## 24.17 Sheaves of differential graded bimodules and tensor-hom adjunction

This section is the analogue of part of Differential Graded Algebra, Section 22.12.

Definition 24.17.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A *differential graded $(\mathcal{A}, \mathcal{B})$-bimodule* is given by a complex $\mathcal{M}^\bullet $ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

\[ \mathcal{M}^ n \times \mathcal{B}^ m \to \mathcal{M}^{n + m},\quad (x, b) \longmapsto xb \]

and

\[ \mathcal{A}^ n \times \mathcal{M}^ m \to \mathcal{M}^{n + m},\quad (a, x) \longmapsto ax \]

called the multiplication maps with the following properties

multiplication satisfies $a(a'x) = (aa')x$ and $(xb)b' = x(bb')$,

$(ax)b = a(xb)$,

$\text{d}(ax) = \text{d}(a) x + (-1)^{\deg (a)}a \text{d}(x)$ and $\text{d}(xb) = \text{d}(x) b + (-1)^{\deg (x)}x \text{d}(b)$,

the identity section $1$ of $\mathcal{A}^0$ acts as the identity by multiplication, and

the identity section $1$ of $\mathcal{B}^0$ acts as the identity by multiplication.

We often denote such a structure $\mathcal{M}$ and sometimes we write ${}_\mathcal {A}\mathcal{M}_\mathcal {B}$. A *homomorphism of differential graded $(\mathcal{A}, \mathcal{B})$-bimodules* $f : \mathcal{M} \to \mathcal{N}$ is a map of complexes $f : \mathcal{M}^\bullet \to \mathcal{N}^\bullet $ of $\mathcal{O}$-modules compatible with the multiplication maps.

Given a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule $\mathcal{M}$ and an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we use the notation

\[ \mathcal{M}(U) = \Gamma (U, \mathcal{M}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{M}^ n(U) \]

This is a differential graded $(\mathcal{A}(U), \mathcal{B}(U))$-bimodule.

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

Lemma 24.17.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{N}$ be a right differential graded $\mathcal{B}$-module. There is a $1$-to-$1$ correspondence between $(\mathcal{A}, \mathcal{B})$-bimodule structures on $\mathcal{N}$ compatible with the given differential graded $\mathcal{B}$-module structure and homomorphisms

\[ \mathcal{A} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {B}(\mathcal{N}, \mathcal{N}) \]

of differential graded $\mathcal{O}$-algebras.

**Proof.**
Omitted.
$\square$

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right differential graded $\mathcal{A}$-module and let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. In this case the differential graded tensor product defined in Section 24.15

\[ \mathcal{M} \otimes _\mathcal {A} \mathcal{N} \]

is a right differential graded $\mathcal{B}$-module with multiplication maps as in Section 24.8. This construction defines a functor and a functor of graded categories

\[ \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{B}, \text{d}) \quad \text{and}\quad \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \]

by sending homomorphisms of degree $n$ from $\mathcal{M} \to \mathcal{M}'$ to the induced map of degree $n$ from $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ to $\mathcal{M}' \otimes _\mathcal {A} \mathcal{N}$.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right differential graded $\mathcal{B}$-module. In this case the differential graded internal hom defined in Section 24.16

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L}) \]

is a right differential graded $\mathcal{A}$-module where the right graded $\mathcal{A}$-module structure is the one defined in Section 24.8. Another way to define the multiplication is the use the composition

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L}) \otimes _\mathcal {O} \mathcal{A} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L}) \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{N}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L}) \]

where the first arrow comes from Lemma 24.17.2 and the second arrow is the composition of Section 24.16. Since these arrows are both compatible with differentials, we conclude that we indeed obtain a differential graded $\mathcal{A}$-module. This construction defines a functor and a functor of differential graded categories

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, -) : \textit{Mod}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{A}) \quad \text{and}\quad \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, -) : \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \]

by sending homomorphisms of degree $n$ from $\mathcal{L} \to \mathcal{L}'$ to the induced map of degree $n$ from $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L})$ to $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L}')$.

Lemma 24.17.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a right differential graded $\mathcal{A}$-module. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Let $\mathcal{L}$ be a right differential graded $\mathcal{B}$-module. With conventions as above we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{B}, \text{d})}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{L}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}( \mathcal{M}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L})) \]

and

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {A}^{dg}( \mathcal{M}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, \mathcal{L})) \]

functorially in $\mathcal{M}$, $\mathcal{N}$, $\mathcal{L}$.

**Proof.**
Omitted. Hint: On the graded level we have seen this is true in Lemma 24.8.2. Thus it suffices to check the isomorphisms are compatible with differentials which can be done by a computation on the level of local sections.
$\square$

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. As a special case of the above, suppose we are given a homomorphism $\varphi : \mathcal{A} \to \mathcal{B}$ of differential graded $\mathcal{O}$-algebras. Then we obtain a functor and a functor of differential graded categories

\[ \otimes _{\mathcal{A}, \varphi } \mathcal{B} : \textit{Mod}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{B}, \text{d}) \quad \text{and}\quad \otimes _{\mathcal{A}, \varphi } \mathcal{B} : \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \]

On the other hand, we have the restriction functors

\[ res_\varphi : \textit{Mod}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}(\mathcal{A}, \text{d}) \quad \text{and}\quad res_\varphi : \textit{Mod}^{dg}(\mathcal{B}, \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \]

We can use the lemma above to show these functors are adjoint to each other (as usual with restriction and base change). Namely, let us write ${}_\mathcal {A}\mathcal{B}_\mathcal {B}$ for $\mathcal{B}$ viewed as a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Then for any right differential graded $\mathcal{B}$-module $\mathcal{L}$ we have

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}({}_\mathcal {A}\mathcal{B}_\mathcal {B}, \mathcal{L}) = res_\varphi (\mathcal{L}) \]

as right differential graded $\mathcal{A}$-modules. Thus Lemma 24.8.2 tells us that we have a functorial isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{B}, \text{d})}( \mathcal{M} \otimes _{\mathcal{A}, \varphi } \mathcal{B}, \mathcal{L}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}( \mathcal{M}, res_\varphi (\mathcal{L})) \]

We usually drop the dependence on $\varphi $ in this formula if it is clear from context. In the same manner we obtain the equality

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {B}( \mathcal{M} \otimes _\mathcal {A} \mathcal{B}, \mathcal{L}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {A}^{dg}(\mathcal{M}, \mathcal{L}) \]

of graded $\mathcal{O}$-modules.

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