The Stacks project

Lemma 15.95.4. Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. There is an additive functor1 $L\eta _ f : D(A) \to D(A)$ such that if $M \in D(A)$ is represented by a complex $M^\bullet $ of $f$-torsion free $A$-modules, then $L\eta _ fM = \eta _ fM^\bullet $ and similarly for morphisms.

Proof. Denote $\mathcal{T} \subset \text{Mod}_ A$ the full subcategory of $f$-torsion free $A$-modules. We have a corresponding inclusion

\[ K(\mathcal{T}) \quad \subset \quad K(\text{Mod}_ A) = K(A) \]

of $K(\mathcal{T})$ as a full triangulated subcategory of $K(A)$. Let $S \subset \text{Arrows}(K(\mathcal{T}))$ be the quasi-isomorphisms. We will apply Derived Categories, Lemma 13.5.7 to show that the map

\[ S^{-1}K(\mathcal{T}) \longrightarrow D(A) \]

is an equivalence of triangulated categories. The lemma shows that it suffices to prove: given a complex $M^\bullet $ of $A$-modules, there exists a quasi-isomorphism $K^\bullet \to M^\bullet $ with $K^\bullet $ a complex of $f$-torsion free modules. By Lemma 15.59.10 we can find a quasi-isomorphism $K^\bullet \to M^\bullet $ such that the complex $K^\bullet $ is K-flat (we won't use this) and consists of flat $A$-modules $K^ i$. In particular, $f$ is a nonzerodivisor on $K^ i$ for all $i$ as desired.

With these preliminaries out of the way we can define $L\eta _ f$. Namely, by the discussion at the start of this section we have already a well defined functor

\[ K(\mathcal{T}) \xrightarrow {\eta _ f} K(\mathcal{T}) \to K(A) \to D(A) \]

which according to Lemma 15.95.3 sends quasi-isomorphisms to quasi-isomorphisms. Hence this functor factors over $S^{-1}K(\mathcal{T}) = D(A)$ by Categories, Lemma 4.27.8. $\square$

[1] Beware that this functor isn't exact, i.e., does not transform distinguished triangles into distinguished triangles. See Example 15.95.1.

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