The Stacks project

Lemma 99.3.3. In Situation 99.3.1. Let $T$ be an algebraic space over $S$. We have

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \mathit{Hom}(\mathcal{F}, \mathcal{G})) = \{ (h, u) \mid h : T \to B, u : \mathcal{F}_ T \to \mathcal{G}_ T\} \]

where $\mathcal{F}_ T, \mathcal{G}_ T$ denote the pullbacks of $\mathcal{F}$ and $\mathcal{G}$ to the algebraic space $X \times _{B, h} T$.

Proof. Choose a scheme $U$ and a surjective ├ętale morphism $p : U \to T$. Let $R = U \times _ T U$ with projections $t, s : R \to U$.

Let $v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})$ be a natural transformation. Then $v(p)$ corresponds to a pair $(h_ U, u_ U)$ over $U$. As $v$ is a transformation of functors we see that the pullbacks of $(h_ U, u_ U)$ by $s$ and $t$ agree. Since $T = U/R$ (Spaces, Lemma 65.9.1), we obtain a morphism $h : T \to B$ such that $h_ U = h \circ p$. Then $\mathcal{F}_ U$ is the pullback of $\mathcal{F}_ T$ to $X_ U$ and similarly for $\mathcal{G}_ U$. Hence $u_ U$ descends to a $\mathcal{O}_{X_ T}$-module map $u : \mathcal{F}_ T \to \mathcal{G}_ T$ by Descent on Spaces, Proposition 74.4.1.

Conversely, let $(h, u)$ be a pair over $T$. Then we get a natural transformation $v : T \to \mathit{Hom}(\mathcal{F}, \mathcal{G})$ by sending a morphism $a : T' \to T$ where $T'$ is a scheme to $(h \circ a, a^*u)$. We omit the verification that the construction of this and the previous paragraph are mutually inverse. $\square$


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