Lemma 109.73.1. Counter examples to algebraization of coherent sheaves.

Grothendieck's existence theorem as stated in Cohomology of Schemes, Theorem 30.27.1 is false if we drop the assumption that $X \to \mathop{\mathrm{Spec}}(A)$ is separated.

The stack of coherent sheaves $\mathcal{C}\! \mathit{oh}_{X/B}$ of Quot, Theorems 98.6.1 and 98.5.12 is in general not algebraic if we drop the assumption that $X \to S$ is separated

The functor $\mathrm{Quot}_{\mathcal{F}/X/B}$ of Quot, Proposition 98.8.4 is not an algebraic space in general if we drop the assumption that $X \to B$ is separated.

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