3 added 685 characters in body

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? pff, not a chance. There are various ideas about stacks I would like to test out, but the sites I am most familiar with have few application-rich topologies. (Smooth, finite-dimensional manifolds are particularly boring in this respect, and topological spaces are not much better)

What I'm after is a table listing the well-known/common topologies on $Sch$ and their relative 'fineness'. Or, if you like, containment. We of course have the canonical topology - is there a characterisation of that in terms of schematic properties, as opposed to the obvious categorical definition?

And furthermore, one expects that for nice schemes, various topologies will coalesce, say one sort of covers becoming cofinal in another, when restricted to a subcategory of $Sch$. Say those schemes which are Noetherian, smooth or even just varieties.

Then there are things like when categories of sheaves, or 2-categories of stacks, are equivalent. But maybe this is asking too much.

Maybe I'm after something like 'Counterexamples in Grothendieck topologies'. Does such a thing exist, all in one place? I'm sure it is all there in SGA, or the stacks project, or in Vakil's Foundations of Algebraic Geometry, but I'm after the distilled essence.

PS I am interested in things which are (pre)topologies even if they are not usually used as such for the purposes of sheaves.

EDIT: I'm not merely after examples of Grothendieck topologies on $Sch$, even though that is handy. I want a reference, if there is one, or just a straight-out answer, that compares the various topologies on $Sch$, and under which circumstances (restricting $Sch$ to a subcategory) they coincide.

For example, does an fppf cover of a variety have local sections over an etale cover? Do the fppf and fpqc topologies give rise to the same sheaves over a nicely behaved scheme? Is the etale topology strictly 'weaker' than some other topology no matter what schemes one looks at? Does one get the same Deligne-Mumford stacks for topology A and topology B?

(Grumble over)

2 added 1 characters in body

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? pff, not a chance. There are various ideas about stacksI stacks I would like to test out, but the sites I am most familiar with have few application-rich topologies. (Smooth, finite-dimensional manifolds are particularly boring in this respect, and topological spaces are not much better)

What I'm after is a table listing the well-known/common topologies on $Sch$ and their relative 'fineness'. Or, if you like, containment. We of course have the canonical topology - is there a characterisation of that in terms of schematic properties, as opposed to the obvious categorical definition?

And furthermore, one expects that for nice schemes, various topologies will coalesce, say one sort of covers becoming cofinal in another, when restricted to a subcategory of $Sch$. Say those schemes which are Noetherian, smooth or even just varieties.

Then there are things like when categories of sheaves, or 2-categories of stacks, are equivalent. But maybe this is asking too much.

Maybe I'm after something like 'Counterexamples in Grothendieck topologies'. Does such a thing exist, all in one place? I'm sure it is all there in SGA, or the stacks project, or in Vakil's Foundations of Algebraic Geometry, but I'm after the distilled essence.

PS I am interested in things which are (pre)topologies even if they are not usually used as such for the purposes of sheaves.

1

A bestiary of topologies on Sch

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? pff, not a chance. There are various ideas about stacksI would like to test out, but the sites I am most familiar with have few application-rich topologies. (Smooth, finite-dimensional manifolds are particularly boring in this respect, and topological spaces are not much better)

What I'm after is a table listing the well-known/common topologies on $Sch$ and their relative 'fineness'. Or, if you like, containment. We of course have the canonical topology - is there a characterisation of that in terms of schematic properties, as opposed to the obvious categorical definition?

And furthermore, one expects that for nice schemes, various topologies will coalesce, say one sort of covers becoming cofinal in another, when restricted to a subcategory of $Sch$. Say those schemes which are Noetherian, smooth or even just varieties.

Then there are things like when categories of sheaves, or 2-categories of stacks, are equivalent. But maybe this is asking too much.

Maybe I'm after something like 'Counterexamples in Grothendieck topologies'. Does such a thing exist, all in one place? I'm sure it is all there in SGA, or the stacks project, or in Vakil's Foundations of Algebraic Geometry, but I'm after the distilled essence.

PS I am interested in things which are (pre)topologies even if they are not usually used as such for the purposes of sheaves.