The Stacks project

Lemma 110.74.1. Counter examples to algebraization of coherent sheaves.

  1. 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.

  2. The stack of coherent sheaves $\mathcal{C}\! \mathit{oh}_{X/B}$ of Quot, Theorems 99.6.1 and 99.5.12 is in general not algebraic if we drop the assumption that $X \to S$ is separated

  3. The functor $\mathrm{Quot}_{\mathcal{F}/X/B}$ of Quot, Proposition 99.8.4 is not an algebraic space in general if we drop the assumption that $X \to B$ is separated.

Proof. Part (1) we saw above. This shows that $\textit{Coh}_{X/A}$ fails axiom [4] of Artin's Axioms, Section 98.14. Hence it cannot be an algebraic stack by Artin's Axioms, Lemma 98.9.5. In this way we see that (2) is true. To see (3), note that there are compatible surjections $\mathcal{O}_{X_ n} \to \mathcal{F}_ n$ for all $n$. Thus we see that $\mathrm{Quot}_{\mathcal{O}_ X/X/A}$ fails axiom [4] and we see that (3) is true as before. $\square$

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