-1}(x,C(1_ x))$. Therefore $a=a\pi d(s)=d(s)$ using formula (5) of Lemma 22.20.1, and $b=-d(\pi )sb=-d(\pi )$ by formula (6) of Lemma 22.20.1. Hence

\[ 1_{C(1_ x)} = a\pi + sb = d(s)\pi - sd(\pi ) = d(s\pi ) \]

since $s$ is of degree $-1$. $\square$


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