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Differential graded algebras, or DGAs, are a basic object of study in many areas of modern mathematics. While they were present (implicitly at least) since the start of modern differential geometry, I would like to know where the abstract definition of a DGA was first written down, and by whom?

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Search for the earliest appearance of "differential graded algebra" and "DGA" on MathSciNet. The earliest hit is DGA in a review of a 1954 paper of Cartan where DGA (or more precisely, the redundant term "DGA-algebra") is defined: "Sur les groupes d'Eilenberg-Mac Lane $H(\Pi,n)$ I. Méthode des constructions," Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 467–471. In the same year there is an entry from the Cartan seminar: Séminaire Henri Cartan de l'Ecole Normale Supérieure, 1954/1955. Algèbres d'Eilenberg-MacLane et homotopie.

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    $\begingroup$ I don't think "DGA-algebra" is redundant, as the A there stands for "associative" (while in DGA, it stands for "algebra"). $\endgroup$
    – Aleksandar Milivojević
    Commented Aug 31, 2020 at 23:20
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    $\begingroup$ @AleksandarMilivojevic fair enough, I was not reading the review closely. $\endgroup$
    – KConrad
    Commented Aug 31, 2020 at 23:26
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    $\begingroup$ In Cartan's usage, the A stands for "augmented", not "associative". A DGA-algebra is a differential graded augmented algebra. $\endgroup$
    – John Rognes
    Commented Sep 1, 2020 at 19:29
  • $\begingroup$ @JohnRognes Then it probably means "augmented" everywhere I've seen DGA-algebra. Thank you for the correction. $\endgroup$
    – Aleksandar Milivojević
    Commented Sep 1, 2020 at 21:11

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