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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

\[ \text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}} : (\mathcal{M} \otimes _\mathcal {A} \mathcal{N})^ n \longrightarrow (\mathcal{M} \otimes _\mathcal {A} \mathcal{N})^{n + 1} \]

defined by the rule that

\[ \text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}}(x \otimes y) = \text{d}_\mathcal {M}(x) \otimes y + (-1)^{\deg (x)}x \otimes \text{d}_\mathcal {N}(y) \]

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

\begin{align*} & \text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}}( xa \otimes y - x \otimes ay) \\ & = \text{d}_\mathcal {M}(xa) \otimes y + (-1)^{\deg (x) + \deg (a)} xa \otimes \text{d}_\mathcal {N}(y) -\text{d}_\mathcal {M}(x) \otimes ay - (-1)^{\deg (x)} x \otimes \text{d}_\mathcal {N}(ay) \\ & = \text{d}_\mathcal {M}(x)a \otimes y + (-1)^{\deg (x)}x\text{d}(a) \otimes y + (-1)^{\deg (x) + \deg (a)} xa \otimes \text{d}_\mathcal {N}(y) \\ & -\text{d}_\mathcal {M}(x) \otimes ay - (-1)^{\deg (x)} x \otimes \text{d}(a)y - (-1)^{\deg (x) + \deg (a)} x\otimes a\text{d}_\mathcal {N}(y) \end{align*}

then we observe that the elements

\[ \text{d}_\mathcal {M}(x)a \otimes y - \text{d}_\mathcal {M}(x) \otimes ay,\quad x\text{d}(a) \otimes y - x \otimes \text{d}(a)y,\quad \text{and}\quad xa \otimes \text{d}_\mathcal {N}(y) - x\otimes a\text{d}_\mathcal {N}(y) \]

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

\[ - \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}(\mathcal{A}, \text{d}) \longrightarrow \text{Comp}(\mathcal{O}) \]

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

\[ - \otimes _\mathcal {A} \mathcal{N} : \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \longrightarrow \text{Comp}^{dg}(\mathcal{O}) \]

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

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{M}') \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O})}( \mathcal{M} \otimes _\mathcal {A} \mathcal{N}, \mathcal{M}' \otimes _\mathcal {A} \mathcal{N}) \]

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

\[ (\text{d}_{\mathcal{M}'} \circ f - (-1)^ n f \circ \text{d}_\mathcal {M}) \otimes \text{id}_\mathcal {N} \]

is equal to

\[ \text{d}_{\mathcal{M}' \otimes _\mathcal {A} \mathcal{N}} \circ (f \otimes \text{id}_\mathcal {N}) - (-1)^ n (f \otimes \text{id}_\mathcal {N}) \circ \text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}} \]

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

\[ (\text{d}_{\mathcal{M}'}(f(x)) - (-1)^ n f(\text{d}_\mathcal {M}(x))) \otimes y \]

and for the second we obtain

\begin{align*} & \text{d}_{\mathcal{M}' \otimes _\mathcal {A} \mathcal{N}}(f(x) \otimes y) - (-1)^ n (f \otimes \text{id}_\mathcal {N})( \text{d}_{\mathcal{M} \otimes _\mathcal {A} \mathcal{N}}(x \otimes y) \\ & = \text{d}_{\mathcal{M}'}(f(x)) \otimes y + (-1)^{\deg (x) + n}f(x) \otimes \text{d}_\mathcal {N}(y) \\ & -(-1)^ n f(\text{d}_\mathcal {M}(x)) \otimes y -(-1)^ n (-1)^{\deg (x)}f(x) \otimes \text{d}_\mathcal {N}(y) \end{align*}

which is indeeed the same local section.

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