Was Grothendieck familiar with Buchsbaum's exact categories while writing Tohoku? I'm trying to situate Tohoku historically.
Several sources discussing Tohoku I've found either (i) do not mention Buchsbaum's definition of exact categories, or (ii) briefly mention it but do not indicate whether Grothendieck was aware of his work. At the same time, many sources suggest an innovation of Tohoku was that it correctly identified the 'right' categories in which homological algebra made sense. But this is what Buchsbaum did in his thesis. I know Grothendieck did not have access to Eilenberg and Cartan's Homological Algebra (which, I think, cites Buchsbaum's thesis). Did he also develop his abelian categories independently?
 A: In a letter dated July 13th,1955 Serre informed Grothendieck that Buchsbaum (a Samuel Eilenberg Ph.D student) had developed a "system very similar to your [Grothendieck's] abelian classes", to be published in the Transactions of the AMS.
Serre adds that he doesn't know if Grothendieck is aware of that, but that it doesn't matter.
However Buchsbaum was unable to prove the capital fact that sheaves possess sufficiently many injectives.
Then Serre communicates to Grothendieck Eilenberg's proposition  that Grothendieck publish the interesting part of his research in the Transactions too, using Buchsbaum "...for all the trivial results..." so that "...you  would basically only need to write up the interesting part..."
The rest is history: as we all know that suggestion was not followed by Grothendieck, who decided to publish his ground-breaking theory in the Tohoku Mathematical Journal.
You can read all that on pages 18 and 19 of the bilingual (French-English) Correspondance Grothendieck-Serre, one of the most fascinating books I have ever seen on the history of mathematics.
