Subject: categories: Deligne on Govorov-Lazard, proof and history

From: Colin McLarty <[email-masked]>
Date: Wed, 17 Mar 2004 10:29:56 -0500

Deligne's proof that inverse image functors preserve flat modules is to
internalize the Govorov-Lazard proof that a module is flat iff it is a
projective colimit of finite free modules, and to show inverse image
functors preserve these internalized colimits (SGA 4, Springer-Verlag
edition, Exp. V appendix).

The classical proof (say, Eisenbud COMMUTATIVE ALGEBRA appendix 6, theorem
A6.6) looks good for topos logic though I admit I have not worked through
internalizing it.  Also, I have still not fought through the definitions in
Deligne's appendix to SGA 4.  I would still appreciate any reference to a
cleaner treatment if anyone knows one.  It seems likely that Deligne's
"local inductive limits" are internal colimits in the modern sense but I
have not read the thing closely enough yet to really say that.

This seems to me a strikingly sophisticated lifting of a theorem, from SET
to any topos, for 1972.  The current methods for routine lifting were just
being created over here in elementary topos theory at that time.  I believe
Deligne did not know of that.  Or am I making too much of this proof?  I'd
appreciate any historical reflections on relations between such lifting in
the Grothendieck school, and methods of elementary topos theory.

Colin