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*}

10.97 Taking limits of modules

In this section we discuss what happens when we take a limit of modules.

Lemma 10.97.1. Let $A$ be a ring. Let $I \subset A$ be an ideal. Let $(M_ n)$ be an inverse system of $A$-modules. Set $M = \mathop{\mathrm{lim}}\nolimits M_ n$. If $M_ n = M_{n + 1}/I^ nM_{n + 1}$ and $I$ is finitely generated then $M/I^ nM = M_ n$ and $M$ is $I$-adically complete.

Proof. As $M_{n + 1} \to M_ n$ is surjective, the map $M \to M_1$ is surjective. Pick $x_ t \in M$, $t \in T$ mapping to generators of $M_1$. This gives a map $\bigoplus _{t \in T} A \to M$. Note that the images of $x_ t$ in $M_ n$ generate $M_ n$ for all $n$ too. Consider the exact sequences

\[ 0 \to K_ n \to \bigoplus \nolimits _{t \in T} A/I^ n \to M_ n \to 0 \]

We claim the map $K_{n + 1} \to K_ n$ is surjective. Namely, if $y \in K_ n$ choose a lift $y' \in \bigoplus _{t \in T} A/I^{n + 1}$. Then $y'$ maps to an element of $I^ n M_{n + 1}$ by our assumption $M_ n = M_{n + 1}/I^ nM_{n + 1}$. Hence we can modify our choice of $y'$ by an element of $\bigoplus _{t \in T} I^ n/I^{n + 1}$ so that $y'$ maps to zero in $M_{n + 1}$. Then $y' \in K_{n +1}$ maps to $y$. Hence $(K_ n)$ is a sequence of modules with surjective transition maps and we obtain an exact sequence

\[ 0 \to \mathop{\mathrm{lim}}\nolimits K_ n \to \left(\bigoplus \nolimits _{t \in T} A\right)^\wedge \to M \to 0 \]

by Lemma 10.86.1. Fix an integer $m$. As $I$ is finitely generated, the completion with respect to $I$ is complete and $(\bigoplus _{t \in T} A)^\wedge / I^ m(\bigoplus _{t \in T} A)^\wedge = \bigoplus _{t \in T} A/I^ m$ (Lemma 10.95.3). We obtain a short exact sequence

\[ (\mathop{\mathrm{lim}}\nolimits K_ n)/I^ m(\mathop{\mathrm{lim}}\nolimits K_ n) \to \bigoplus \nolimits _{t \in T} A/I^ m \to M/I^ mM \to 0 \]

Since $\mathop{\mathrm{lim}}\nolimits K_ n \to K_ m$ is surjective we conclude that $M/I^ mM = M_ m$. It follows in particular that $M$ is $I$-adically complete. $\square$

Lemma 10.97.2. Let $A$ be a Noetherian graded ring. Let $I \subset A_+$ be a homogeneous ideal. Let $(N_ n)$ be an inverse system of finite graded $A$-modules with $N_ n = N_{n + 1}/I^ n N_{n + 1}$. Then there is a finite graded $A$-module $N$ such that $N_ n = N/I^ nN$ as graded modules for all $n$.

Proof. Pick $r$ and homogeneous elements $x_{1, 1}, \ldots , x_{1, r} \in N_1$ of degrees $d_1, \ldots , d_ r$ generating $N_1$. Since the transition maps are surjective, we can pick a compatible system of homogeneous elements $x_{n, i} \in N_ n$ lifting $x_{1, i}$. By the graded Nakayama lemma (Lemma 10.55.1) we see that $N_ n$ is generated by the elements $x_{n, 1}, \ldots , x_{n, r}$ sitting in degrees $d_1, \ldots , d_ r$. Thus for $m \leq n$ we see that $N_ n \to N_ n/I^ m N_ n$ is an isomorphism in degrees $< \min (d_ i) + m$ (as $I^ mN_ n$ is zero in those degrees). Thus the inverse system of degree $d$ parts

\[ \ldots = N_{2 + d - \min (d_ i), d} = N_{1 + d - \min (d_ i), d} = N_{d - \min (d_ i), d} \to N_{-1 + d - \min (d_ i), d} \to \ldots \]

stabilizes as indicated. Let $N$ be the graded $A$-module whose $d$th graded part is this stabilization. In particular, we have the elements $x_ i = \mathop{\mathrm{lim}}\nolimits x_{n, i}$ in $N$. We claim the $x_ i$ generate $N$: any $x \in N_ d$ is a linear combination of $x_1, \ldots , x_ r$ because we can check this in $N_{d - \min (d_ i), d}$ where it holds as $x_{d - \min (d_ i), i}$ generate $N_{d - \min (d_ i)}$. Finally, the reader checks that the surjective map $N/I^ nN \to N_ n$ is an isomorphism by checking to see what happens in each degree as before. Details omitted. $\square$

Lemma 10.97.3. Let $A$ be a graded ring. Let $I \subset A_+$ be a homogeneous ideal. Denote $A' = \mathop{\mathrm{lim}}\nolimits A/I^ n$. Let $(G_ n)$ be an inverse system of graded $A$-modules with $G_ n$ annihilated by $I^ n$. Let $M$ be a graded $A$-module and let $\varphi _ n : M \to G_ n$ be a compatible system of graded $A$-module maps. If the induced map

\[ \varphi : M \otimes _ A A' \longrightarrow \mathop{\mathrm{lim}}\nolimits G_ n \]

is an isomorphism, then $M_ d \to \mathop{\mathrm{lim}}\nolimits G_{n, d}$ is an isomorphism for all $d \in \mathbf{Z}$.

Proof. By convention graded rings are in degrees $\geq 0$ and graded modules may have nonzero parts of any degree, see Section 10.55. The map $\varphi $ exists because $\mathop{\mathrm{lim}}\nolimits G_ n$ is a module over $A'$ as $G_ n$ is annihilated by $I^ n$. Another useful thing to keep in mind is that we have

\[ \bigoplus \nolimits _{d \in \mathbf{Z}} \mathop{\mathrm{lim}}\nolimits G_{n, d} \subset \mathop{\mathrm{lim}}\nolimits G_ n \subset \prod \nolimits _{d \in \mathbf{Z}} \mathop{\mathrm{lim}}\nolimits G_{n, d} \]

where a subscript ${\ }_ d$ indicates the $d$th graded part.

Injective. Let $x \in M_ d$. If $x \mapsto 0$ in $\mathop{\mathrm{lim}}\nolimits G_{n, d}$ then $x \otimes 1 = 0$ in $M \otimes _ A A'$. Then we can find a finitely generated submodule $M' \subset M$ with $x \in M'$ such that $x \otimes 1$ is zero in $M' \otimes _ A A'$. Say $M'$ is generated by homogeneous elements sitting in degrees $d_1, \ldots , d_ r$. Let $n = d - \min (d_ i) + 1$. Since $A'$ has a map to $A/I^ n$ and since $A \to A/I^ n$ is an isomorphism in degrees $\leq n - 1$ we see that $M' \to M' \otimes _ A A'$ is injective in degrees $\leq n - 1$. Thus $x = 0$ as desired.

Surjective. Let $y \in \mathop{\mathrm{lim}}\nolimits G_{n, d}$. Choose a finite sum $\sum x_ i \otimes f'_ i$ in $M \otimes _ A A'$ mapping to $y$. We may assume $x_ i$ is homogeneous, say of degree $d_ i$. Observe that although $A'$ is not a graded ring, it is a limit of the graded rings $A/I^ nA$ and moreover, in any given degree the transition maps eventually become isomorphisms (see above). This gives

\[ A = \bigoplus \nolimits _{d \geq 0} A_ d \subset A' \subset \prod \nolimits _{d \geq 0} A_ d \]

Thus we can write

\[ f'_ i = \sum \nolimits _{j = 0, \ldots , d - d_ i - 1} f_{i, j} + f_ i + g'_ i \]

with $f_{i, j} \in A_ j$, $f_ i \in A_{d - d_ i}$, and $g'_ i \in A'$ mapping to zero in $\prod _{j \leq d - d_ i} A_ j$. Now if we compute $\varphi _ n(\sum _{i, j} f_{i, j}x_ i) \in G_ n$, then we get a sum of homogeneous elements of degree $< d$. Hence $\varphi (\sum x_ i \otimes f_{i, j})$ maps to zero in $\mathop{\mathrm{lim}}\nolimits G_{n, d}$. Similarly, a computation shows the element $\varphi (\sum x_ i \otimes g'_ i)$ maps to zero in $\prod _{d' \leq d} \mathop{\mathrm{lim}}\nolimits G_{n, d'}$. Since we know that $\varphi (\sum x_ i \otimes f'_ i)$ is $y$, we conclude that $\sum f_ ix_ i \in M_ d$ maps to $y$ as desired. $\square$


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