\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*}

The Stacks project

Lemma 46.23.5. Let $X/A$ with dualizing module $\omega _ X$ be as in Example 46.23.1. Let $d = \dim (X_ s)$ be the dimension of the closed fibre. If $\dim (X) = d + \dim (A)$, then the dualizing module $\omega _ X$ represents the functor

\[ \mathcal{F} \longmapsto \mathop{\mathrm{Hom}}\nolimits _ A(H^ d(X, \mathcal{F}), \omega _ A) \]

on the category of coherent $\mathcal{O}_ X$-modules.

Proof. We have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, \omega _ X) & = \mathop{\mathrm{Ext}}\nolimits ^{-\dim (X)}_ X(\mathcal{F}, \omega _ X^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}[\dim (X)], \omega _ X^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}[\dim (X)], f^!(\omega _ A^\bullet )) \\ & = \mathop{\mathrm{Hom}}\nolimits _ S(Rf_*\mathcal{F}[\dim (X)], \omega _ A^\bullet ) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(H^ d(X, \mathcal{F}), \omega _ A) \end{align*}

The first equality because $H^ i(\omega _ X^\bullet ) = 0$ for $i < -\dim (X)$, see Lemma 46.23.4 and Derived Categories, Lemma 13.27.3. The second equality is follows from the definition of Ext groups. The third equality is our choice of $\omega _ X^\bullet $. The fourth equality holds because $f^!$ is the right adjoint of Lemma 46.3.1 for $f$, see Section 46.20. The final equality holds because $R^ if_*\mathcal{F}$ is zero for $i > d$ (Cohomology of Schemes, Lemma 29.20.9) and $H^ j(\omega _ A^\bullet )$ is zero for $j < -\dim (A)$. $\square$


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