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Every category admits a Grothendieck topology, called canonical, which is the finest topology which makes representable functor into sheaves.

Is there a concrete description of the canonical topology on the category of schemes? By Grothendieck's results on descent this is at least as fine as the fpqc topology, but I don't even know if the two actually coincide. If not, what is known about it?

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    $\begingroup$ SGA IV Exp 2 gives a caracterisation of the canonical topology in terms of universally strict epimorphisms. I'm not sure what these are in the category of schemes though. $\endgroup$
    – AFK
    Commented Dec 23, 2009 at 0:16
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    $\begingroup$ For the category of affine schemes I think (but am not certain) that the canonical topology is the flat topology (no finite presentation necessary); I think this is in Knutson. $\endgroup$
    – David Zureick-Brown
    Commented Dec 23, 2009 at 3:14
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    $\begingroup$ The flat site is not even subcanonical according to Vistoli's notes. $\endgroup$
    – Harry Gindi
    Commented Dec 23, 2009 at 8:30
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    $\begingroup$ Page 37 remark 2.56 for reference. $\endgroup$
    – Harry Gindi
    Commented Dec 23, 2009 at 8:39

2 Answers 2

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Proposition 3.4 in Orlov's paper

Quasicoherent sheaves in commutative and noncommutative geometry. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 3, 119--138; translation in Izv. Math. 67 (2003), no. 3, 535--554.

describes the canonical topology and the universally strict epimorphisms on the category of affine schemes. In the same paper Orlov characterizes the quasicoherent sheaves on the small Zariski site in terms of the canonical topology. He also studies an important subcanonical topology - the effective descent topology - which seems to be finer than the flat topology.

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    $\begingroup$ Thank you. I'm now home for Christmas holidays. Do you have any online reference? $\endgroup$
    – Andrea Ferretti
    Commented Dec 23, 2009 at 17:49
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    $\begingroup$ mpim-bonn.mpg.de/preprints/send?bid=56 $\endgroup$
    – Tony Pantev
    Commented Dec 23, 2009 at 18:28
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    $\begingroup$ But this studies only the canonical topology on the category of affine schemes, right? $\endgroup$
    – Martin Brandenburg
    Commented May 24, 2011 at 16:15
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following the answers given by Pantev, I will give you more example of canonical topology on some category(universally strict epimorphism)

if C is an abelian category or a topos, then canonical topology consists of all epimorphism if C is a quasi abelian category,then canonical topology consists of all strict epimorphism(Note that strict epimorphism is subcanonical in general)

if C is a category of associative unital k-algebras(opposite category of affine schemes,not necessarily commutative). Canonical topology consists of all strict epimorphism which are precisely surjective morphism of algebras. In this case, univerally strict epimorphism coincides with strict epimorphism.

Rosenberg has a very detailed treatment for a category and 2-category(taken as category of spaces)with canonical topology(he called right exact structure).It is in MPIM preprint series," Homological algebra of noncommutative 'space' I"

What pantev mentioned is related to my answers in another question: Does sheafification preserve sheaves for a different topology?

The effective descent topology is finer that fpqc topology, fppf topology,smooth topology(in Kontsevich-Rosenberg sense)

The descent topology on category of affine schemes(not necessarily commutative) coincides with subcanonical topology

The reference is Orlov's paper and Kontsevich-Rosenberg MPIM preprint series. Noncommutative stack and Noncommutative grassmannian and related construction

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