The Stacks project

Lemma 18.28.5. Colimits and tensor product.

  1. A filtered colimit of flat presheaves of modules is flat. A direct sum of flat presheaves of modules is flat.

  2. A filtered colimit of flat sheaves of modules is flat. A direct sum of flat sheaves of modules is flat.

Proof. Part (1) follows from Lemma 18.27.7 and Algebra, Lemma 10.8.8 by looking at sections over objects. To see part (2), use Lemma 18.27.7 and the fact that a filtered colimit of exact complexes is an exact complex (this uses that sheafification is exact and commutes with colimits). Some details omitted. $\square$

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