Remark 99.8.5 (Quot via Artin's axioms). Let $S$ be a Noetherian scheme all of whose local rings are G-rings. Let $X$ be an algebraic space over $S$ whose structure morphism $f : X \to S$ is of finite presentation and separated. Let $\mathcal{F}$ be a finitely presented quasi-coherent sheaf on $X$ flat over $S$. In this remark we sketch how one can use Artin's axioms to prove that $\mathrm{Quot}_{\mathcal{F}/X/S}$ is an algebraic space locally of finite presentation over $S$ and avoid using the algebraicity of the stack of coherent sheaves as was done in the proof of Proposition 99.8.4.
We check the conditions listed in Artin's Axioms, Proposition 98.16.1. Representability of the diagonal of $\mathrm{Quot}_{\mathcal{F}/X/S}$ can be seen as follows: suppose we have two quotients $\mathcal{F}_ T \to \mathcal{Q}_ i$, $i = 1, 2$. Denote $\mathcal{K}_1$ the kernel of the first one. Then we have to show that the locus of $T$ over which $u : \mathcal{K}_1 \to \mathcal{Q}_2$ becomes zero is representable. This follows for example from Flatness on Spaces, Lemma 77.8.6 or from a discussion of the $\mathit{Hom}$ sheaf earlier in this chapter. Axioms [0] (sheaf), [1] (limits), [2] (Rim-Schlessinger) follow from Lemmas 99.8.2, 99.7.7, and 99.7.8 (plus some extra work to deal with the properness condition). Axiom [3] (finite dimensionality of tangent spaces) follows from the description of the infinitesimal deformations in Remark 99.7.10 and finiteness of cohomology of coherent sheaves on proper algebraic spaces over fields (Cohomology of Spaces, Lemma 69.20.2). Axiom [4] (effectiveness of formal objects) follows from Grothendieck's existence theorem (More on Morphisms of Spaces, Theorem 76.42.11). As usual, the trickiest to verify is axiom [5] (openness of versality). One can for example use the obstruction theory described in Remark 99.7.9 and the description of deformations in Remark 99.7.10 to do this using the criterion in Artin's Axioms, Lemma 98.22.2. Please compare with the second proof of Lemma 99.5.11.
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