Section 22.28 (0FQG): Bimodules—The Stacks project

22.28 Bimodules

We continue the discussion started in Section 22.12.

Definition 22.28.1. Bimodules. Let $R$ be a ring.

Let $A$ and $B$ be $R$ -algebras. An $(A, B)$ -bimodule is an $R$ -module $M$ equippend with $R$ -bilinear maps

$A \times M \to M, (a, x) \mapsto ax \quad \text{and}\quad M \times B \to M, (x, b) \mapsto xb$

such that the following hold
1. $a'(ax) = (a'a)x$ and $(xb)b' = x(bb')$ ,
2. $a(xb) = (ax)b$ , and
3. $1 x = x = x 1$ .
Let $A$ and $B$ be $\mathbf{Z}$ -graded $R$ -algebras. A graded $(A, B)$ -bimodule is an $(A, B)$ -bimodule $M$ which has a grading $M = \bigoplus M^ n$ such that $A^ n M^ m \subset M^{n + m}$ and $M^ n B^ m \subset M^{n + m}$ .
Let $A$ and $B$ be differential graded $R$ -algebras. A differential graded $(A, B)$ -bimodule is a graded $(A, B)$ -bimodule which comes equipped with a differential $\text{d} : M \to M$ homogeneous of degree $1$ such that $\text{d}(ax) = \text{d}(a)x + (-1)^{\deg (a)}a\text{d}(x)$ and $\text{d}(xb) = \text{d}(x)b + (-1)^{\deg (x)}x\text{d}(b)$ for homogeneous elements $a \in A$ , $x \in M$ , $b \in B$ .

Observe that a differential graded $(A, B)$ -bimodule $M$ is the same thing as a right differential graded $B$ -module which is also a left differential graded $A$ -module such that the grading and differentials agree and such that the $A$ -module structure commutes with the $B$ -module structure. Here is a precise statement.

Lemma 22.28.2. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$ . Let $M$ be a right differential graded $B$ -module. There is a $1$ -to- $1$ correspondence between $(A, B)$ -bimodule structures on $M$ compatible with the given differential graded $B$ -module structure and homomorphisms

$A \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, M)$

of differential graded $R$ -algebras.

Proof. Let $\mu : A \times M \to M$ define a left differential graded $A$ -module structure on the underlying complex of $R$ -modules $M^\bullet$ of $M$ . By Lemma 22.13.1 the structure $\mu$ corresponds to a map $\gamma : A \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , M^\bullet )$ of differential graded $R$ -algebras. The assertion of the lemma is simply that $\mu$ commutes with the $B$ -action, if and only if $\gamma$ ends up inside

$\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(M, M) \subset \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , M^\bullet )$

We omit the detailed calculation. $\square$

Let $M$ be a differential graded $(A, B)$ -bimodule. Recall from Section 22.11 that the left differential graded $A$ -module structure corresponds to a right differential graded $A^{opp}$ -module structure. Since the $A$ and $B$ module structures commute this gives $M$ the structure of a differential graded $A^{opp} \otimes _ R B$ -module:

$x \cdot (a \otimes b) = (-1)^{\deg (a)\deg (x)} axb$

Conversely, if we have a differential graded $A^{opp} \otimes _ R B$ -module $M$ , then we can use the formula above to get a differential graded $(A, B)$ -bimodule.

Lemma 22.28.3. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$ . The construction above defines an equivalence of categories

$\begin{matrix} \text{differential graded} \\ (A, B)\text{-bimodules} \end{matrix} \longleftrightarrow \begin{matrix} \text{right differential graded } \\ A^{opp} \otimes _ R B\text{-modules} \end{matrix}$

Proof. Immediate from discussion the above. $\square$

Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$ -algebras. Let $P$ be a differential graded $(A, B)$ -bimodule. We say $P$ has property (P) if it there exists a filtration

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

by differential graded $(A, B)$ -bimodules such that

$P = \bigcup F_ pP$ ,
the inclusions $F_ iP \to F_{i + 1}P$ are split as graded $(A, B)$ -bimodule maps,
the quotients $F_{i + 1}P/F_ iP$ are isomorphic as differential graded $(A, B)$ -bimodules to a direct sum of $(A \otimes _ R B)[k]$ .

Lemma 22.28.4. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$ -algebras. Let $M$ be a differential graded $(A, B)$ -bimodule. There exists a homomorphism $P \to M$ of differential graded $(A, B)$ -bimodules which is a quasi-isomorphism such that $P$ has property (P) as defined above.

Proof. Immediate from Lemmas 22.28.3 and 22.20.4. $\square$

Lemma 22.28.5. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$ -algebras. Let $P$ be a differential graded $(A, B)$ -bimodule having property (P) with corresponding filtration $F_\bullet$ , then we obtain a short exact sequence

$0 \to \bigoplus \nolimits F_ iP \to \bigoplus \nolimits F_ iP \to P \to 0$

of differential graded $(A, B)$ -bimodules which is split as a sequence of graded $(A, B)$ -bimodules.

Proof. Immediate from Lemmas 22.28.3 and 22.20.1. $\square$

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22.28 Bimodules

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