Solution

The affine module sheaf construction makes the remaining statements explicit. For $A=k[V]$ and an $A$-module $M$, defineon the basis of principal open subsets. Restrictions are the natural localization maps. Equivalently, a section on an arbitrary open set is a family $s_P\in M_{{\mathfrak{m}}_P}$ which near each point is represented by a single fraction $m/f^r$, with $f$ nonvanishing throughout that neighbourhood. This locally representable family construction gives a sheaf; the usual localization gluing verifies its quasi-coherence. Its stalk is $M_{{\mathfrak{m}}_P}$ andThis last identification is precisely the standard module-localization gluing on the affine basis, not an assertion that arbitrary sheaves are determined by their global sections.

Now write $A=k[X]$, $B=k[Y]$ and $\alpha ={\varphi }^{\#}:A\to B$. Let $\mathcal{G}=\tilde{M}$, $\mathcal{H}=\tilde{N}$ for $A$-modules $M,N$, and let $\mathcal{F}=\tilde{L}$ for a $B$-module $L$. The affine versions of the three constructions areHere ${}_{A}L$ denotes restriction of scalars along $\alpha$, whereas the last associated sheaf lives on $Y$. To verify the tensor-product assertion, localize at every prime: both stalks are $M_{\mathfrak{m}}{\otimes }_{A_{\mathfrak{m}}}{N}_{\mathfrak{m}}$. For the direct image, ${\varphi }^{-1}D(a)=D(\alpha (a))$, andThese equalities identify the sheaves on a basis and respect restrictions. For the pullback of a sheaf of modules, its stalk at $Q$ is $B_{{\mathfrak{m}}_Q}{\otimes }_{A}N$, exactly the stalk of $\widetilde{B{\otimes }_{A}N}$. The local tensor maps are compatible, so these stalk identifications come from an actual sheaf isomorphism. Thus all three constructions are quasi-coherent in the stated affine setting.

Finally construct the natural projection formula for sheaves map by sending a local tensor $m\otimes b$ to $b(1\otimes m)$ in the pulled-back module, followed by direct image. In the affine description, the target ${\varphi }_{\ast }{\varphi }^{\ast }\tilde{M}$ is the sheaf associated with the $A$-module ${}_A(B{\otimes }_{A}M)$, while $\tilde{M}{\otimes }_{{\mathcal{O}}_X}{\varphi }_{\ast }{\mathcal{O}}_Y$ corresponds to $M{\otimes }_{A}B$. The mapsare inverse $A$-linear maps. Localizing them commutes with every restriction, provingNo flatness assumption on $A\to B$ or finiteness assumption on $M$ is needed.