Subject: categories: Topos cohomology, context and technical questions

From: Colin McLarty <[email-masked]>
Date: Mon, 15 Mar 2004 09:23:11 -0500

Thanks to Christopher Townsend and Carsten Butz for help on cohomology in
an elementary topos.  It seems the general theory is not much advanced
beyond what it was in Johnstone 1977.

My question came out of a conversation with algebraic geometers several
years ago, which I have taken up again lately.  Deligne, for example,
describes toposes as one of Grothendieck's great ideas (one of
Grothendieck's four "idees maitresses").  But for him and many other
geometers their value lies in organizing cohomology.  Insofar as
Grothendieck toposes support a simple general theory of cohomology, and
elementary toposes do not, these people find only Grothendieck toposes
interesting.

Certainly there is a lot to say for elementary topos theory even from that
perspective:  The elementary topos axioms organize the theory of
Grothendieck toposes.  Elementary toposes have some cohomology theory
though not so simple and general.  And elementary toposes have other
roles.

What interests me, now, is how far elementary topos theory helps with
cohomology per se.

One approach is to notice:  The elementary theory of "a topos whose
Abelian groups have enough injectives" supports a considerable general
theory of cohomology via injective resolutions.  But I have not worked out
how far it really goes.  (People with foundational interests will notice
the exact result depends on whether and how this theory works with
infinite complexes.  There are various approaches depending on what you
mean by "elementary".)

This raises my first technical question:

SGA 4 proves inverse image functors preserve flat modules, but the
transparent proof assumes enough points (Exp. V Prop. 1.7).  Deligne gives
a far from transparent proof, for all (Grothendieck) toposes, in an
appendix on "local inductive limits".  He urges the reader "to avoid, as a
matter of principle, reading this appendix".  Is the result proved more
simply somewhere?  Do "local inductive limits" survive today in some form?
In short, can we follow Deligne's advice on not reading this appendix, and
still prove his result?  I have made no progress on the appendix yet, as
the opening definition is full of typos.  If there is a cleaner exposition
I'd rather start with that.

The second question:

The IHES version of SGA 4 gives a faulty proof that, in every
(Grothendieck) topos, rings admit a standard kind of resolution over any
cover by tensoring with a resolution of the integers.  This is Prop. 1.4
of Expose V.  The Springer-Verlag version corrects the mistake by proving
the result only when the topos has enough points (Prop 1.11 Exp. V).
Johnstone 1977 recovers the theorem for the case of a presheaf topos
(Lemma 8.2)  which is the case of interest and easily extends to any topos
with enough points.

Is that version optimal, in some easy to prove sense?  Is there an easy
example of a ring in a Grothendieck topos where the resolution
fails?           Is it known to be optimal in any sense?

best, Colin