Lemma of the day

Let C be a site. Let O’ —> O be a surjection of sheaves of rings whose kernel I is an ideal of square zero. Let F’ be an O’-module and set F = F’/I F’. The following are equivalent

  1. F’ is a flat O’-module, and
  2. F is a flat O-module and I ⊗O F —> F’ is injective.

See Tag 08M4.