## 22.29 Resolutions of differential graded algebras

Let $R$ be a ring. Under our assumptions the free $R$-algebra $R\langle S \rangle $ on a set $S$ is the algebra with $R$-basis the expressions

\[ s_1 s_2 \ldots s_ n \]

where $n \geq 0$ and $s_1, \ldots , s_ n \in S$ is a sequence of elements of $S$. Multiplication is given by concatenation

\[ (s_1 s_2 \ldots s_ n) \cdot (s'_1 s'_2 \ldots s'_ m) = s_1 \ldots s_ n s'_1 \ldots s'_ m \]

This algebra is characterized by the property that the map

\[ \mathop{Mor}\nolimits _{R\text{-alg}}(R\langle S \rangle , A) \to \text{Map}(S, A),\quad \varphi \longmapsto (s \mapsto \varphi (s)) \]

is a bijection for every $R$-algebra $A$.

In the category of graded $R$-algebras our set $S$ should come with a grading, which we think of as a map $\deg : S \to \mathbf{Z}$. Then $R\langle S\rangle $ has a grading such that the monomials have degree

\[ \deg (s_1 s_2 \ldots s_ n) = \deg (s_1) + \ldots + \deg (s_ n) \]

In this setting we have

\[ \mathop{Mor}\nolimits _{\text{graded }R\text{-alg}}(R\langle S \rangle , A) \to \text{Map}_{\text{graded sets}}(S, A),\quad \varphi \longmapsto (s \mapsto \varphi (s)) \]

is a bijection for every graded $R$-algebra $A$.

If $A$ is a graded $R$-algebra and $S$ is a graded set, then we can similarly form $A\langle S \rangle $. Elements of $A\langle S \rangle $ are sums of elements of the form

\[ a_0 s_1 a_1 s_2 \ldots a_{n - 1} s_ n a_ n \]

with $a_ i \in A$ modulo the relations that these expressions are $R$-multilinear in $(a_0, \ldots , a_ n)$. Thus for every sequence $s_1, \ldots , s_ n$ of elements of $S$ there is an inclusion

\[ A \otimes _ R \ldots \otimes _ R A \subset A\langle S \rangle \]

and the algebra is the direct sum of these. With this definition the reader shows that the map

\[ \mathop{Mor}\nolimits _{\text{graded }R\text{-alg}}(A\langle S \rangle , B) \to \mathop{Mor}\nolimits _{\text{graded }R\text{-alg}}(A, B) \times \text{Map}_{\text{graded sets}}(S, B), \]

sending $\varphi $ to $(\varphi |_ A, (s \mapsto \varphi (s)))$ is a bijection for every graded $R$-algebra $A$. We observe that if $A$ was a free graded $R$-algebra, then so is $A\langle S \rangle $.

Suppose that $A$ is a differential graded $R$-algebra and that $S$ is a graded set. Suppose moreover for every $s \in S$ we are given a homogeneous element $f_ s \in A$ with $\deg (f_ s) = \deg (s) + 1$ and $\text{d}f_ s = 0$. Then there exists a unique structure of differential graded algebra on $A\langle S \rangle $ with $\text{d}(s) = f_ s$. For example, given $a, b, c \in A$ and $s, t \in S$ we would define

\begin{align*} \text{d}(asbtc) & = \text{d}(a)sbtc + (-1)^{\deg (a)}a f_ s b t c + (-1)^{\deg (a) + \deg (s)} as\text{d}(b)tc \\ & + (-1)^{\deg (a) + \deg (s) + \deg (b)} asb f_ t c + (-1)^{\deg (a) + \deg (s) + \deg (b) + \deg (t)} asbt\text{d}(c) \end{align*}

We omit the details.

Lemma 22.29.1. Let $R$ be a ring. Let $(B, \text{d})$ be a differential graded $R$-algebra. There exists a quasi-isomorphism $(A, \text{d}) \to (B, \text{d})$ of differential graded $R$-algebras with the following properties

$A$ is K-flat as a complex of $R$-modules,

$A$ is a free graded $R$-algebra.

**Proof.**
First we claim we can find $(A_0, \text{d}) \to (B, \text{d})$ having (1) and (2) inducing a surjection on cohomology. Namely, take a graded set $S$ and for each $s \in S$ a homogeneous element $b_ s \in \mathop{\mathrm{Ker}}(d : B \to B)$ of degree $\deg (s)$ such that the classes $\overline{b}_ s$ in $H^*(B)$ generate $H^*(B)$ as an $R$-module. Then we can set $A_0 = R\langle S \rangle $ with zero differential and $A_0 \to B$ given by mapping $s$ to $b_ s$.

Given $A_0 \to B$ inducing a surjection on cohomology we construct a sequence

\[ A_0 \to A_1 \to A_2 \to \ldots B \]

by induction. Given $A_ n \to B$ we set $S_ n$ be a graded set and for each $s \in S_ n$ we let $a_ s \in \mathop{\mathrm{Ker}}(A_ n \to A_ n)$ be a homogeneous element of degree $\deg (s) + 1$ mapping to a class $\overline{a}_ s$ in $H^*(A_ n)$ which maps to zero in $H^*(B)$. We choose $S_ n$ large enough so that the elements $\overline{a}_ s$ generate $\mathop{\mathrm{Ker}}(H^*(A_ n) \to H^*(B))$ as an $R$-module. Then we set

\[ A_{n + 1} = A_ n\langle S_ n \rangle \]

with differential given by $\text{d}(s) = a_ s$ see discussion above. Then each $(A_ n, \text{d})$ satisfies (1) and (2), we omit the details. The map $H^*(A_ n) \to H^*(B)$ is surjective as this was true for $n = 0$.

It is clear that $A = \mathop{\mathrm{colim}}\nolimits A_ n$ is a free graded $R$-algebra. It is K-flat by More on Algebra, Lemma 15.57.10. The map $H^*(A) \to H^*(B)$ is an isomorphism as it is surjective and injective: every element of $H^*(A)$ comes from an element of $H^*(A_ n)$ for some $n$ and if it dies in $H^*(B)$, then it dies in $H^*(A_{n + 1})$ hence in $H^*(A)$.
$\square$

As an application we prove the “correct” version of Lemma 22.25.2.

Lemma 22.29.2. Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. Assume $A \otimes _ R C$ represents $A \otimes ^\mathbf {L}_ R C$ in $D(R)$. Let $N$ be a differential graded $A^{opp} \otimes _ R B$-module. Let $N'$ be a differential graded $B^{opp} \otimes _ R C$-module. Then the composition

\[ \xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes _ A^\mathbf {L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes _ B^\mathbf {L} N'} & & D(C, \text{d}) } \]

is isomorphic to $- \otimes _ A^\mathbf {L} N''$ for some differential graded $A^{opp} \otimes _ R C$-module $N''$.

**Proof.**
Using Lemma 22.29.1 we choose a quasi-isomorphism $(B', \text{d}) \to (B, \text{d})$ with $B'$ K-flat as a complex of $R$-modules. By Lemma 22.28.1 the functor $-\otimes ^\mathbf {L}_{B'} B : D(B', \text{d}) \to D(B, \text{d})$ is an equivalence with quasi-inverse given by restriction. Note that restriction is canonically isomorphic to the functor $- \otimes ^\mathbf {L}_ B B : D(B, \text{d}) \to D(B', \text{d})$ where $B$ is viewed as a $(B, B')$-bimodule. Thus it suffices to prove the lemma for the compositions

\[ D(A) \to D(B) \to D(B'),\quad D(B') \to D(B) \to D(C),\quad D(A) \to D(B') \to D(C). \]

The first one is Lemma 22.25.3 because $B'$ is K-flat as a complex of $R$-modules. The second one is true because $B \otimes _ B^\mathbf {L} N' = N' = B \otimes _ B N'$ and hence Lemma 22.25.1 applies. Thus we reduce to the case where $B$ is K-flat as a complex of $R$-modules.

Assume $B$ is K-flat as a complex of $R$-modules. It suffices to show that (22.25.1.1) is an isomorphism, see Lemma 22.25.2. Choose a quasi-isomorphism $L \to A$ where $L$ is a differential graded $R$-module which has property (P). Then it is clear that $P = L \otimes _ R B$ has property (P) as a differential graded $B$-module. Hence we have to show that $P \to A \otimes _ R B$ induces a quasi-isomorphism

\[ P \otimes _ B (B \otimes _ R C) \longrightarrow (A \otimes _ R B) \otimes _ B (B \otimes _ R C) \]

We can rewrite this as

\[ P \otimes _ R B \otimes _ R C \longrightarrow A \otimes _ R B \otimes _ R C \]

Since $B$ is K-flat as a complex of $R$-modules, it follows from More on Algebra, Lemma 15.57.4 that it is enough to show that

\[ P \otimes _ R C \to A \otimes _ R C \]

is a quasi-isomorphism, which is exactly our assumption.
$\square$

The following lemma does not really belong in this section, but there does not seem to be a good natural spot for it.

Lemma 22.29.3. Let $(A, \text{d})$ be a differential graded algebra with $H^ i(A)$ countable for each $i$. Let $M$ be an object of $D(A, \text{d})$. Then the following are equivalent

$M = \text{hocolim} E_ n$ with $E_ n$ compact in $D(A, \text{d})$, and

$H^ i(M)$ is countable for each $i$.

**Proof.**
Assume (1) holds. Then we have $H^ i(M) = \mathop{\mathrm{colim}}\nolimits H^ i(E_ n)$ by Derived Categories, Lemma 13.31.8. Thus it suffices to prove that $H^ i(E_ n)$ is countable for each $n$. By Proposition 22.27.4 we see that $E_ n$ is isomorphic in $D(A, \text{d})$ to a direct summand of a differential graded module $P$ which has a finite filtration $F_\bullet $ by differential graded submodules such that $F_ jP/F_{j - 1}P$ are finite direct sums of shifts of $A$. By assumption the groups $H^ i(F_ jP/F_{j - 1}P)$ are countable. Arguing by induction on the length of the filtration and using the long exact cohomology sequence we conclude that (2) is true. The interesting implication is the other one.

We claim there is a countable differential graded subalgebra $A' \subset A$ such that the inclusion map $A' \to A$ defines an isomorphism on cohomology. To construct $A'$ we choose countable differential graded subalgebras

\[ A_1 \subset A_2 \subset A_3 \subset \ldots \]

such that (a) $H^ i(A_1) \to H^ i(A)$ is surjective, and (b) for $n > 1$ the kernel of the map $H^ i(A_{n - 1}) \to H^ i(A_ n)$ is the same as the kernel of the map $H^ i(A_{n - 1}) \to H^ i(A)$. To construct $A_1$ take any countable collection of cochains $S \subset A$ generating the cohomology of $A$ (as a ring or as a graded abelian group) and let $A_1$ be the differential graded subalgebra of $A$ generated by $S$. To construct $A_ n$ given $A_{n - 1}$ for each cochain $a \in A_{n - 1}^ i$ which maps to zero in $H^ i(A)$ choose $s_ a \in A^{i - 1}$ with $\text{d}(s_ a) = a$ and let $A_ n$ be the differential graded subalgebra of $A$ generated by $A_{n - 1}$ and the elements $s_ a$. Finally, take $A' = \bigcup A_ n$.

By Lemma 22.28.1 the restriction map $D(A, \text{d}) \to D(A', \text{d})$, $M \mapsto M_{A'}$ is an equivalence. Since the cohomology groups of $M$ and $M_{A'}$ are the same, we see that it suffices to prove the implication (2) $\Rightarrow $ (1) for $(A', \text{d})$.

Assume $A$ is countable. By the exact same type of argument as given above we see that for $M$ in $D(A, \text{d})$ the following are equivalent: $H^ i(M)$ is countable for each $i$ and $M$ can be represented by a countable differential graded module. Hence in order to prove the implication (2) $\Rightarrow $ (1) we reduce to the situation described in the next paragraph.

Assume $A$ is countable and that $M$ is a countable differential graded module over $A$. We claim there exists a homomorphism $P \to M$ of differential graded $A$-modules such that

$P \to M$ is a quasi-isomorphism,

$P$ has property (P), and

$P$ is countable.

Looking at the proof of the construction of P-resolutions in Lemma 22.13.4 we see that it suffices to show that we can prove Lemma 22.13.3 in the setting of countable differential graded modules. This is immediate from the proof.

Assume that $A$ is countable and that $M$ is a countable differential graded module with property (P). Choose a filtration

\[ 0 = F_{-1}P \subset F_0P \subset F_1P \subset \ldots \subset P \]

by differential graded submodules such that we have

$P = \bigcup F_ pP$,

$F_ iP \to F_{i + 1}P$ is an admissible monomorphism,

isomorphisms of differential graded modules $F_ iP/F_{i - 1}P \to \bigoplus _{j \in J_ i} A[k_ j]$ for some sets $J_ i$ and integers $k_ j$.

Of course $J_ i$ is countable for each $i$. For each $i$ and $j \in J_ i$ choose $x_{i, j} \in F_ iP$ of degree $k_ j$ whose image in $F_ iP/F_{i - 1}P$ generates the summand corresponding to $j$.

Claim: Given $n$ and finite subsets $S_ i \subset J_ i$, $i = 1, \ldots , n$ there exist finite subsets $S_ i \subset T_ i \subset J_ i$, $i = 1, \ldots , n$ such that $P' = \bigoplus _{i \leq n} \bigoplus _{j \in T_ i} Ax_{i, j}$ is a differential graded submodule of $P$. This was shown in the proof of Lemma 22.27.3 but it is also easily shown directly: the elements $x_{i, j}$ freely generate $P$ as a right $A$-module. The structure of $P$ shows that

\[ \text{d}(x_{i, j}) = \sum \nolimits _{i' < i} x_{i', j'}a_{i', j'} \]

where of course the sum is finite. Thus given $S_0, \ldots , S_ n$ we can first choose $S_0 \subset S'_0, \ldots , S_{n - 1} \subset S'_{n - 1}$ with $\text{d}(x_{n, j}) \in \bigoplus _{i' < n, j' \in S'_{i'}} x_{i', j'}A$ for all $j \in S_ n$. Then by induction on $n$ we can choose $S'_0 \subset T_0, \ldots , S'_{n - 1} \subset T_{n - 1}$ to make sure that $\bigoplus _{i' < n, j' \in T_{i'}} x_{i', j'}A$ is a differential graded $A$-submodule. Setting $T_ n = S_ n$ we find that $P' = \bigoplus _{i \leq n, j \in T_ i} x_{i, j}A$ is as desired.

From the claim it is clear that $P = \bigcup P'_ n$ is a countable rising union of $P'_ n$ as above. By construction each $P'_ n$ is a differential graded module with property (P) such that the filtration is finite and the succesive quotients are finite direct sums of shifts of $A$. Hence $P'_ n$ defines a compact object of $D(A, \text{d})$, see for example Proposition 22.27.4. Since $P = \text{hocolim} P'_ n$ in $D(A, \text{d})$ by Lemma 22.16.2 the proof of the implication (2) $\Rightarrow $ (1) is complete.
$\square$

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