Dual and base change for finite locally free sheaves

Cover criterion: if $\phi :\mathcal{F}\to \mathcal{G}$ is a morphism of ${\mathcal{O}}_Y$-modules and $Y={\bigcup }_i{W}_i$ is an open cover such that each restriction $\phi {\mathrm{\mid }}_{W_i}:\mathcal{F}{\mathrm{\mid }}_{W_i}\to \mathcal{G}{\mathrm{\mid }}_{W_i}$ is an isomorphism, then $\phi$ is an isomorphism; indeed injectivity is local by [F5], and for $t\in \mathcal{G}(V)$ the sections $s_i\in \mathcal{F}(V\cap W_i)$ with $\phi (s_i)=t{\mathrm{\mid }}_{V\cap W_i}$ agree on overlaps because $\phi (s_i-s_j)=0$ and $\phi$ is injective on $V\cap W_i\cap W_j$, so they glue by [F5] to $s\in \mathcal{F}(V)$ with $\phi (s)=t$ by [F5] again.

For every ringed space $(Z,{\mathcal{O}}_Z)$ and every $r\ge 0$ the morphism $\alpha :{\mathcal{O}}_Z^r\to \mathcal{H}o{m}_{{\mathcal{O}}_Z}({\mathcal{O}}_Z^r,{\mathcal{O}}_Z)$ that on sections over $V$ sends $(a_1,\dots ,a_r)$ to the homomorphism $(x_1,\dots ,x_r)\mapsto {\sum }_i{x}_i{a}_i$ is an isomorphism: its inverse sends $\psi \in {\mathrm{Hom}}_{{\mathcal{O}}_V}({\mathcal{O}}_V^r,{\mathcal{O}}_V)$ to $(\psi (e_1),\dots ,\psi (e_r))$, where $e_1,\dots ,e_r$ are the standard basis sections, and both maps are natural in $V$, so they define mutually inverse isomorphisms of sheaves; for $r=0$ both sides are the zero sheaf, since ${\mathcal{O}}_Z^0=0$ and $\mathrm{Hom}(0,{\mathcal{O}}_V)=0$, so the case of rank zero is included.

Let $\mathcal{G}$ be an ${\mathcal{O}}_X$-module and let $U\subseteq X$ be open, with $f_U:f^{-1}U\to U$ the restriction of $f$. Restricting the defining formula of [F3] to the open $f^{-1}U$ gives a canonical isomorphism $\rho :(f^{\ast }\mathcal{G}){\mathrm{\mid }}_{f^{-1}U}\to f_U^{\ast }(\mathcal{G}{\mathrm{\mid }}_U)$, because $f^{-1}(\mathcal{G}{\mathrm{\mid }}_U)=(f^{-1}\mathcal{G}){\mathrm{\mid }}_{f^{-1}U}$ and ${\mathcal{O}}_{f^{-1}U}={\mathcal{O}}_Y{\mathrm{\mid }}_{f^{-1}U}$; the isomorphisms $\rho$ are natural in $\mathcal{G}$ and in $U$: they are compatible with restrictions to smaller opens and with morphisms $u:\mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}$.

Let $g:Z\to W$ be a morphism of ringed spaces and $r\ge 0$: then there is a canonical isomorphism $g^{\ast }({\mathcal{O}}_W^r)\cong {\mathcal{O}}_Z^r$. Indeed, by [F3] $g^{\ast }({\mathcal{O}}_W^r)={\mathcal{O}}_Z{\otimes }_{g^{-1}{\mathcal{O}}_W}{g}^{-1}({\mathcal{O}}_W^r)$; the inverse image of a finite direct sum is the direct sum of the inverse images, because on the colimit presheaf the construction is sectionwise and finite direct sums of presheaves are computed sectionwise, so $g^{-1}({\mathcal{O}}_W^r)=(g^{-1}{\mathcal{O}}_W{)}^r$; tensoring over the ring $g^{-1}{\mathcal{O}}_W$ distributes over finite direct sums and ${\mathcal{O}}_Z{\otimes }_{g^{-1}{\mathcal{O}}_W}{g}^{-1}{\mathcal{O}}_W\cong {\mathcal{O}}_Z$ because $g^{-1}{\mathcal{O}}_W\to {\mathcal{O}}_Z$ is a ring map making ${\mathcal{O}}_Z$ a module over $g^{-1}{\mathcal{O}}_W$ and tensoring a module by the ring itself returns the module; hence $g^{\ast }({\mathcal{O}}_W^r)\cong ({\mathcal{O}}_Z{)}^r$, with the case $r=0$ giving the zero sheaf.

Let $U\subseteq X$ be open with an isomorphism $\theta :\mathcal{E}{\mathrm{\mid }}_U\to {\mathcal{O}}_U^r$. Then ${\mathcal{E}}^{\vee }{\mathrm{\mid }}_U=\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{E},{\mathcal{O}}_X){\mathrm{\mid }}_U\cong \mathcal{H}o{m}_{{\mathcal{O}}_U}(\mathcal{E}{\mathrm{\mid }}_U,{\mathcal{O}}_U)\cong \mathcal{H}o{m}_{{\mathcal{O}}_U}({\mathcal{O}}_U^r,{\mathcal{O}}_U)\cong {\mathcal{O}}_U^r$, the first isomorphism by [F2], the second induced by $\theta$, and the last by step 1.2; hence ${\mathcal{E}}^{\vee }$ is finite locally free and agrees with $\mathcal{E}$ in rank on every chart.

For each open $U\subseteq X$ define ${\mathrm{\Psi }}_U:{\mathrm{Hom}}_{{\mathcal{O}}_U}(\mathcal{E}{\mathrm{\mid }}_U,{\mathcal{O}}_U)\to {\mathrm{Hom}}_{{\mathcal{O}}_{f^{-1}U}}(f^{\ast }\mathcal{E}{\mathrm{\mid }}_{f^{-1}U},{\mathcal{O}}_{f^{-1}U})$ by $\phi \mapsto {\rho }_{\mathcal{O}}^{-1}\circ f_U^{\ast }(\phi )\circ {\rho }_{\mathcal{E}}$, where ${\rho }_{\mathcal{E}},{\rho }_{\mathcal{O}}$ are the isomorphisms of step 1.3 for $\mathcal{G}=\mathcal{E}$ and $\mathcal{G}={\mathcal{O}}_X$; the maps ${\mathrm{\Psi }}_U$ are compatible with restrictions in $U$ by the naturality of step 1.3, hence define a morphism of ${\mathcal{O}}_X$-modules $\mathrm{\Psi }:{\mathcal{E}}^{\vee }\to f_{\ast }\mathcal{H}o{m}_{{\mathcal{O}}_Y}(f^{\ast }\mathcal{E},{\mathcal{O}}_Y)$; by the adjunction [F4] the morphism $\mathrm{\Psi }$ has a transpose $c:f^{\ast }({\mathcal{E}}^{\vee })⟶(f^{\ast }\mathcal{E}{)}^{\vee },$ the canonical comparison morphism, natural in $\mathcal{E}$ and $f$.

The evaluation morphism $\mathrm{ev}:\mathcal{E}\to {\mathcal{E}}^{\vee \vee }$ given on sections over $V$ by $e\mapsto (\phi \mapsto \phi (e))$ is well defined and ${\mathcal{O}}_X$-linear, because $\phi \in {\mathcal{E}}^{\vee }(V)$ is a homomorphism $\mathcal{E}{\mathrm{\mid }}_V\to {\mathcal{O}}_V$ by [F2] and the assignment is additive and ${\mathcal{O}}_V$-linear in $e$ and compatible with restrictions; on a chart $U$ of step 2.1 with trivialization $\theta$ and basis $e_1,\dots ,e_r$ of $\mathcal{E}{\mathrm{\mid }}_U$ mapping to the standard basis, the dual basis ${\phi }_j={\theta }_j$ of ${\mathcal{E}}^{\vee }{\mathrm{\mid }}_U$ satisfies $\mathrm{ev}(e_j)(\phi )=\phi (e_j)$, so under the identifications ${\mathcal{E}}^{\vee }{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ and ${\mathcal{E}}^{\vee \vee }{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ supplied by step 1.2 the map $\mathrm{ev}{\mathrm{\mid }}_U$ corresponds to the identity matrix and is an isomorphism; by the cover criterion of step 1.1 applied to a trivialising cover of $X$, $\mathrm{ev}$ is an isomorphism.

Let $U\subseteq X$ be a chart of $\mathcal{E}$ as in step 2.1. On $f^{-1}U$ the morphism $c$ is computed by steps 1.3 and 1.2 as follows: the identifications $f^{\ast }({\mathcal{E}}^{\vee }){\mathrm{\mid }}_{f^{-1}U}\cong f_U^{\ast }({\mathcal{E}}^{\vee }{\mathrm{\mid }}_U)\cong f_U^{\ast }({\mathcal{O}}_U^r)\cong {\mathcal{O}}_{f^{-1}U}^r$ and $(f^{\ast }\mathcal{E}{)}^{\vee }{\mathrm{\mid }}_{f^{-1}U}\cong \mathcal{H}o{m}_{{\mathcal{O}}_{f^{-1}U}}(f_U^{\ast }(\mathcal{E}{\mathrm{\mid }}_U),{\mathcal{O}}_{f^{-1}U})\cong \mathcal{H}o{m}_{{\mathcal{O}}_{f^{-1}U}}({\mathcal{O}}_{f^{-1}U}^r,{\mathcal{O}}_{f^{-1}U})\cong {\mathcal{O}}_{f^{-1}U}^r$ hold by step 1.4 and steps 1.3, 1.2, and the transpose construction of step 2.2 sends the pulled-back $j$-th basis functional ${\phi }_j$ — a local section of $f^{\ast }({\mathcal{E}}^{\vee })$ — to $f_U^{\ast }({\phi }_j)$, which under these identifications is the $j$-th coordinate functional of ${\mathcal{O}}_{f^{-1}U}^r$, that is, the $j$-th basis element; therefore $c{\mathrm{\mid }}_{f^{-1}U}$ corresponds to the identity matrix and is an isomorphism.

Choosing a trivialising open cover of $X$, the opens $f^{-1}U$ cover $Y$ and $c$ restricts to an isomorphism on each of them by step 3.2, so $c$ is an isomorphism by the cover criterion of step 1.1: this proves claim 3, while claim 1 is step 2.1 and claim 2 is step 3.1. Every morphism constructed is canonical and all identifications involve only finitely many standard basis elements of a free module of finite rank, so no choice principle is used.