Lemma 15.59.10. Let R be a ring. For any complex M^\bullet there exists a K-flat complex K^\bullet whose terms are flat R-modules and a quasi-isomorphism K^\bullet \to M^\bullet which is termwise surjective.
Proof. Let \mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\text{Mod}_ R) be the class of flat R-modules. By Derived Categories, Lemma 13.29.1 there exists a system K_1^\bullet \to K_2^\bullet \to \ldots and a diagram
\xymatrix{ K_1^\bullet \ar[d] \ar[r] & K_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}M^\bullet \ar[r] & \tau _{\leq 2}M^\bullet \ar[r] & \ldots }
with the properties (1), (2), (3) listed in that lemma. These properties imply each complex K_ i^\bullet is a bounded above complex of flat modules. Hence K_ i^\bullet is K-flat by Lemma 15.59.7. The induced map \mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet \to M^\bullet is a quasi-isomorphism and termwise surjective by construction. The complex \mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet is K-flat by Lemma 15.59.8. The terms \mathop{\mathrm{colim}}\nolimits K_ i^ n are flat because filtered colimits of flat modules are flat, see Algebra, Lemma 10.39.3. \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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: