Lemma 99.4.2 (0D3T)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 99: Quot and Hilbert Spaces
Section 99.4: The Isom functor
Lemma 99.4.2 (cite)

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Lemma 99.4.2. In Situation 99.3.1. Let $T$ be an algebraic space over $S$. We have

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

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. Observe that the left and right hand side of the equality are subsets of the left and right hand side of the equality in Lemma 99.3.3. We omit the verification that these subsets correspond under the identification given in the proof of that lemma. $\square$

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