Lemma 24.35.1. In the situation above, suppose that $\mathcal{A} = (A_ n)$ and $\mathcal{B} = (B_ n)$ are inverse systems of differential graded $R$-algebras. If $\varphi : (A_ n) \to (B_ n)$ is an isomorphism of pro-objects, then the functor $\mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{B}, \text{d})$ constructed above is an equivalence.
24.35 Inverse systems of differential graded algebras
In this section we consider the following special case of the situation discussed in Section 24.33:
$\mathcal{C}$ is the category $\mathbf{N}$ with a unique morphism $i \to j$ if and only if $i \leq j$,
$\mathcal{O}$ is the constant (pre)sheaf of rings with value a given ring $R$.
In this setting a sheaf $\mathcal{A}$ of differential graded $\mathcal{O}$-algebras is the same thing as an inverse system $(A_ n)$ of differential graded $R$-algebras. A sheaf $\mathcal{M}$ of differential graded $\mathcal{A}$-modules is the same thing as an inverse system $(M_ n)$ where $M_ n$ is a differential graded $A_ n$-module and the transition maps $M_{n + 1} \to M_ n$ are $A_{n + 1}$-module maps.
Suppose that $\mathcal{B} = (B_ n)$ is a second inverse system of differential graded $R$-algebras. Given a morphism $\varphi : (A_ n) \to (B_ n)$ of pro-objects we will construct an exact functor from $\mathit{QC}(\mathcal{A}, \text{d})$ to $\mathit{QC}(\mathcal{B}, \text{d})$. Namely, according to Categories, Example 4.22.6 the morphism $\varphi $ is given by a sequence $\ldots \geq m(3) \geq m(2) \geq m(1)$ of integers and a commutative diagram
\[ \xymatrix{ \ldots \ar[r] & A_{m(3)} \ar[d]^{\varphi _3} \ar[r] & A_{m(2)} \ar[d]^{\varphi _2} \ar[r] & A_{m(1)} \ar[d]^{\varphi _1} \\ \ldots \ar[r] & B_3 \ar[r] & B_2 \ar[r] & B_1 } \]
of differential graded $R$-algebras. Then given a good sheaf of differential graded $\mathcal{A}$-modules $\mathcal{M} = (M_ n)$ representing an object of $\mathit{QC}(\mathcal{A}, \text{d})$ we can set
\[ N_ n = M_{m(n)} \otimes _{A_{m(n)}} B_ n \]
This inverse system determines an object of $\mathit{QC}(\mathcal{B}, \text{d})$ because the $A_{m(n)}$-modules $M_{m(n)}$ are K-flat; details omitted. We also leave it to the reader to show that the resulting functor is independent of the choices made in its construction.
Proof. Let $\psi : (B_ n) \to (A_ n)$ be a morphism of pro-objects which is inverse to $\varphi $. According to the discussion in Categories, Example 4.22.6 we may assume that $\varphi $ is given by a system of maps as above and $\psi $ is given $n(1) < n(2) < \ldots $ and a commutative diagram
\[ \xymatrix{ \ldots \ar[r] & B_{n(3)} \ar[d]^{\psi _3} \ar[r] & B_{n(2)} \ar[d]^{\psi _2} \ar[r] & B_{n(1)} \ar[d]^{\psi _1} \\ \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 } \]
of differential graded $R$-algebras. Since $\varphi \circ \psi = \text{id}$ we may, after possibly increasing the values of the functions $n(\cdot )$ and $m(\cdot )$ assume that $B_{n(m(i))} \to A_{m(i)} \to B_ i$ is the identity. It follows that the composition of the functors
\[ \mathit{QC}(\mathcal{B}, \text{d}) \to \mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{B}, \text{d}) \]
sends a good sheaf of differential graded $\mathcal{B}$-modules $\mathcal{N} = (N_ n)$ to the inverse system $\mathcal{N}' = (N'_ i)$ with values
\[ N'_ i = N_{n(m(i))} \otimes _{B_{n(m(i))}} B_ i \]
which is canonically quasi-isomorphic to $\mathcal{N}$ exactly because $\mathcal{N}$ is an object of $\mathit{QC}(\mathcal{B}, \text{d})$ and because $N_ j$ is a K-flat differential graded module for all $j$. Since the same is true for the composition the other way around we conclude. $\square$
Let $\mathcal{C} = \mathbf{N}$ and $\mathcal{O}$ the constant sheaf with value a ring $R$ and let $\mathcal{A}$ be given by an inverse system $(A_ n)$ of differential graded $R$-algebras. Suppose given two left differential graded $\mathcal{A}$-modules $\mathcal{N}$ and $\mathcal{N}'$ given by inverse systems $(N_ n)$ and $(N'_ n)$. Thus each $N_ n$ and $N'_ n$ is a left differential graded $A_ n$-module. Let us temporarily say that $(N_ n)$ and $(N'_ n)$ are pro-isomorphic in the derived category if there exist a sequence of integers
\[ 1 = n_0 < n_1 < n_2 < n_3 < \ldots \]
and maps
\[ N_{n_{2i}} \to N'_{n_{2i - 1}} \quad \text{in}\quad D(A_{n_{2i}}^{opp}, \text{d}) \]
and
\[ N'_{n_{2i + 1}} \to N'_{n_{2i}} \quad \text{in}\quad D(A_{n_{2i + 1}}^{opp}, \text{d}) \]
such that the compositions $N_{n_{2i}} \to N_{n_{2i - 2}}$ and $N'_{n_{2i + 1}} \to N'_{2i - 1}$ are given by the transition maps of the respective systems.
Lemma 24.35.2. If $(N_ n)$ and $(N'_ n)$ are pro-isomorphic in the derived category as defined above, then for every object $(M_ n)$ of $D(\mathbf{N}, \mathcal{A})$ we have in $D(R)$.
\[ R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{A_ n}^\mathbf {L} N_ n) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{A_ n}^\mathbf {L} N'_ n) \]
Proof. The assumption implies that the inverse system $(M_ n \otimes _{A_ n}^\mathbf {L} N_ n)$ of $D(R)$ is pro-isomorphic (in the usual sense) to the inverse system $(M_ n \otimes _{A_ n}^\mathbf {L} N'_ n)$ of $D(R)$. Hence the result follows from the fact that taking $R\mathop{\mathrm{lim}}\nolimits $ is well defined for inverse systems in the derived category, see discussion in More on Algebra, Section 15.87. $\square$
Lemma 24.35.3. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$. Let $K_ n$ be the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ viewed as a differential graded $R$-algebra. Let $(M_ n)$ be an object of $D(\mathbf{N}, (K_ n))$. Then for any $t \geq 1$ we have in $D(R)$.
\[ R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _ R^\mathbf {L} K_ t) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{K_ n}^\mathbf {L} K_ t) \]
Proof. We fix $t \geq 1$. For $n \geq t$ let us denote ${}_ nK_ t$ the differential graded $R$-algebra $K_ t$ viewed as a left differential graded $K_ n$-module. Observe that
\[ M_ n \otimes _ R^\mathbf {L} K_ t = M_ n \otimes _{K_ n}^\mathbf {L} (K_ n \otimes _ R^\mathbf {L} K_ t) = M_ n \otimes _{K_ n}^\mathbf {L} (K_ n \otimes _ R K_ t) \]
Hence by Lemma 24.35.2 it suffices to show that $({}_ nK_ t)$ and $(K_ n \otimes _ R K_ t)$ are pro-isomorphic in the derived category. The multiplication maps
\[ K_ n \otimes _ R K_ t \longrightarrow {}_ nK_ t \]
are maps of left differential graded $K_ n$-modules. Thus to finish the proof it suffices to show that for all $n \geq 1$ there exists an $N > n$ and a map
\[ {}_ NK_ t \longrightarrow {}_ NK_ n \otimes _ R K_ t \]
in $D(K_ N^{opp}, \text{d})$ whose composition with the multiplication map is the transition map (in either direction). This is done in Divided Power Algebra, Lemma 23.12.4 by an explicit construction. $\square$
Proposition 24.35.4. Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal. The following three categories are canonically equivalent:
Let $\mathcal{A}$ be the sheaf of $R$-algebras on $\mathbf{N}$ corresponding to the inverse system of $R$-algebras $A_ n = R/I^ n$. The category $\mathit{QC}(\mathcal{A})$.
Choose generators $f_1, \ldots , f_ r$ of $I$. Let $\mathcal{B}$ be the sheaf of differential graded $R$-algebras on $\mathbf{N}$ corresponding to the inverse system of Koszul algebras on $f_1^ n, \ldots , f_ r^ n$. The category $\mathit{QC}(\mathcal{B})$.
The full subcategory $D_{comp}(R, I) \subset D(R)$ of derived complete objects, see More on Algebra, Definition 15.91.4 and text following.
Proof. Consider the obvious morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathbf{N}), \mathcal{A}) \to (\mathop{\mathit{Sh}}\nolimits (pt), R)$ of ringed topoi and let us consider the adjoint functors $Lf^*$ and $Rf_*$. The first restricts to a functor
\[ F : D_{comp}(R, I) \longrightarrow \mathit{QC}(\mathcal{A}) \]
which sends an object $K$ of $D_{comp}(R, I)$ represented by a K-flat complex $K^\bullet $ to the object $(K^\bullet \otimes _ R R/I^ n)$ of $\mathit{QC}(\mathcal{A})$. The second restricts to a functor
\[ G : \mathit{QC}(\mathcal{A}) \longrightarrow D_{comp}(R, I) \]
which sends an object $(M_ n^\bullet )$ of $\mathit{QC}(\mathcal{A})$ to $R\mathop{\mathrm{lim}}\nolimits M_ n^\bullet $. The output is derived complete for example by More on Algebra, Lemma 15.91.14. Also, it follows from More on Algebra, Proposition 15.94.2 that $G \circ F = \text{id}$. Thus to see that $F$ and $G$ are quasi-inverse equivalences it suffices to see that the kernel of $G$ is zero (see Derived Categories, Lemma 13.7.2). However, it does not appear easy to show this directly!
In this paragraph we will show that $\mathit{QC}(\mathcal{A})$ and $\mathit{QC}(\mathcal{B})$ are equivalent. Write $\mathcal{B} = (B_ n)$ where $B_ n$ is the Koszul complex viewed as a cochain complex in degrees $-r, -r + 1, \ldots , 0$. By Divided Power Algebra, Remark 23.12.2 (but with chain complexes turned into cochain complexes) we can find $1 < n_1 < n_2 < \ldots $ and maps of differential graded $R$-algebras $B_{n_ i} \to E_ i \to R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})$ and $E_ i \to B_{n_{i - 1}}$ such that
\[ \xymatrix{ B_{n_1} \ar[d] & B_{n_2} \ar[d] \ar[l] & B_{n_3} \ar[d] \ar[l] & \ldots \ar[l] \\ E_1 \ar[d] & E_2 \ar[l] \ar[d] & E_3 \ar[l] \ar[d] & \ldots \ar[l] \\ B_1 & B_{n_1} \ar[l] & B_{n_2} \ar[l] & \ldots \ar[l] } \]
is a commutative diagram of differential graded $R$-algebras and such that $E_ i \to R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})$ is a quasi-isomorphism. We conclude
there is an equivalence between $\mathit{QC}(\mathcal{B})$ and $\mathit{QC}((E_ i))$,
there is an equivalence between $\mathit{QC}((E_ i))$ and $\mathit{QC}((R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})))$,
there is an equivalence between $\mathit{QC}((R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})))$ and $\mathit{QC}(\mathcal{A})$.
Namely, for (1) we can apply Lemma 24.35.1 to the diagram above which shows that $(E_ i)$ and $(B_ n)$ are pro-isomorphic. For (2) we can apply Lemma 24.34.2 to the inverse system of quasi-isomorphisms $E_ i \to R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})$. For (3) we can apply Lemma 24.35.1 and the elementary fact that the inverse systems $(R/I^ n)$ and $(R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})$ are pro-isomorphic.
Exactly as in the first paragraph of the proof we can define adjoint functors1
\[ F' : D_{comp}(R, I) \longrightarrow \mathit{QC}(\mathcal{B}) \quad \text{and}\quad G' : \mathit{QC}(\mathcal{B}) \longrightarrow D_{comp}(R, I). \]
The first sends an object $K$ of $D_{comp}(R, I)$ represented by a K-flat complex $K^\bullet $ to the object $(K^\bullet \otimes _ R B_ n)$ of $\mathit{QC}(\mathcal{B})$. The second sends an object $(M_ n)$ of $\mathit{QC}(\mathcal{B})$ to $R\mathop{\mathrm{lim}}\nolimits M_ n$. Arguing as above it suffices to show that the kernel of $G'$ is zero. So let $\mathcal{M} = (M_ n)$ be a good sheaf of differential graded modules over $\mathcal{B}$ which represents an object of $\mathit{QC}(\mathcal{B})$ in the kernel of $G'$. Then
\[ 0 = R\mathop{\mathrm{lim}}\nolimits M_ n \Rightarrow 0 = (R\mathop{\mathrm{lim}}\nolimits M_ n) \otimes _ R^\mathbf {L} B_ t = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _ R^\mathbf {L} B_ t) \]
By Lemma 24.35.3 we have $R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _ R^\mathbf {L} B_ t) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{B_ n}^\mathbf {L} B_ t)$. Since $(M_ n)$ is an object of $\mathit{QC}(\mathcal{B})$ we see that the inverse system $M_ n \otimes _{B_ n}^\mathbf {L} B_ t$ is eventually constant with value $M_ t$. Hence $M_ t = 0$ as desired. $\square$
Remark 24.35.5. Let $R$ be a ring and let $f_1, \ldots , f_ r \in R$ be a sequence of elements generating an ideal $I$. Let $K_ n$ be the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ viewed as a differential graded $R$-algebra. We say $f_1, \ldots , f_ r$ is a weakly proregular sequence if for all $n$ there is an $m > n$ such that $K_ m \to K_ n$ induces the zero map on cohomology except in degree $0$. If so, then the arguments in the proof of Proposition 24.35.4 continue to work even when $R$ is not Noetherian. In particular we see that $\mathit{QC}(\{ R/I^ n\} )$ is equivalent as an $R$-linear triangulated category to the category $D_{comp}(R, I)$ of derived complete objects, provided $I$ can be generated by a weakly proregular sequence. If the need arises, we will precisely state and prove this here.
[1] It can be shown that these functors are, via the equivalences above, compatible with $F$ and $G$ defined before.
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