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.

  1. 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. $1 x = x = x 1$.

  2. 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 $M^ n B^ m \subset M^{n + m}$.

  3. 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

  1. $P = \bigcup F_ pP$,

  2. the inclusions $F_ iP \to F_{i + 1}P$ are split as graded $(A, B)$-bimodule maps,

  3. 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.

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.

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