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24.32 Miscellany

Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a sheaf of differential graded $\mathcal{O}$-algebras. Using the composition1

\[ \mathcal{A} \otimes _\mathcal {O}^\mathbf {L} \mathcal{A} \longrightarrow \mathcal{A} \otimes _\mathcal {O} \mathcal{A} \longrightarrow \mathcal{A} \]

and the relative cup product (see Cohomology on Sites, Remark 21.19.7 and Section 21.33) we obtain a multiplication2

\[ \mu : Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \longrightarrow Rf_*\mathcal{A} \]

in $D(\mathcal{O}')$. This multiplication is associative in the sense that the diagram

\[ \xymatrix{ Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]_-{\mu \otimes 1} \ar[d]_{1 \otimes \mu } & & Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[d]^\mu \\ Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]^-\mu & & Rf_*\mathcal{A} } \]

commutes in $D(\mathcal{O}')$; this follows from Cohomology on Sites, Lemma 21.33.2. In exactly the same way, given a right differential graded $\mathcal{A}$-module $\mathcal{M}$ we obtain a multiplication

\[ \mu _\mathcal {M} : Rf_*\mathcal{M} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \longrightarrow Rf_*\mathcal{M} \]

in $D(\mathcal{O}')$. This multiplication is compatible with $\mu $ above in the sense that the diagram

\[ \xymatrix{ Rf_*\mathcal{M} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]_-{\mu _\mathcal {M} \otimes 1} \ar[d]_{1 \otimes \mu } & & Rf_*\mathcal{M} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[d]^{\mu _\mathcal {M}} \\ Rf_*\mathcal{M} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]^-{\mu _\mathcal {M}} & & Rf_*\mathcal{M} } \]

commutes in $D(\mathcal{O}')$; again this follows from Cohomology on Sites, Lemma 21.33.2.

A particular example of the above is when one takes $f$ to be the morphism to the punctual topos $\mathop{\mathit{Sh}}\nolimits (pt)$. In that case $\mu $ is just the cup product map

\[ R\Gamma (\mathcal{C}, \mathcal{A}) \otimes _{\Gamma (\mathcal{C}, \mathcal{O})}^\mathbf {L} R\Gamma (\mathcal{C}, \mathcal{A}) \longrightarrow R\Gamma (\mathcal{C}, \mathcal{A}), \quad \eta \otimes \theta \mapsto \eta \cup \theta \]

and similarly $\mu _\mathcal {M}$ is the cup product map

\[ R\Gamma (\mathcal{C}, \mathcal{M}) \otimes _{\Gamma (\mathcal{C}, \mathcal{O})}^\mathbf {L} R\Gamma (\mathcal{C}, \mathcal{A}) \longrightarrow R\Gamma (\mathcal{C}, \mathcal{M}), \quad \eta \otimes \theta \mapsto \eta \cup \theta \]

In general, via the identifications

\[ R\Gamma (\mathcal{C}, \mathcal{A}) = R\Gamma (\mathcal{C}', Rf_*\mathcal{A}) \quad \text{and}\quad R\Gamma (\mathcal{C}, \mathcal{M}) = R\Gamma (\mathcal{C}', Rf_*\mathcal{M}) \]

of Cohomology on Sites, Remark 21.14.4 the map $\mu _\mathcal {M}$ induces the cup product on cohomology. To see this use Cohomology on Sites, Lemma 21.33.4 where the second morphism of topoi is the morphism from $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ to the punctual topos as above.

If $\mathcal{M}_1 \to \mathcal{M}_2$ is a homomorphism of right differential graded $\mathcal{A}$-modules, then the diagram

\[ \xymatrix{ Rf_*\mathcal{M}_1 \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]_-{\mu _{\mathcal{M}_1}} \ar[d] & & Rf_*\mathcal{M}_1 \ar[d] \\ Rf_*\mathcal{M}_2 \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]^-{\mu _{\mathcal{M}_2}} & & Rf_*\mathcal{M}_2 } \]

commutes in $D(\mathcal{O}')$; this follows from the fact that the relative cup product is functorial. Suppose we have a short exact sequence

\[ 0 \to \mathcal{M}_1 \xrightarrow {a} \mathcal{M}_2 \to \mathcal{M}_3 \to 0 \]

of right differential graded $\mathcal{A}$-modules. Then we claim that the diagram

\[ \xymatrix{ Rf_*\mathcal{M}_3 \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]_-{\mu _{\mathcal{M}_3}} \ar[d]_{Rf_*\delta \otimes \text{id}} & & Rf_*\mathcal{M}_3 \ar[d]^{Rf_*\delta } \\ Rf_*\mathcal{M}_1[1] \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]^-{\mu _{\mathcal{M}_1[1]}} & & Rf_*\mathcal{M}_1[1] } \]

commutes in $D(\mathcal{O}')$ where $\delta : \mathcal{M}_3 \to \mathcal{M}_1[1]$ is the morphism of $D(\mathcal{O})$ coming from the given short exact sequence (see Derived Categories, Section 13.12). This is clear if our sequence is split as a sequence of graded right $\mathcal{A}$-modules, because in this case $\delta $ can be represented by a map of right $\mathcal{A}$-modules and the discussion above applies. In general we argue using the cone on $a$ and the diagram

\[ \xymatrix{ \mathcal{M}_1 \ar[r]_ a \ar[d] & \mathcal{M}_2 \ar[r]_ i \ar[d] & C(a) \ar[r]_{-p} \ar[d]^ q & \mathcal{M}_1[1] \ar[d] \\ \mathcal{M}_1 \ar[r] & \mathcal{M}_2 \ar[r] & \mathcal{M}_3 \ar[r]^\delta & \mathcal{M}_1[1] } \]

where the right square is commutative in $D(\mathcal{O})$ by the definition of $\delta $ in Derived Categories, Lemma 13.12.1. Now the cone $C(a)$ has the structure of a right differential graded $\mathcal{A}$-module such that $i$, $p$, $q$ are homomorphisms of right differential graded $\mathcal{A}$-modules, see Definition 24.22.2. Hence by the above we know that the corresponding diagrams commute for the morphisms $q$ and $-p$. Since $q$ is an isomorphism in $D(\mathcal{O})$ we conclude the same is true for $\delta $ as desired.

In the situation above given a right differential graded $\mathcal{A}$-module $\mathcal{M}$ let

\[ \xi \in H^ n(\mathcal{C}, \mathcal{M}) \]

In other words, $\xi $ is a degree $n$ cohomology class in the cohomology of $\mathcal{M}$ viewed as a complex of $\mathcal{O}$-modules. By Lemma 24.29.9 we can construct maps

\[ x : \mathcal{A} \rightarrow \mathcal{M}'[n] \quad \text{and}\quad s : \mathcal{M} \to \mathcal{M}' \]

of right differential graded $\mathcal{A}$-modules where $s$ is a quasi-isomorphism and such that $\xi $ is the image of $1 \in H^0(\mathcal{C}, \mathcal{A})$ via the morphism $s[n]^{-1} \circ x$ in the derived category $D(\mathcal{A}, \text{d})$ and a fortiori in the derived category $D(\mathcal{O})$. It follows that the corresponding map

\[ \xi ' = (s[n])^{-1} \circ x : \mathcal{A} \longrightarrow \mathcal{M}[n] \]

in $D(\mathcal{O})$ is uniquely characterized by the following two properties

  1. $\xi '$ can be lifted to a morphism in $D(\mathcal{A}, \text{d})$, and

  2. $\xi = \xi '(1)$ in $H^0(\mathcal{C}, \mathcal{M}[n]) = H^ n(\mathcal{C}, \mathcal{M})$.

Using the compatibilities of $x$ and $s$ with the relative cup product discussed above it follows that for every3 morphism of ringed topoi $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ the derived pushforward

\[ Rf_*\xi ' : Rf_*\mathcal{A} \longrightarrow Rf_*\mathcal{M}[n] \]

of $\xi '$ is compatible with the maps $\mu $ and $\mu _{\mathcal{M}[n]}$ constructed above in the sense that the diagram

\[ \xymatrix{ Rf_*\mathcal{A} \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]_-\mu \ar[d]_{Rf_*\xi ' \otimes \text{id}} & & Rf_*\mathcal{A} \ar[d]^{Rf_*\xi '} \\ Rf_*\mathcal{M}[n] \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*\mathcal{A} \ar[rr]^-{\mu _{\mathcal{M}[n]}} & & Rf_*\mathcal{M}[n] } \]

commutes in $D(\mathcal{O}')$. Using this compatibility for the map to the punctual topos, we see in particular that

\[ \xymatrix{ R\Gamma (\mathcal{C}, \mathcal{A}) \otimes _{\Gamma (\mathcal{C}, \mathcal{O})}^\mathbf {L} R\Gamma (\mathcal{C}, \mathcal{A}) \ar[d]_{\xi ' \otimes \text{id}} \ar[r] & R\Gamma (\mathcal{C}, \mathcal{A}) \ar[d]^{\xi '} \\ R\Gamma (\mathcal{C}, \mathcal{M}[n]) \otimes _{\Gamma (\mathcal{C}, \mathcal{O})}^\mathbf {L} R\Gamma (\mathcal{C}, \mathcal{A}) \ar[r] & R\Gamma (\mathcal{C}, \mathcal{M}[n]) } \]

commutes. Combined with $\xi '(1) = \xi $ this implies that the induced map on cohomology

\[ \xi ' : R\Gamma (\mathcal{C}, \mathcal{A}) \to R\Gamma (\mathcal{C}, \mathcal{M}[n]), \quad \eta \mapsto \xi \cup \eta \]

is given by left cup product by $\xi $ as indicated.

[1] It would be more precise to write $F(\mathcal{A}) \otimes _\mathcal {O}^\mathbf {L} F(\mathcal{A}) \to F(\mathcal{A} \otimes _\mathcal {O} \mathcal{A}) \to F(\mathcal{A})$ were $F$ denotes the forgetful functor to complexes of $\mathcal{O}$-modules. Also, note that $\mathcal{A} \otimes _\mathcal {O} \mathcal{A}$ indicates the tensor product of Section 24.15 so that $F(\mathcal{A} \otimes _\mathcal {O} \mathcal{A}) = \text{Tot}(F(\mathcal{A}) \otimes _\mathcal {O} F(\mathcal{A}))$. The first arrow of the sequence is the canonical map from the derived tensor product of two complexes of $\mathcal{O}$-modules to the usual tensor product of complexes of $\mathcal{O}$-modules.
[2] Here and below $Rf_* : D(\mathcal{O}) \to D(\mathcal{O}')$ is the derived functor studied in Cohomology on Sites, Section 21.19 ff.
[3] For example the identity morphism.

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