The Stacks project

Lemma 15.59.16. Let $R$ be a ring. Let $a : K^\bullet \to L^\bullet $ be a map of complexes of $R$-modules. If $K^\bullet $ is K-flat, then there exist a complex $N^\bullet $ and maps of complexes $b : K^\bullet \to N^\bullet $ and $c : N^\bullet \to L^\bullet $ such that

  1. $N^\bullet $ is K-flat,

  2. $c$ is a quasi-isomorphism,

  3. $a$ is homotopic to $c \circ b$.

If the terms of $K^\bullet $ are flat, then we may choose $N^\bullet $, $b$, and $c$ such that the same is true for $N^\bullet $.

Proof. We will use that the homotopy category $K(R)$ is a triangulated category, see Derived Categories, Proposition 13.10.3. Choose a distinguished triangle $K^\bullet \to L^\bullet \to C^\bullet \to K^\bullet [1]$. Choose a quasi-isomorphism $M^\bullet \to C^\bullet $ with $M^\bullet $ K-flat with flat terms, see Lemma 15.59.10. By the axioms of triangulated categories, we may fit the composition $M^\bullet \to C^\bullet \to K^\bullet [1]$ into a distinguished triangle $K^\bullet \to N^\bullet \to M^\bullet \to K^\bullet [1]$. By Lemma 15.59.5 we see that $N^\bullet $ is K-flat. Again using the axioms of triangulated categories, we can choose a map $N^\bullet \to L^\bullet $ fitting into the following morphism of distinghuised triangles

\[ \xymatrix{ K^\bullet \ar[r] \ar[d] & N^\bullet \ar[r] \ar[d] & M^\bullet \ar[r] \ar[d] & K^\bullet [1] \ar[d] \\ K^\bullet \ar[r] & L^\bullet \ar[r] & C^\bullet \ar[r] & K^\bullet [1] } \]

Since two out of three of the arrows are quasi-isomorphisms, so is the third arrow $N^\bullet \to L^\bullet $ by the long exact sequences of cohomology associated to these distinguished triangles (or you can look at the image of this diagram in $D(R)$ and use Derived Categories, Lemma 13.4.3 if you like). This finishes the proof of (1), (2), and (3). To prove the final assertion, we may choose $N^\bullet $ such that $N^ n \cong M^ n \oplus K^ n$, see Derived Categories, Lemma 13.10.7. Hence we get the desired flatness if the terms of $K^\bullet $ are flat. $\square$

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