## 24.15 Tensor product for sheaves of differential graded modules

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

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf 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 left differential graded $\mathcal{A}$-module. In this situation we define the *tensor product* $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ as follows. As a graded $\mathcal{O}$-module it is given by the construction in Section 24.6. It comes endowed with a differential

defined by the rule that

for homogeneous local sections $x$ and $y$ of $\mathcal{M}$ and $\mathcal{N}$. To see that this is well defined we have to show that $\text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}}$ annihilates elements of the form $xa \otimes y - x \otimes ay$ for homogeneous local sections $x$, $a$, $y$ of $\mathcal{M}$, $\mathcal{A}$, $\mathcal{N}$. We compute

then we observe that the elements

map to zero in $\mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ and we conclude. We omit the verification that $\text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}} \circ \text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}} = 0$.

If we fix the left differential graded $\mathcal{A}$-module $\mathcal{N}$ we obtain a functor

where on the right hand side we have the category of complexes of $\mathcal{O}$-modules. This can be upgraded to a functor of differential graded categories

On underlying graded objects, we send a homomorphism $f : \mathcal{M} \to \mathcal{M}'$ of degree $n$ to the degree $n$ map $f \otimes \text{id}_\mathcal {N} : \mathcal{M} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M}' \otimes _\mathcal {A} \mathcal{N}$, because this is what we did in Section 24.6. To show that this works, we have to verify that the map

is compatible with differentials. To see this for $f$ as above we have to show that

is equal to

Let us compute the effect of these operators on a local section of the form $x \otimes y$ with $x$ and $y$ homogeneous local sections of $\mathcal{M}$ and $\mathcal{N}$. For the first we obtain

and for the second we obtain

which is indeeed the same local section.

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