The Stacks project

Lemma 70.10.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume

  1. every weakly associated point of $X$ is a point of codimension $0$, and

  2. $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 66.49.1.

  3. $X$ is representable by a scheme $X_0$ (awkward but temporary notation).

Then the sheaf of meromorphic functions $\mathcal{K}_ X$ is the quasi-coherent sheaf of $\mathcal{O}_ X$-algebras associated to the quasi-coherent sheaf of meromorphic functions $\mathcal{K}_{X_0}$.

Proof. For the equivalence between $\mathit{QCoh}(\mathcal{O}_ X)$ and $\mathit{QCoh}(\mathcal{O}_{X_0})$, please see Properties of Spaces, Section 65.29. The lemma is true because $\mathcal{K}_ X$ and $\mathcal{K}_{X_0}$ are quasi-coherent and have the same value on corresponding affine opens of $X$ and $X_0$ by Lemma 70.10.4 and Divisors, Lemma 31.23.6. $\square$


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