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*}

15.85 Taking limits of complexes

In this section we discuss what happens when we have a “formal deformation” of a complex and we take its limit. We will consider two cases

  1. we have a limit $A = \mathop{\mathrm{lim}}\nolimits A_ n$ of an inverse system of rings whose transition maps are surjective with locally nilpotent kernels and objects $K_ n \in D(A_ n)$ which fit together in the sense that $K_ n = K_{n + 1} \otimes _{A_{n + 1}}^\mathbf {L} A_ n$, or

  2. we have a ring $A$, an ideal $I$, and objects $K_ n \in D(A/I^ n)$ which fit together in the sense that $K_ n = K_{n + 1} \otimes _{A/I^{n + 1}}^\mathbf {L} A/I^ n$.

Under additional hypotheses we can show that $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ reproduces the system in the sense that $K_ n = K \otimes _ A^\mathbf {L} A_ n$ or $K_ n = K \otimes _ A^\mathbf {L} A/I^ n$.

Lemma 15.85.1. Let $A = \mathop{\mathrm{lim}}\nolimits A_ n$ be a limit of an inverse system $(A_ n)$ of rings. Suppose given $K_ n \in D(A_ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A_{n + 1})$. Assume

  1. the transition maps $A_{n + 1} \to A_ n$ are surjective with locally nilpotent kernels,

  2. $K_1$ is pseudo-coherent, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A_{n + 1}}^\mathbf {L} A_ n \to K_ n$.

Then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a pseudo-coherent object of $D(A)$ and $K \otimes _ A^\mathbf {L} A_ n \to K_ n$ is an isomorphism for all $n$.

Proof. By assumption we can find a bounded above complex of finite free $A_1$-modules $P_1^\bullet $ representing $K_1$, see Definition 15.62.1. By Lemma 15.70.4 we can, by induction on $n > 1$, find complexes $P_ n^\bullet $ of finite free $A_ n$-modules representing $K_ n$ and maps $P_ n^\bullet \to P_{n - 1}^\bullet $ representing the maps $K_ n \to K_{n - 1}$ inducing isomorphisms (!) of complexes $P_ n^\bullet \otimes _{A_ n} A_{n - 1} \to P_{n - 1}^\bullet $. Thus $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is represented by $P^\bullet = \mathop{\mathrm{lim}}\nolimits P_ n^\bullet $, see Lemma 15.78.1 and Remark 15.78.6. Since $P_ n^ i$ is a finite free $A_ n$-module for each $n$ and $A = \mathop{\mathrm{lim}}\nolimits A_ n$ we see that $P^ i$ is finite free of the same rank as $P_1^ i$ for each $i$. This means that $K$ is pseudo-coherent. It also follows that $K \otimes _ A^\mathbf {L} A_ n$ is represented by $P^\bullet \otimes _ A A_ n = P_ n^\bullet $ which proves the final assertion. $\square$

Lemma 15.85.2. Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_ n \in D(A/I^ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A/I^{n + 1})$. Assume

  1. $A$ is $I$-adically complete,

  2. $K_1$ is pseudo-coherent, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A/I^{n + 1}}^\mathbf {L} A/I^ n \to K_ n$.

Then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a pseudo-coherent, derived complete object of $D(A)$ and $K \otimes _ A^\mathbf {L} A/I^ n \to K_ n$ is an isomorphism for all $n$.

Proof. We already know that $K$ is pseudo-coherent and that $K \otimes _ A^\mathbf {L} A/I^ n \to K_ n$ is an isomorphism for all $n$, see Lemma 15.85.1. To finish the proof it suffices to show that $K$ is derived complete. This follows from Lemma 15.82.8. $\square$

reference

Lemma 15.85.3. Let $A = \mathop{\mathrm{lim}}\nolimits A_ n$ be a limit of an inverse system $(A_ n)$ of rings. Suppose given $K_ n \in D(A_ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A_{n + 1})$. Assume

  1. the transition maps $A_{n + 1} \to A_ n$ are surjective with locally nilpotent kernels,

  2. $K_1$ is a perfect object, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A_{n + 1}}^\mathbf {L} A_ n \to K_ n$.

Then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a perfect object of $D(A)$ and $K \otimes _ A^\mathbf {L} A_ n \to K_ n$ is an isomorphism for all $n$.

Proof. We already know that $K$ is pseudo-coherent and that $K \otimes _ A^\mathbf {L} A_ n \to K_ n$ is an isomorphism for all $n$ by Lemma 15.85.1. Thus it suffices to show that $H^ i(K \otimes _ A^\mathbf {L} \kappa ) = 0$ for $i \ll 0$ and every surjective map $A \to \kappa $ whose kernel is a maximal ideal $\mathfrak m$, see Lemma 15.71.7. Any element of $A$ which maps to a unit in $A_1$ is a unit in $A$ by Algebra, Lemma 10.31.4 and hence $\mathop{\mathrm{Ker}}(A \to A_1)$ is contained in the Jacobson radical of $A$ by Algebra, Lemma 10.18.1. Hence $A \to \kappa $ factors as $A \to A_1 \to \kappa $. Hence

\[ K \otimes _ A^\mathbf {L} \kappa = K \otimes _ A^\mathbf {L} A_1 \otimes _{A_1}^\mathbf {L} \kappa = K_1 \otimes _{A_1}^\mathbf {L} \kappa \]

and we get what we want as $K_1$ has finite tor dimension by Lemma 15.69.2. $\square$

Lemma 15.85.4. Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_ n \in D(A/I^ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A/I^{n + 1})$. Assume

  1. $A$ is $I$-adically complete,

  2. $K_1$ is a perfect object, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A/I^{n + 1}}^\mathbf {L} A/I^ n \to K_ n$.

Then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a perfect, derived complete object of $D(A)$ and $K \otimes _ A^\mathbf {L} A/I^ n \to K_ n$ is an isomorphism for all $n$.

Proof. Combine Lemmas 15.85.3 and 15.85.2 (to get derived completeness). $\square$

The following lemma is more of a curiosity. It is rather subtle and esoteric and should be used with care. We do not know if it holds for unbounded complexes.

Lemma 15.85.5. Let $A$ be a ring and $I \subset A$ an ideal. Suppose given $K_ n \in D(A/I^ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A/I^{n + 1})$. If

  1. $A$ is Noetherian,

  2. $K_1$ is bounded above, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A/I^{n + 1}}^\mathbf {L} A/I^ n \to K_ n$,

then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a derived complete object of $D^-(A)$ and $K \otimes _ A^\mathbf {L} A/I^ n \to K_ n$ is an isomorphism for all $n$.

Proof. Suppose that $H^ i(K_1) = 0$ for $i > b$. Then we can find a complex of free $A/I$-modules $P_1^\bullet $ representing $K_1$ with $P_1^ i = 0$ for $i > b$. By Lemma 15.70.3 we can, by induction on $n > 1$, find complexes $P_ n^\bullet $ of free $A/I^ n$-modules representing $K_ n$ and maps $P_ n^\bullet \to P_{n - 1}^\bullet $ representing the maps $K_ n \to K_{n - 1}$ inducing isomorphisms (!) of complexes $P_ n^\bullet /I^{n - 1}P_ n^\bullet \to P_{n - 1}^\bullet $.

Thus we have arrived at the situation where $R\mathop{\mathrm{lim}}\nolimits K_ n$ is represented by $P^\bullet = \mathop{\mathrm{lim}}\nolimits P_ n^\bullet $, see Lemma 15.78.1 and Remark 15.78.6. The complexes $P_ n^\bullet $ are uniformly bounded above complexes of flat $A/I^ n$-modules and the transition maps are termwise surjective. Then $P^\bullet $ is a bounded above complex of flat $A$-modules by Lemma 15.27.4. It follows that $K \otimes _ A^\mathbf {L} A/I^ t$ is represented by $P^\bullet \otimes _ A A/I^ t$. We have $P^\bullet \otimes _ A A/I^ t = \mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t$ termwise by Lemma 15.27.4. The transition maps $P_{n + 1}^\bullet \otimes _ A A/I^ t \to P_ n^\bullet \otimes _ A A/I^ t$ are isomorphisms for $n \geq t$. Hence we have $\mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t = R\mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t$. By assumption and our choice of $P_ n^\bullet $ the complex $P_ n^\bullet \otimes _ A A/I^ t = P_ n^\bullet \otimes _{A/I^ n} A/I^ t$ represents $K_ n \otimes _{A/I^ n}^\mathbf {L} A/I^ t = K_ t$ for all $n \geq t$. We conclude

\[ P^\bullet \otimes _ A A/I^ t = R\mathop{\mathrm{lim}}\nolimits P_ n^\bullet \otimes _ A A/I^ t = R\mathop{\mathrm{lim}}\nolimits K_ t = K_ t \]

In other words, we have $K \otimes _ A^\mathbf {L} A/I^ t = K_ t$. This proves the lemma as it follows that $K$ is derived complete by Proposition 15.84.2. $\square$

Here is a different type of result.

reference

Lemma 15.85.6 (Kollár-Kovács). Let $I$ be an ideal of a Noetherian ring $A$. Let $K \in D(A)$. Set $K_ n = K \otimes _ A^\mathbf {L} A/I^ n$. Assume for all $i \in \mathbf{Z}$ we have

  1. $H^ i(K)$ is a finite $A$-module, and

  2. the system $H^ i(K_ n)$ satisfies Mittag-Leffler.

Then $\mathop{\mathrm{lim}}\nolimits H^ i(K)/I^ nH^ i(K)$ is equal to $\mathop{\mathrm{lim}}\nolimits H^ i(K_ n)$ for all $i \in \mathbf{Z}$.

Proof. Recall that $K^\wedge = R\mathop{\mathrm{lim}}\nolimits K_ n$ is the derived completion of $K$, see Proposition 15.84.2. By Lemma 15.84.4 we have $H^ i(K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ i(K)/I^ nH^ i(K)$. By Lemma 15.78.4 we get short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{i - 1}(K_ n) \to H^ i(K^\wedge ) \to \mathop{\mathrm{lim}}\nolimits H^ i(K_ n) \to 0 \]

The Mittag-Leffler condition guarantees that the left terms are zero (Lemma 15.78.1) and we conclude the lemma is true. $\square$


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