The Stacks project

Lemma 99.3.9. In Situation 99.3.1 assume that

  1. $B$ is a Noetherian algebraic space,

  2. $f$ is locally of finite type and quasi-separated,

  3. $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, and

  4. $\mathcal{G}$ is a finite type $\mathcal{O}_ X$-module, flat over $B$, with support proper over $B$.

Then the functor $\mathit{Hom}(\mathcal{F}, \mathcal{G})$ is an algebraic space affine and of finite presentation over $B$.

Proof. We may replace $X$ by a quasi-compact open neighbourhood of the support of $\mathcal{G}$, hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to \mathcal{F}$ by a perfect complex $P$ of the triple $(X, \mathcal{F}, -1)$, see Derived Categories of Spaces, Definition 75.14.1 and Theorem 75.14.7). Then the induced map

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P, \mathcal{G}) \]

is an isomorphism because $P \to \mathcal{F}$ induces an isomorphism $H^0(P) \to \mathcal{F}$ and $H^ i(P) = 0$ for $i > 0$. Moreover, for any morphism $g : T \to B$ denote $h : X_ T = T \times _ B X \to X$ the projection and set $P_ T = Lh^*P$. Then it is equally true that

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ T}}(\mathcal{F}_ T, \mathcal{G}_ T) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ T})}(P_ T, \mathcal{G}_ T) \]

is an isomorphism, as $P_ T = Lh^*P \to Lh^*\mathcal{F} \to \mathcal{F}_ T$ induces an isomorphism $H^0(P_ T) \to \mathcal{F}_ T$ (because $h^*$ is right exact and $H^ i(P) = 0$ for $i > 0$). Thus it suffices to prove the result for the functor

\[ T \longmapsto \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ T})}(P_ T, \mathcal{G}_ T). \]

By the Leray spectral sequence (see Cohomology on Sites, Remark 21.14.4) we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ T})}(P_ T, \mathcal{G}_ T) = H^0(X_ T, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)) = H^0(T, Rf_{T, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)) \]

where $f_ T : X_ T \to T$ is the base change of $f$. By Derived Categories of Spaces, Lemma 75.21.5 we have

\[ Rf_{T, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T) = Lg^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P, \mathcal{G}). \]

By Derived Categories of Spaces, Lemma 75.22.3 the object $K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P, \mathcal{G})$ of $D(\mathcal{O}_ B)$ is perfect. This means we can apply Lemma 99.3.8 as long as we can prove that the cohomology sheaf $H^ i(Lg^*K)$ is $0$ for all $i < 0$ and $g : T \to B$ as above. This is clear from the last displayed formula as the cohomology sheaves of $Rf_{T, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)$ are zero in negative degrees due to the fact that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)$ has vanishing cohomology sheaves in negative degrees as $P_ T$ is perfect with vanishing cohomology sheaves in positive degrees. $\square$

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