The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 7.19.3. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of categories. Assume that $v$ is right adjoint to $u$. Then we have

  1. $u^ ph_ V = h_{v(V)}$ for any $V$ in $\mathcal{D}$,

  2. the category $\mathcal{I}^ v_ U$ has an initial object,

  3. the category ${}_ V^ u\mathcal{I}$ has a final object,

  4. ${}_ pu = v^ p$, and

  5. $u^ p = v_ p$.

Proof. Proof of (1). Let $V$ be an object of $\mathcal{D}$. We have $u^ ph_ V = h_{v(V)}$ because $u^ ph_ V(U) = \mathop{Mor}\nolimits _\mathcal {D}(u(U), V) = \mathop{Mor}\nolimits _\mathcal {C}(U, v(V))$ by assumption.

Proof of (2). Let $U$ be an object of $\mathcal{C}$. Let $\eta : U \to v(u(U))$ be the map adjoint to the map $\text{id} : u(U) \to u(U)$. Then we claim $(u(U), \eta )$ is an initial object of $\mathcal{I}_ U^ v$. Namely, given an object $(V, \phi : U \to v(V))$ of $\mathcal{I}_ U^ v$ the morphism $\phi $ is adjoint to a map $\psi : u(U) \to V$ which then defines a morphism $(u(U), \eta ) \to (V, \phi )$.

Proof of (3). Let $V$ be an object of $\mathcal{D}$. Let $\xi : u(v(V)) \to V$ be the map adjoint to the map $\text{id} : v(V) \to v(V)$. Then we claim $(v(V), \xi )$ is a final object of ${}_ V^ u\mathcal{I}$. Namely, given an object $(U, \psi : u(U) \to V)$ of ${}_ V^ u\mathcal{I}$ the morphism $\psi $ is adjoint to a map $\phi : U \to v(V)$ which then defines a morphism $(U, \psi ) \to (v(V), \xi )$.

Hence for any presheaf $\mathcal{F}$ on $\mathcal{C}$ we have

\begin{eqnarray*} v^ p\mathcal{F}(V) & = & \mathcal{F}(v(V)) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{v(V)}, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ ph_ V, \mathcal{F}) \\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{D})}(h_ V, {}_ pu\mathcal{F}) \\ & = & {}_ pu\mathcal{F}(V) \end{eqnarray*}

which proves part (2). Part (3) follows by the uniqueness of adjoint functors. $\square$


Comments (1)

Comment #1778 by Kiran Kedlaya on

Typo in the third paragraph: "calim".


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