Restriction of tensor product of $\mathcal{O}_X$-modules to an open subset [duplicate]

Question: My question is simple : Let $X$ be a scheme and $\mathcal{F}$ and $\mathcal{G}$ are (quasi-coherent) $\mathcal{O}_X$-modules. And let $U$ be an open subset of $X$. Then

$$(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{G})|_U \cong \mathcal{F}|_U \otimes_{\mathcal{O}_{X}|_{U}} \mathcal{G}|_U ?$$

Answer: You may prove the claim directly by looking at presheaves: If $U \subseteq X$ is an open subset and if $E,F$ are quasi coherent sheaves on $X$ it follows $E_U,F_U$ (the restriction of $E,F$ to $U$) are quasi coherent sheaves on $U$.

Define the following presheaf on $X$: $T(E,F)(V):=E(V)\otimes_{\mathcal{O}_X(V)}F(V)$ for any open set $V \subseteq X$. It follows the tensor product $E\otimes_{\mathcal{O}}F\cong T(E,F)^+$ is the sheafification of $T(E,F)$. You may define a similar presheaf on $U$:

$$T_U(E_U,F_U)(V):=E_U(V)\otimes_{\mathcal{O}_U(V)}F_U(V)$$

with $T_U(E_U,F_U)^+ :=E_U\otimes_{\mathcal{O}_U}F_U$.

You may restrict $T(E,F)$ to $U$, denoted $T(E,F)_U$, by letting $V \subseteq U$. It follows

$$T_U(E_U,F_U) \cong T(E,F)_U$$

are isomorphic as presheaves on $U$, hence

$$ (E \otimes_{\mathcal{O}_X}F)_U \cong E_U\otimes_{\mathcal{O}_U}F_U$$

Comment: "Wow, there could be such a strategy. Key is the sufficiency of isomorphic presheaves and TU(EU,FU)≅T(E,F)U. I feel that I learn some kind of new technique(?). Thanks. – Plantation 9 hours ago"

If you read Proposition I.1.2 in Hartshorne you will find that the association

$$F \rightarrow F^+$$

where you construct a sheaf from a presheaf, satisfies a universal property. For any sheaf $G$ and any morphism of presheaves $\phi: F \rightarrow G$ there is a unique extended morphism $\phi^+: F^+ \rightarrow G$: There is an equality of sets

$$Hom_{psh}(F,G) \cong Hom_{sh}(E^+,G)$$

where the first is maps of presheaves, the second is maps of sheaves.

Also it follows that two presheaves $E \cong F$ are isomorphic iff $E^+ \cong F^+$ are isomorphic sheaves. For this reason: When defining maps between $E^+$ and $F^+$ you may define these maps between $E$ and $F$.

This is another approach (this was deleted for unknown reasons) using properties of the topological pull back: If $f:X \rightarrow Y$ is a map of schemes let $\tilde{f}:f^{-1}(\mathcal{O}_Y) \rightarrow \mathcal{O}_X$ be the corresponding map of sheaves. By definition

$$f^*(E):=\mathcal{O}_X\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(E).$$

You may check that

$$(*)\text{ }f^{-1}(E\otimes_{\mathcal{O}_Y}F)\cong f^{-1}(E)\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F).$$

There are two presheaves on $X$. Define for any open set $U \subseteq X$

$$F(E,F)(U):= lim_{f(U) \subseteq V} E(V)\otimes_{\mathcal{O}_Y(V)} F(V).$$

Define

$$G(f^{-1}(E)\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F))(U):= f^{-1}(E)(U)\otimes_{f^{-1}(\mathcal{O}_Y)(U)} f^{-1}(F)(U).$$

It follows

$$F(E,F)^+ \cong f^{-1}(E\otimes_{\mathcal{O}_Y} F)$$ and

$$G(f^{-1}(E)\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F))^+\cong f^{-1}(E)\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F).$$

It follows

$$G(f^{-1}(E)\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F))(U):=$$

$$(*)\text{ }lim_{f(U) \subseteq V} E(V)\otimes_{lim_{f(U) \subseteq V} \mathcal{O}_Y(V) }lim_{f(U) \subseteq V} F(V) \cong$$

$$lim_{f(U) \subseteq V }E(V)\otimes_{\mathcal{O}_Y(V)} F(V) \cong$$

$$F(E\otimes_{\mathcal{O}_Y} F)(U).$$

Note: This is a direct limit in a generalized sense (you may find it defined in Bourbaki's algebra or commutative algebra books). In Matsumura they define the direct limit $lim_i M_i$ of a directed set of $A$-modules $M_i$. Given a directed set of rings $A_i, i\in I$ and a pair of directed sets of abelian groups $M_i, N_i,i\in I$ with each $M_i,N_i$ an $A_i$-module and for each map $f_{i,j}:M_i \rightarrow M_j$ and $\phi_{i,j}: A_i \rightarrow A_j$ it follows $f_{i,j}(a_im_i)=\phi_{i,j}(a_i)f_{i,j}(m_i)$. Let $A:=lim_I A_i$. It follows $lim_I(M_i\otimes_{A_i} N_i),lim_I M_i, lim_I N_i$ are left $A$-modules and there is an isomorphism of $A$-modules

$$lim_I (M_i\otimes_{A_i} N_i) \cong (lim_I M_i)\otimes_A (lim_I N_i).$$

Hence

$$G(f^{-1}(E)\otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F))^+ \cong F(E,F)^+ .$$

It follows

$$f^*(E\otimes_{\mathcal{O}_Y} F) \cong \mathcal{O}_X \otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(E\otimes_{\mathcal{O}_Y}F) \cong$$

$$\mathcal{O}_X \otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(E) \otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F)) \cong \mathcal{O}_X \otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(E) \otimes_{\mathcal{O}_X} \mathcal{O}_X \otimes_{f^{-1}(\mathcal{O}_Y)} f^{-1}(F)$$

$$\cong f^*(E)\otimes_{\mathcal{O}_X} f^*(F).$$

Many of the elementary properties of the functor $f^*$ can be proved directly using basic properties of $f^{-1}$ - one wants to avoid using abstract machinery such as "monoidal categories" or "representable functors".

Example: If $f:U\rightarrow X$ is an inclusion of an open set $U \subseteq X$ it follows

$$f^*(E\otimes_{\mathcal{O}_X}F) \cong f^*(E)\otimes_{\mathcal{O}_U}f^*(F) \cong E_U \otimes_{\mathcal{O}_U} F_U$$

where $E_U,F_U$ are the restrictions to $U$.

If we take for granted the claims in Cor. 7.19 of Gortz--Wedhorn then there is essentially not much to show anymore. To prove the claim that $(\mathcal{F} \otimes \mathcal{G})|_{U} \cong \mathcal{F}|_{U} \otimes \mathcal{G}|_{U}$, it suffices to check that they have the same values on affine open subsets. So let's take an affine open subset $\mathrm{Spec}(A)$ of $U$: $$\begin{split} \Gamma\big(\mathrm{Spec}(A), \mathcal{F}|_{U} \otimes \mathcal{G}|_{U}\big) &\cong \Gamma\big(\mathrm{Spec}(A),\mathcal{F}|_{U}\big) \otimes \Gamma\big(\mathrm{Spec}(A),\mathcal{G}|_{U}\big) \\ &\cong \Gamma\big(\mathrm{Spec}(A),\mathcal{F}\big) \otimes \Gamma\big(\mathrm{Spec}(A),\mathcal{G}\big)\end{split}$$ where the first step is Cor. 7.19(d) and the second step is by definition of restriction. On the other hand, $$\begin{split}\Gamma\big(\mathrm{Spec}(A),(\mathcal{F} \otimes\mathcal{G})|_{U}\big) &\cong \Gamma\big(\mathrm{Spec}(A),\mathcal{F} \otimes \mathcal{G}\big) \\ &\cong \Gamma\big(\mathrm{Spec}(A),\mathcal{F}\big) \otimes \Gamma\big(\mathrm{Spec}(A),\mathcal{G})\big)\end{split}$$ where this time the first step is by definition of restriction, and the second step is that same Cor. 7.19(d).

In fact, more generally, if $f \colon X\to Y$ is a morphism of schemes, then the pullback functor $f^*$ on (quasi-coherent) sheaves always commutes with tensor product. Our claim is the special case where $f$ is an open immersion.