Corollary 1

Tag 0E1I, permanent. Cite it as [Bourbaki, Tag 0E1I].

Algebra, chapter X, § 8. Dimension homologique

Let $M$ be a graded $A$ -module of finite type. If the $A_0$ -module $M/J_0M$ is projective (resp. graded free, resp. flat) and if $\operatorname{Tor}_1^A(A_0,M)=0$ , then the $A$ -module $M$ is projective (resp. graded free, resp. flat).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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