Lemma 15.95.4. Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. There is an additive functor^{1} $L\eta _ f : D(A) \to D(A)$ 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

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 13.5.8 to show that the map

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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: