= Solution
The opening <change-of-rings tensor quotient> sends $m\otimes_Rn$ to $m\otimes_An$. The target pairing is <bilinear> and $R$-balanced because $\theta(r)m\otimes_An=m\otimes_A\theta(r)n$. On the source, let $A$ act through the first factor; commutativity makes that action compatible with the $R$-balancing relations, and the map is $A$-linear. It is <surjective>: the target is the quotient imposing the additional relations $am\otimes_Rn=m\otimes_Ran$ for every $a\in A$. The construction commutes with <module homomorphisms> in both variables.
The <pullback-direct-image adjunction unit> is obtained locally by pulling a section $h\in\mathcal H(V)$ to $\phi^{-1}V$ and sending it to $1\otimes h$ in $\phi^*\mathcal H(\phi^{-1}V)$. These maps respect restrictions and the $\mathcal O_Y$-actions, hence define
$$
\eta_{\mathcal H}:\mathcal H\longrightarrow\phi_*\phi^*\mathcal H.
$$
For $\mathcal H=\mathcal O_Y$, multiplication identifies $\phi^*\mathcal O_Y$ with $\mathcal O_X$, and $\eta_{\mathcal O_Y}$ is precisely the structure morphism $\phi^\#: \mathcal O_Y\to\phi_*\mathcal O_X$.
To construct the <direct-image tensor comparison>, on $V\subseteq Y$ send $s\otimes t$, with $s\in\mathcal F(\phi^{-1}V)$ and $t\in\mathcal G(\phi^{-1}V)$, to its tensor section of $\mathcal F\otimes_{\mathcal O_X}\mathcal G$. This pairing is $\mathcal O_Y(V)$-balanced through $\phi^\#$ and is compatible with restrictions. The <universal property of the tensor product of modules> and <sheafification> therefore give
$$
\phi_*\mathcal F\otimes_{\mathcal O_Y}\phi_*\mathcal G\longrightarrow\phi_*(\mathcal F\otimes_{\mathcal O_X}\mathcal G).
$$
Apply this with $\mathcal G=\phi^*\mathcal H$ after tensoring the unit $\eta_{\mathcal H}$ with $\phi_*\mathcal F$. The composite is the map in the <projection formula for sheaves>. If $\mathcal H|_V\cong\mathcal O_V^r$, then $\phi^*\mathcal H|_{\phi^{-1}V}\cong\mathcal O_{\phi^{-1}V}^r$, and the composite identifies with
$$
(\phi_*\mathcal F|_V)^{\oplus r}\longrightarrow\phi_*(\mathcal F^{\oplus r})|_V,
$$
the identity on the $r$ components. Thus it is an <isomorphism> for a <locally free sheaf> of finite rank. Neither $\mathcal F$ nor $\mathcal G$ was assumed <quasi-coherent>.
Back to article page
