In complex geometry, there is the a lemma, analogous to the Poincaré lemma in (real) differential geometry, which states that a $(p,q)$-form that is $\bar\partial$-closed is locally $\bar\partial$-exact. In the book Complex Geometry by Huybrechts, we find the following remark:
The [...] proposition and its corollary are known as the Grothendieck-Poincaré lemma. The first proof of it is due to Grothendieck and was presented by Serre in the Séminaire Cartan in 1958.
The claim that this lemma was (first) proven by Grothendieck is also backed up by these notes, in section 5. Now, I would like to see the original presentation of the proof, so I tried to find the source that Huybrechts refers to. The obvious thing to do is go through the Séminaire Cartan scripts. However, in the volumes corresponding to the year 1958 there is no contribution by Serre.
In fact, I don't think there is anything related to the Grothendieck-Poincaré lemma in the entire collection (I convinced myself of this by skipping through all of Serre's contributions, which can be found with a little bit of effort, e.g. on this page.
Therefore, I would like to know the following:
- Did Serre present this proof by Grothendieck anywhere (else)? Where, and when?
- If not, where and when was the proof due to Grothendieck first published or otherwise made public?