tensor products of sheaves of modules

DEFINITION 9.6   Let $(X,\mathcal A)$ be a sheaf of algebras (possibly non commutative). Let $\mathcal F$ be a right $\mathcal A$-module. Let $\mathcal G$ be a left $\mathcal A$-module. Then the tensor product $\mathcal F\otimes _{\mathcal A}\mathcal G$ is the sheafication of the presheaf defined by

$$U\mapsto \mathcal F(U)\otimes_ {\mathcal A(U)}\mathcal G(U).$$

DEFINITION 9.7   Let $f:X\to Y$ be a morphism between locally ringed spaces. Let $\mathcal F$ be a sheaf of $\mathcal{O}_Y$-modules on $Y$. Then the inverse image of $\mathcal F$ as an $\mathcal{O}$-module with respect to $f$ as a sheaf of $\mathcal{O}$-modules is defined as

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