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*}

10.86 Inverse systems

In many papers (and in this section) the term inverse system is used to indicate an inverse system over the partially ordered set $(\mathbf{N}, \geq )$. We briefly discuss such systems in this section. This material will be discussed more broadly in Homology, Section 12.28. Suppose we are given a ring $R$ and a sequence of $R$-modules

\[ M_1 \xleftarrow {\varphi _2} M_2 \xleftarrow {\varphi _3} M_3 \leftarrow \ldots \]

with maps as indicated. By composing successive maps we obtain maps $\varphi _{ii'} : M_ i \to M_{i'}$ whenever $i \geq i'$ such that moreover $\varphi _{ii''} = \varphi _{i'i''} \circ \varphi _{i i'}$ whenever $i \geq i' \geq i''$. Conversely, given the system of maps $\varphi _{ii'}$ we can set $\varphi _ i = \varphi _{i(i-1)}$ and recover the maps displayed above. In this case

\[ \mathop{\mathrm{lim}}\nolimits M_ i = \{ (x_ i) \in \prod M_ i \mid \varphi _ i(x_ i) = x_{i - 1}, \ i = 2, 3, \ldots \} \]

compare with Categories, Section 4.15. As explained in Homology, Section 12.28 this is actually a limit in the category of $R$-modules, as defined in Categories, Section 4.14.

Lemma 10.86.1. Let $R$ be a ring. Let $0 \to K_ i \to L_ i \to M_ i \to 0$ be short exact sequences of $R$-modules, $i \geq 1$ which fit into maps of short exact sequences

\[ \xymatrix{ 0 \ar[r] & K_ i \ar[r] & L_ i \ar[r] & M_ i \ar[r] & 0 \\ 0 \ar[r] & K_{i + 1} \ar[r] \ar[u] & L_{i + 1} \ar[r] \ar[u] & M_{i + 1} \ar[r] \ar[u] & 0} \]

If for every $i$ there exists a $c = c(i) \geq i$ such that $\mathop{\mathrm{Im}}(K_ c \to K_ i) = \mathop{\mathrm{Im}}(K_ j \to K_ i)$ for all $j \geq c$, then the sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits K_ i \to \mathop{\mathrm{lim}}\nolimits L_ i \to \mathop{\mathrm{lim}}\nolimits M_ i \to 0 \]

is exact.

Proof. This is a special case of the more general Lemma 10.85.4. $\square$


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 03C9. Beware of the difference between the letter 'O' and the digit '0'.