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Charles Doran, wrote an annoted bibliography on deformation theory that is available from him via email (I have asked him the question below via email). In it one finds the quotation:

Grothendieck's fundamental observation was that the ring-theoretic analogue of geometric “jet-space” constructions is a representation of the corresponding functor by Artinian rings.

I am thinking that this refers to the second installment of Grothendieck's FGA Bourbaki Seminaire note: http://www.numdam.org/numdam-bin/fitem?id=SB_1958-1960__5__369_0 ? I guess I am looking for a kind GAGA like dictionary that motivates Grothendieck's introduction of pro-representable functors as a substitute for the analytic Kodaira-Spencer theory?

The question Deformation theory and differential graded Lie algebras looks vaguely useful.

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  • $\begingroup$ You can also have a look at Grothendieck's exposes at Seminaire Henri Cartan (13), exp. 7-17, 1960-61, Techniques de construction en geometrie analytique I-X. $\endgroup$
    – Peter Dalakov
    Commented Aug 14, 2016 at 16:13
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    $\begingroup$ Thanks Peter! I actually am in the process of examining those documents. I gather they deal with Grothendieck's viewpoint on Teichm\"uller theory? $\endgroup$
    – mathdude
    Commented Aug 14, 2016 at 20:25

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