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The Stacks project

Lemma 71.10.5. 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 67.49.1.

  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 66.29. 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 71.10.4 and Divisors, Lemma 31.23.6. \square


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