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18.26 Tensor product

In Sections 18.9 and 18.11 we defined the change of rings functor by a tensor product construction. To be sure this construction makes sense also to define the tensor product of presheaves of $\mathcal{O}$-modules. To be precise, suppose $\mathcal{C}$ is a category, $\mathcal{O}$ is a presheaf of rings, and $\mathcal{F}$, $\mathcal{G}$ are presheaves of $\mathcal{O}$-modules. In this case we define $\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}$ to be the presheaf

\[ U \longmapsto (\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G})(U) = \mathcal{F}(U) \otimes _{\mathcal{O}(U)} \mathcal{G}(U) \]

If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings and $\mathcal{F}$, $\mathcal{G}$ are sheaves of $\mathcal{O}$-modules then we define

\[ \mathcal{F} \otimes _\mathcal {O} \mathcal{G} = (\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G})^\# \]

to be the sheaf of $\mathcal{O}$-modules associated to the presheaf $\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}$.

Here are some formulas which we will use below without further mention:

\[ (\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}) \otimes _{p, \mathcal{O}} \mathcal{H} = \mathcal{F} \otimes _{p, \mathcal{O}} (\mathcal{G} \otimes _{p, \mathcal{O}} \mathcal{H}), \]

and similarly for sheaves. If $\mathcal{O}_1 \to \mathcal{O}_2$ is a map of presheaves of rings, then

\[ (\mathcal{F} \otimes _{p, \mathcal{O}_1} \mathcal{G}) \otimes _{p, \mathcal{O}_1} \mathcal{O}_2 = (\mathcal{F} \otimes _{p, \mathcal{O}_1} \mathcal{O}_2) \otimes _{p, \mathcal{O}_2} (\mathcal{G} \otimes _{p, \mathcal{O}_1} \mathcal{O}_2), \]

and similarly for sheaves. These follow from their algebraic counterparts and sheafification.

Lemma 18.26.1. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings. Let $\mathcal{F}$, $\mathcal{G}$ be presheaves of $\mathcal{O}$-modules. Then $\mathcal{F}^\# \otimes _{\mathcal{O}^\# } \mathcal{G}^\# $ is equal to $(\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G})^\# $.

Proof. Omitted. Hint: use the characterization of tensor product in terms of bilinear maps below and use the universal property of sheafification. $\square$

Let $\mathcal{C}$ be a site, let $\mathcal{O}$ be a sheaf of rings and let $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ be sheaves of $\mathcal{O}$-modules. In this case we define

\[ \text{Bilin}_\mathcal {O}(\mathcal{F} \times \mathcal{G}, \mathcal{H}) = \{ \varphi \in \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}( \mathcal{F} \times \mathcal{G}, \mathcal{H}) \mid \varphi \text{ is }\mathcal{O}\text{-bilinear}\} . \]

With this definition we have

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O} (\mathcal{F} \otimes _\mathcal {O} \mathcal{G}, \mathcal{H}) = \text{Bilin}_\mathcal {O}(\mathcal{F} \times \mathcal{G}, \mathcal{H}). \]

In other words $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$ represents the functor which associates to $\mathcal{H}$ the set of bilinear maps $\mathcal{F} \times \mathcal{G} \to \mathcal{H}$. In particular, since the notion of a bilinear map makes sense for a pair of modules on a ringed topos, we see that the tensor product of sheaves of modules is intrinsic to the topos (compare the discussion in Section 18.18). In fact we have the following.

Lemma 18.26.2. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_\mathcal {D}$-modules. Then $f^*(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{G}) = f^*\mathcal{F} \otimes _{\mathcal{O}_\mathcal {C}} f^*\mathcal{G}$ functorially in $\mathcal{F}$, $\mathcal{G}$.

Proof. For a sheaf $\mathcal{H}$ of $\mathcal{O}_\mathcal {C}$ modules we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( f^*(\mathcal{F} \otimes _\mathcal {O} \mathcal{G}), \mathcal{H}) & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}( \mathcal{F} \otimes _\mathcal {O} \mathcal{G}, f_*\mathcal{H}) \\ & = \text{Bilin}_{\mathcal{O}_\mathcal {D}}( \mathcal{F} \times \mathcal{G}, f_*\mathcal{H}) \\ & = \text{Bilin}_{f^{-1}\mathcal{O}_\mathcal {D}}( f^{-1}\mathcal{F} \times f^{-1}\mathcal{G}, \mathcal{H}) \\ & = \mathop{\mathrm{Hom}}\nolimits _{f^{-1}\mathcal{O}_\mathcal {D}}( f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_\mathcal {D}} f^{-1}\mathcal{G}, \mathcal{H}) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( f^*\mathcal{F} \otimes _{f^*\mathcal{O}_\mathcal {D}} f^*\mathcal{G}, \mathcal{H}) \end{align*}

The interesting “$=$” in this sequence of equalities is the third equality. It follows from the definition and adjointness of $f_*$ and $f^{-1}$ (as discussed in previous sections) in a straightforward manner. $\square$

Lemma 18.26.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}$-modules.

  1. If $\mathcal{F}$, $\mathcal{G}$ are locally free, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  2. If $\mathcal{F}$, $\mathcal{G}$ are finite locally free, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  3. If $\mathcal{F}$, $\mathcal{G}$ are locally generated by sections, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  4. If $\mathcal{F}$, $\mathcal{G}$ are of finite type, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  5. If $\mathcal{F}$, $\mathcal{G}$ are quasi-coherent, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  6. If $\mathcal{F}$, $\mathcal{G}$ are of finite presentation, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

  7. If $\mathcal{F}$ is of finite presentation and $\mathcal{G}$ is coherent, then $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$ is coherent.

  8. If $\mathcal{F}$, $\mathcal{G}$ are coherent, so is $\mathcal{F} \otimes _\mathcal {O} \mathcal{G}$.

Proof. Omitted. Hint: Compare with Sheaves of Modules, Lemma 17.16.6. $\square$

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