Let $ X $ be a topological space and let $ \mathscr A $ be a sheaf of commutative rings on $ X $. For my purposes $ X $ could well be a smooth real manifold and $ \mathscr A $ could be the sheaf of smooth real-valued functions on $ X $.
Let $ \mathscr F $ be an $ \mathscr A $-module. I'm trying to make sense of the exterior power $ \bigwedge_{\mathscr A}^k \mathscr F $ of $ \mathscr F $.
My book says that $ \bigwedge_{\mathscr A}^k \mathscr F $ is the sheafification of the presheaf $ P $ on $ X $ defined by putting $$ P(U) = \bigwedge\nolimits_{\mathscr A(U)}^k\mathscr F(U) $$ for every $ U\subset X $ open.
I tried to make sense of $ P $ as a presheaf but I haven't been able to define the restriction mappings $$ \bigwedge\nolimits_{\mathscr A(\color{red}{U})}^k\mathscr F(U)\to \bigwedge\nolimits_{\mathscr A(\color{red}{V})}^k\mathscr F(V) $$ for $ V\subset U\subset X $ open sets. Notice that the ring over which the exterior power is taken is changing, since $ \mathscr F(W) $ is in general only an $ \mathscr A(W) $-module for $ W\subset X $ open.
Given those $ V\subset U $ there's of course a map (which is a ring homomorphism) $$ \rho_V^U\colon \mathscr A(U)\to \mathscr A(V) $$ so I tried to mess things up with the restriction/extension-of-scalars adjunction $ (\rho_V^U)_*\dashv (\rho_V^U)^* $. Unfortunately I always stepped in some compatibility requirement (of $ \bigwedge^k $ with $ (\rho_V^U)_* $ or $ (\rho_V^U)^* $) that apparently didn't hold.
How are those restrictions defined?
Let $V \subseteq U$. The map $F(U)^k \to F(V)^k \to \wedge^k F(V)$, $(a_1,\dotsc,a_n) \mapsto a_1|_V \wedge \cdots \wedge a_k|_V$ is alternating and $A(U)$-multilinear (where the $A(U)$-module structure on $\wedge^k F(V)$ comes from restriction of scalars along $A(U) \to A(V)$), hence lifts to a $A(U)$-linear map $\wedge^k F(U) \to \wedge^k F(V)$ (universal property of exterior power).