The Stacks project

Lemma 109.73.1. 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 98.6.1 and 98.5.12 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 98.8.4 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 97.14. Hence it cannot be an algebraic stack by Artin's Axioms, Lemma 97.9.5. 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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