Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 16 / 1

Construct the tensor product of sheaves by first forming the presheaf $U\mapsto \mathcal{F}(U){\otimes }_{{\mathcal{O}}_X(U)}\mathcal{G}(U)$, with restriction maps induced by those of the two sheaves of modules, and then applying sheafification. The local module actions are compatible with restrictions and therefore give the sheaf an ${\mathcal{O}}_X$-module structure. Its stalks areIndeed, a finite collection of germs of sheaf sections can be represented on a common neighbourhood, and every finite tensor relation holds on a sufficiently small neighbourhood. Equivalently, this construction represents bilinear maps of sheaves of modules that are balanced over the structure sheaf. Sections of the resulting sheaf need not themselves be tensors of global sections: that is why the sheafification step matters.

The direct image sheaf is defined on an open set $V\subseteq Y$ byRestrictions are those of $\mathcal{F}$, and the sheaf gluing axiom follows by taking inverse images of an open cover. The morphism of ringed spaces supplies ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$. Thus $a\in {\mathcal{O}}_Y(V)$ acts on $s\in \mathcal{F}({\varphi }^{-1}V)$ by ${\varphi }^{\#}(a)s$, making ${\varphi }_{\ast }\mathcal{F}$ a sheaf of modules over ${\mathcal{O}}_Y$. This definition uses no quasi-coherence.

First form the inverse image sheaf ${\varphi }^{-1}\mathcal{H}$ by applying sheafification toIt is a sheaf of modules over ${\varphi }^{-1}{\mathcal{O}}_Y$. The morphism of ringed spaces gives a ring map ${\varphi }^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$, so the pullback of a sheaf of modules isAt $x\in X$, writing $y=\varphi (x)$, its stalk is ${\mathcal{O}}_{X,x}{\otimes }_{{\mathcal{O}}_{Y,y}}{\mathcal{H}}_y$. The tensor construction makes the action of the structure sheaf explicit; taking the inverse image alone would not give the requested ${\mathcal{O}}_X$-module.

The opening change-of-rings tensor quotient sends $m{\otimes }_{R}n$ to $m{\otimes }_{A}n$. The target pairing is bilinear and $R$-balanced because $\theta (r)m{\otimes }_{A}n=m{\otimes }_A\theta (r)n$. 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 $am{\otimes }_{R}n=m{\otimes }_{R}an$ 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 ${\varphi }^{-1}V$ and sending it to $1\otimes h$ in ${\varphi }^{\ast }\mathcal{H}({\varphi }^{-1}V)$. These maps respect restrictions and the ${\mathcal{O}}_Y$-actions, hence defineFor $\mathcal{H}={\mathcal{O}}_Y$, multiplication identifies ${\varphi }^{\ast }{\mathcal{O}}_Y$ with ${\mathcal{O}}_X$, and ${\eta }_{{\mathcal{O}}_Y}$ is precisely the structure morphism ${\varphi }^{\#}:{\mathcal{O}}_Y\to {\varphi }_{\ast }{\mathcal{O}}_X$.

To construct the direct-image tensor comparison, on $V\subseteq Y$ send $s\otimes t$, with $s\in \mathcal{F}({\varphi }^{-1}V)$ and $t\in \mathcal{G}({\varphi }^{-1}V)$, to its tensor section of $\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G}$. This pairing is ${\mathcal{O}}_Y(V)$-balanced through ${\varphi }^{\#}$ and is compatible with restrictions. The universal property of the tensor product of modules and sheafification therefore giveApply this with $\mathcal{G}={\varphi }^{\ast }\mathcal{H}$ after tensoring the unit ${\eta }_{\mathcal{H}}$ with ${\varphi }_{\ast }\mathcal{F}$. The composite is the map in the projection formula for sheaves. If $\mathcal{H}{|}_V\cong {\mathcal{O}}_V^r$, then ${\varphi }^{\ast }\mathcal{H}{|}_{{\varphi }^{-1}V}\cong {\mathcal{O}}_{{\varphi }^{-1}V}^r$, and the composite identifies withthe 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.