Lemma 13.18.8 (05TG)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.18: Injective resolutions
Lemma 13.18.8 (cite)

Lemma 13.18.8. Let $\mathcal{A}$ be an abelian category. Let $I^\bullet $ be bounded below complex consisting of injective objects. Let $L^\bullet \in K(\mathcal{A})$. Then

\[ \mathop{\mathrm{Mor}}\nolimits _{K(\mathcal{A})}(L^\bullet , I^\bullet ) = \mathop{\mathrm{Mor}}\nolimits _{D(\mathcal{A})}(L^\bullet , I^\bullet ). \]

Proof. Let $a$ be an element of the right hand side. We may represent $a = \gamma \alpha ^{-1}$ where $\alpha : K^\bullet \to L^\bullet $ is a quasi-isomorphism and $\gamma : K^\bullet \to I^\bullet $ is a map of complexes. By Lemma 13.18.6 we can find a morphism $\beta : L^\bullet \to I^\bullet $ such that $\beta \circ \alpha $ is homotopic to $\gamma $. This proves that the map is surjective. Let $b$ be an element of the left hand side which maps to zero in the right hand side. Then $b$ is the homotopy class of a morphism $\beta : L^\bullet \to I^\bullet $ such that there exists a quasi-isomorphism $\alpha : K^\bullet \to L^\bullet $ with $\beta \circ \alpha $ homotopic to zero. Then Lemma 13.18.7 shows that $\beta $ is homotopic to zero also, i.e., $b = 0$. $\square$

Comments (2)

Comment #8417 by Elías Guisado on May 23, 2023 at 09:36

To justify "there exists a quasi-isomorphism $\alpha : K^\bullet \to L^\bullet$ with $\beta \circ \alpha$ homotopic to zero," one could link the result I propose between parentheses in https://stacks.math.columbia.edu/tag/05RW#comment-8372

Comment #8828 by Elías Guisado on February 26, 2024 at 08:48

@#8785 Okay, I just realized: if $\mathcal{A}$ has enough injectives, then the $\delta$ -functor $(\operatorname{Ext}_\mathcal{A}^i(A,-))_{i\geq 0}$ is universal because it is erasable, i.e., it satisfies the hypothesis of 12.12.4: For an object $B\in\mathcal{A}$ , take an injection $B\to I$ into an injective object. Then $\operatorname{Ext}_\mathcal{A}^i(A,I)$ vanishes for all $i>0$ by 13.18.8. Dually, if $\mathcal{A}$ has enough projectives, one can apply the analogous argument to $(\operatorname{Ext}_\mathcal{A}^i(-,B))_{i\geq 0}$ to deduce erasibility (and thus universality).

Is the statement of this result to be found anywhere in the SP?

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