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*}

15.78 Rlim of modules

We briefly discuss $R\mathop{\mathrm{lim}}\nolimits $ on modules. Many of the arguments in this section duplicate the arguments used to construct the cohomological machinery for modules on ringed sites.

Let $(A_ n)$ be an inverse system of rings. We will denote $\textit{Mod}(\mathbf{N}, (A_ n))$ the category of inverse systems $(M_ n)$ of abelian groups such that each $M_ n$ is given the structure of a $A_ n$-module and the transition maps $M_{n + 1} \to M_ n$ are $A_{n + 1}$-module maps. This is an abelian category. Set $A = \mathop{\mathrm{lim}}\nolimits A_ n$. Given an object $(M_ n)$ of $\textit{Mod}(\mathbf{N}, (A_ n))$ the limit $\mathop{\mathrm{lim}}\nolimits M_ n$ is an $A$-module.

Lemma 15.78.1. In the situation above. The functor $\mathop{\mathrm{lim}}\nolimits : \textit{Mod}(\mathbf{N}, (A_ n)) \to \text{Mod}_ A$ has a right derived functor

\[ R\mathop{\mathrm{lim}}\nolimits : D(\textit{Mod}(\mathbf{N}, (A_ n))) \longrightarrow D(A) \]

As usual we set $R^ p\mathop{\mathrm{lim}}\nolimits (K) = H^ p(R\mathop{\mathrm{lim}}\nolimits (K))$. Moreover, we have

  1. for any $(M_ n)$ in $\textit{Mod}(\mathbf{N}, (A_ n))$ we have $R^ p\mathop{\mathrm{lim}}\nolimits M_ n = 0$ for $p > 1$,

  2. the object $R\mathop{\mathrm{lim}}\nolimits M_ n$ of $D(\text{Mod}_ A)$ is represented by the complex

    \[ \prod M_ n \to \prod M_ n,\quad (x_ n) \mapsto (x_ n - f_{n + 1}(x_{n + 1})) \]

    sitting in degrees $0$ and $1$,

  3. if $(M_ n)$ is ML, then $R^1\mathop{\mathrm{lim}}\nolimits M_ n = 0$, i.e., $(M_ n)$ is right acyclic for $\mathop{\mathrm{lim}}\nolimits $,

  4. every $K^\bullet \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ is quasi-isomorphic to a complex whose terms are right acyclic for $\mathop{\mathrm{lim}}\nolimits $, and

  5. if each $K^ p = (K^ p_ n)$ is right acyclic for $\mathop{\mathrm{lim}}\nolimits $, i.e., of $R^1\mathop{\mathrm{lim}}\nolimits _ n K^ p_ n = 0$, then $R\mathop{\mathrm{lim}}\nolimits K$ is represented by the complex whose term in degree $p$ is $\mathop{\mathrm{lim}}\nolimits _ n K_ n^ p$.

Proof. The proof of this is word for word the same as the proof of Lemma 15.77.1. $\square$

Remark 15.78.2. This remark is a continuation of Remark 15.77.5. A sheaf of rings on $\mathbf{N}$ is just an inverse system of rings $(A_ n)$. A sheaf of modules over $(A_ n)$ is exactly the same thing as an object of the category $\textit{Mod}(\mathbf{N}, (A_ n))$ defined above. The derived functor $R\mathop{\mathrm{lim}}\nolimits $ of Lemma 15.78.1 is simply $R\Gamma (\mathbf{N}, -)$ from the derived category of modules to the derived category of modules over the global sections of the structure sheaf. It is true in general that cohomology of groups and modules agree, see Cohomology on Sites, Lemma 21.13.4.

The products in the following lemma can be seen as termwise products of complexes or as products in the derived category $D(A)$, see Derived Categories, Lemma 13.32.2.

Lemma 15.78.3. Let $K = (K_ n^\bullet )$ be an object of $D(\textit{Mod}(\mathbf{N}, (A_ n)))$. There exists a canonical distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits K \to \prod \nolimits _ n K_ n^\bullet \to \prod \nolimits _ n K_ n^\bullet \to R\mathop{\mathrm{lim}}\nolimits K[1] \]

in $D(A)$. In other words, $R\mathop{\mathrm{lim}}\nolimits K$ is a derived limit of the inverse system $(K_ n^\bullet )$ of $D(A)$, see Derived Categories, Definition 13.32.1.

Proof. The proof is exactly the same as the proof of Lemma 15.77.6 using Lemma 15.78.1 in stead of Lemma 15.77.1. $\square$

Lemma 15.78.4. With notation as in Lemma 15.78.3 the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences

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

of $A$-modules.

Proof. The proof is exactly the same as the proof of Lemma 15.77.7 using Lemma 15.78.1 in stead of Lemma 15.77.1. $\square$

Warning. As in the case of abelian groups an object $M = (M_ n^\bullet )$ of $D(\textit{Mod}(\mathbf{N}, (A_ n)))$ is an inverse system of complexes of modules, which is not the same thing as an inverse system of objects in the derived categories. In the following lemma we show how an inverse system of objects in derived categories always lifts to an object of $D(\textit{Mod}(\mathbf{N}, (A_ n)))$.

Lemma 15.78.5. Let $(A_ n)$ be an inverse system of rings. Suppose that we are given

  1. for every $n$ an object $K_ n$ of $D(A_ n)$, and

  2. for every $n$ a map $\varphi _ n : K_{n + 1} \to K_ n$ of $D(A_{n + 1})$ where we think of $K_ n$ as an object of $D(A_{n + 1})$ by restriction via $A_{n + 1} \to A_ n$.

There exists an object $M = (M_ n^\bullet ) \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ and isomorphisms $\psi _ n : M_ n^\bullet \to K_ n$ in $D(A_ n)$ such that the diagrams

\[ \xymatrix{ M_{n + 1}^\bullet \ar[d]_{\psi _{n + 1}} \ar[r] & M_ n^\bullet \ar[d]^{\psi _ n} \\ K_{n + 1} \ar[r]^{\varphi _ n} & K_ n } \]

commute in $D(A_{n + 1})$.

Proof. We write out the proof in detail. For an $A_ n$-module $T$ we write $T_{A_{n + 1}}$ for the same module viewd as an $A_{n + 1}$-module. Suppose that $K_ n^\bullet $ is a complex of $A_ n$-modules representing $K_ n$. Then $K_{n, A_{n + 1}}^\bullet $ is the same complex, but viewed as a complex of $A_{n + 1}$-modules. By the construction of the derived category, the map $\psi _ n$ can be given as

\[ \psi _ n = \tau _ n \circ \sigma _ n^{-1} \]

where $\sigma _ n : L_{n + 1}^\bullet \to K_{n + 1}^\bullet $ is a quasi-isomorphism of complexes of $A_{n + 1}$-modules and $\tau _ n : L_{n + 1}^\bullet \to K_{n, A_{n + 1}}^\bullet $ is a map of complexes of $A_{n + 1}$-modules.

Now we construct the complexes $M_ n^\bullet $ by induction. As base case we let $M_1^\bullet = K_1^\bullet $. Suppose we have already constructed $M_ e^\bullet \to M_{e - 1}^\bullet \to \ldots \to M_1^\bullet $ and maps of complexes $\psi _ i : M_ i^\bullet \to K_ i^\bullet $ such that the diagrams

\[ \xymatrix{ M_{n + 1}^\bullet \ar[d]_{\psi _{n + 1}} \ar[rr] & & M_{n, A_{n + 1}}^\bullet \ar[d]^{\psi _{n, A_{n + 1}}} \\ K_{n + 1}^\bullet & L_{n + 1}^\bullet \ar[l]_{\sigma _ n} \ar[r]^{\tau _ n} & K_{n, A_{n + 1}}^\bullet } \]

above commute in $D(A_{n + 1})$ for all $n < e$. Then we consider the diagram

\[ \xymatrix{ & & M_{e, A_{e + 1}}^\bullet \ar[d]^{\psi _{e, A_{e + 1}}} \\ K_{e + 1}^\bullet & L_{e + 1}^\bullet \ar[r]^{\tau _ e} \ar[l]_{\sigma _ e} & K_{e, A_{e + 1}}^\bullet } \]

in $D(A_{e + 1})$. Because $\psi _ e$ is a quasi-isomorphism, we see that $\psi _{e, A_{e + 1}}$ is a quasi-isomorphism too. By the definition of morphisms in $D(A_{e + 1})$ we can find a quasi-isomorphism $\psi _{e + 1} : M_{e + 1}^\bullet \to K_{e + 1}^\bullet $ of complexes of $A_{e + 1}$-modules such that there exists a morphism of complexes $M_{e + 1}^\bullet \to M_{e, A_{e + 1}}^\bullet $ of $A_{e + 1}$-modules representing the composition $\psi _{e, A_{e + 1}}^{-1} \circ \tau _ e \circ \sigma _ e^{-1}$ in $D(A_{e + 1})$. Thus the lemma holds by induction. $\square$

Remark 15.78.6. With assumptions as in Lemma 15.78.5. A priori there are many isomorphism classes of objects $M$ of $D(\textit{Mod}(\mathbf{N}, (A_ n)))$ which give rise to the system $(K_ n, \varphi _ n)$ of the lemma. For each such $M$ we can consider the complex $R\mathop{\mathrm{lim}}\nolimits M \in D(A)$ where $A = \mathop{\mathrm{lim}}\nolimits A_ n$. By Lemma 15.78.3 we see that $R\mathop{\mathrm{lim}}\nolimits M$ is a derived limit of the inverse system $(K_ n)$ of $D(A)$. Hence we see that the isomorphism class of $R\mathop{\mathrm{lim}}\nolimits M$ in $D(A)$ is independent of the choices made in constructing $M$. In particular, we may apply results on $R\mathop{\mathrm{lim}}\nolimits $ proved in this section to derived limits of inverse systems in $D(A)$. For example, for every $p \in \mathbf{Z}$ there is a canonical short exact sequence

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

because we may apply Lemma 15.78.3 to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma 15.78.3 to the (defining) distinguished triangle $R\mathop{\mathrm{lim}}\nolimits K_ n \to \prod K_ n \to \prod K_ n \to (R\mathop{\mathrm{lim}}\nolimits K_ n)[1]$ of the derived limit.

Lemma 15.78.7. Let $(A_ n)$ be an inverse system of rings. Every $K \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ can be represented by a system of complexes $(M_ n^\bullet )$ such that all the transition maps $M_{n + 1}^\bullet \to M_ n^\bullet $ are surjective.

Proof. Let $K$ be represented by the system $(K_ n^\bullet )$. Set $M_1^\bullet = K_1^\bullet $. Suppose we have constructed surjective maps of complexes $M_ n^\bullet \to M_{n - 1}^\bullet \to \ldots \to M_1^\bullet $ and homotopy equivalences $\psi _ e : K_ e^\bullet \to M_ e^\bullet $ such that the diagrams

\[ \xymatrix{ K_{e + 1}^\bullet \ar[d] \ar[r] & K_ e^\bullet \ar[d] \\ M_{e + 1}^\bullet \ar[r] & M_ e^\bullet } \]

commute for all $e < n$. Then we consider the diagram

\[ \xymatrix{ K_{n + 1}^\bullet \ar[r] & K_ n^\bullet \ar[d] \\ & M_ n^\bullet } \]

By Derived Categories, Lemma 13.9.8 we can factor the composition $K_{n + 1}^\bullet \to M_ n^\bullet $ as $K_{n + 1}^\bullet \to M_{n + 1}^\bullet \to M_ n^\bullet $ such that the first arrow is a homotopy equivalence and the second a termwise split surjection. The lemma follows from this and induction. $\square$

Lemma 15.78.8. Let $(A_ n)$ be an inverse system of rings. Every $K \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ can be represented by a system of complexes $(K_ n^\bullet )$ such that each $K_ n^\bullet $ is K-flat.

Proof. First use Lemma 15.78.7 to represent $K$ by a system of complexes $(M_ n^\bullet )$ such that all the transition maps $M_{n + 1}^\bullet \to M_ n^\bullet $ are surjective. Next, let $K_1^\bullet \to M_1^\bullet $ be a quasi-isomorphism with $K_1^\bullet $ a K-flat complex of $A_1$-modules (Lemma 15.57.12). Suppose we have constructed $K_ n^\bullet \to K_{n - 1}^\bullet \to \ldots \to K_1^\bullet $ and maps of complexes $\psi _ e : K_ e^\bullet \to M_ e^\bullet $ such that

\[ \xymatrix{ K_{e + 1}^\bullet \ar[d] \ar[r] & K_ e^\bullet \ar[d] \\ M_{e + 1}^\bullet \ar[r] & M_ e^\bullet } \]

commutes for all $e < n$. Then we consider the diagram

\[ \xymatrix{ C^\bullet \ar@{..>}[d] \ar@{..>}[r] & K_ n^\bullet \ar[d]^{\psi _ n} \\ M_{n + 1}^\bullet \ar[r]^{\varphi _ n} & M_ n^\bullet } \]

in $D(A_{n + 1})$. As $M_{n + 1}^\bullet \to M_ n^\bullet $ is termwise surjective, the complex $C^\bullet $ fitting into the left upper corner with terms

\[ C^ p = M_{n + 1}^ p \times _{M_ n^ p} K_ n^ p \]

is quasi-isomorphic to $M_{n + 1}^\bullet $ (details omitted). Choose a quasi-isomorphism $K_{n + 1}^\bullet \to C^\bullet $ with $K_{n +1}^\bullet $ K-flat. Thus the lemma holds by induction. $\square$

Lemma 15.78.9. Let $(A_ n)$ be an inverse system of rings. Given $K, L \in D(\textit{Mod}(\mathbf{N}, (A_ n)))$ there is a canonical derived tensor product $K \otimes ^\mathbf {L} L$ in $D(\mathbf{N}, (A_ n))$ compatible with the maps to $D(A_ n)$. The construction is symmetric in $K$ and $L$ and an exact functor of triangulated categories in each variable.

Proof. Choose a representative $(K_ n^\bullet )$ for $K$ such that each $K_ n^\bullet $ is a K-flat complex (Lemma 15.78.8). Then you can define $K \otimes ^\mathbf {L} L$ as the object represented by the system of complexes

\[ (\text{Tot}(K_ n^\bullet \otimes _{A_ n} L_ n^\bullet )) \]

for any choice of representative $(L_ n^\bullet )$ for $L$. This is well defined in both variables by Lemmas 15.57.4 and 15.57.14. Compatibility with the map to $D(A_ n)$ is clear. Exactness follows exactly as in Lemma 15.57.2. $\square$

Remark 15.78.10. Let $A$ be a ring. Let $(E_ n)$ be an inverse system of objects of $D(A)$. We've seen above that a derived limit $R\mathop{\mathrm{lim}}\nolimits E_ n$ exists. Thus for every object $K$ of $D(A)$ also the derived limit $R\mathop{\mathrm{lim}}\nolimits ( K \otimes _ A^\mathbf {L} E_ n )$ exists. It turns out that we can construct these derived limits functorially in $K$ and obtain an exact functor

\[ R\mathop{\mathrm{lim}}\nolimits (- \otimes _ A^\mathbf {L} E_ n) : D(A) \longrightarrow D(A) \]

of triangulated categories. Namely, we first lift $(E_ n)$ to an object $E$ of $D(\mathbf{N}, A)$, see Lemma 15.78.5. (The functor will depend on the choice of this lift.) Next, observe that there is a “diagonal” or “constant” functor

\[ \Delta : D(A) \longrightarrow D(\mathbf{N}, A) \]

mapping the complex $K^\bullet $ to the constant inverse system of complexes with value $K^\bullet $. Then we simply define

\[ R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} E_ n) = R\mathop{\mathrm{lim}}\nolimits (\Delta (K)\otimes ^\mathbf {L} E) \]

where on the right hand side we use the functor $R\mathop{\mathrm{lim}}\nolimits $ of Lemma 15.78.1 and the functor $- \otimes ^\mathbf {L} -$ of Lemma 15.78.9.

Lemma 15.78.11. Let $A$ be a ring. Let $E \to D \to F \to E[1]$ be a distinguished triangle of $D(\mathbf{N}, A)$. Let $(E_ n)$, resp. $(D_ n)$, resp. $(F_ n)$ be the system of objects of $D(A)$ associated to $E$, resp. $D$, resp. $F$. Then for every $K \in D(A)$ there is a canonical distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A D_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A F_ n) \to R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n)[1] \]

in $D(A)$ with notation as in Remark 15.78.10.

Proof. This is clear from the construction in Remark 15.78.10 and the fact that $\Delta : D(A) \to D(\mathbf{N}, A)$, $- \otimes ^\mathbf {L} -$, and $R\mathop{\mathrm{lim}}\nolimits $ are exact functors of triangulated categories. $\square$

Lemma 15.78.12. Let $A$ be a ring. Let $E \to D$ be a morphism of $D(\mathbf{N}, A)$. Let $(E_ n)$, resp. $(D_ n)$ be the system of objects of $D(A)$ associated to $E$, resp. $D$. If $(E_ n) \to (D_ n)$ is an isomorphism of pro-objects, then for every $K \in D(A)$ the corresponding map

\[ R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A E_ n) \longrightarrow R\mathop{\mathrm{lim}}\nolimits (K \otimes ^\mathbf {L}_ A D_ n) \]

in $D(A)$ is an isomorphism (notation as in Remark 15.78.10).

Proof. Follows from the definitions and Lemma 15.77.10. $\square$


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