## 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:

1. $\mathcal{C}$ is the category $\mathbf{N}$ with a unique morphism $i \to j$ if and only if $i \leq j$,

2. $\mathcal{O}$ 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.

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.

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

$R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{A_ n}^\mathbf {L} N_ n) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{A_ n}^\mathbf {L} N'_ n)$

in $D(R)$.

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

$R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _ R^\mathbf {L} K_ t) = R\mathop{\mathrm{lim}}\nolimits (M_ n \otimes _{K_ n}^\mathbf {L} K_ t)$

in $D(R)$.

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:

1. 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 $\mathit{QC}(\mathcal{A})$.

2. 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 $\mathit{QC}(\mathcal{B})$.

3. 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

1. there is an equivalence between $\mathit{QC}(\mathcal{B})$ and $\mathit{QC}((E_ i))$,

2. there is an equivalence between $\mathit{QC}((E_ i))$ and $\mathit{QC}((R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})))$,

3. 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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