The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Email from Juan Pablo Acosta Lopez dated 12/20/14.

Lemma 10.94.1. Let $R \to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If the $S$-module $M \otimes _ R S$ is Mittag-Leffler, then $M$ is Mittag-Leffler.

Proof. Write $M = \mathop{\mathrm{colim}}\nolimits _{i\in I} M_ i$ as a directed colimit of finitely presented $R$-modules $M_ i$. Using Proposition 10.87.6, we see that we have to prove that for each $i \in I$ there exists $i \leq j$, $j\in I$ such that $M_ i\rightarrow M_ j$ dominates $M_ i\rightarrow M$.

Take $N$ the pushout

\[ \xymatrix{ M_ i \ar[r] \ar[d] & M_ j \ar[d] \\ M \ar[r] & N } \]

Then the lemma is equivalent to the existence of $j$ such that $M_ j\rightarrow N$ is universally injective, see Lemma 10.87.4. Observe that the tensorization by $S$

\[ \xymatrix{ M_ i\otimes _ R S \ar[r] \ar[d] & M_ j\otimes _ R S \ar[d] \\ M\otimes _ R S \ar[r] & N\otimes _ R S } \]

Is a pushout diagram. So because $M \otimes _ R S = \mathop{\mathrm{colim}}\nolimits _{i\in I} M_ i \otimes _ R S$ expresses $M\otimes _ R S$ as a colimit of $S$-modules of finite presentation, and $M\otimes _ R S$ is Mittag-Leffler, there exists $j \geq i$ such that $M_ j\otimes _ R S\rightarrow N\otimes _ R S$ is universally injective. So using that $R\rightarrow S$ is faithfully flat we conclude that $M_ j\rightarrow N$ is universally injective too. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05A5. Beware of the difference between the letter 'O' and the digit '0'.