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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 14th 2012
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIf I have a coverage $K$ on $Ring^{op}$ (not the Zariski coverage), is there a canonical way to extend it to $Sch$ such that it agrees with the original one on affine schemes? A similar sort of question could be asked for more general sorts of schemes. Would it be as simple as asking that there is an open cover of a scheme by affines and then a $K$-covering family of each affine? I suppose for this sort of thing to work one needs that $K$-covers descend along Zariski covers in $Ring^{op}$... As to the slightly different question of how to extend a coverage $K$ from a site $(S,J)$ to $Sh(S,J)$ I can't think how to do this in a way which respects the original coverage. Is this even possible

   If I have a coverage $K$ on $Ring^{op}$ (not the Zariski coverage), is there a canonical way to extend it to $Sch$ such that it agrees with the original one on affine schemes? A similar sort of question could be asked for more general sorts of schemes. Would it be as simple as asking that there is an open cover of a scheme by affines and then a $K$-covering family of each affine? I suppose for this sort of thing to work one needs that $K$-covers descend along Zariski covers in $Ring^{op}$…

   As to the slightly different question of how to extend a coverage $K$ from a site $(S,J)$ to $Sh(S,J)$ I can’t think how to do this in a way which respects the original coverage. Is this even possible

2. • CommentRowNumber2.
   • CommentAuthorMarc Hoyois
   • CommentTimeApr 15th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexRegarding the second question, the topology of a site always extends canonically to the category of presheaves: a covering morphism is simply a morphism which becomes an epimorphism after sheafification. It is also the coarsest <del>subcanonical</del> supercanonical topology on the category of presheaves which extends the topology of the site. This is discussed in details in SGA 4-1, II, 5 ("Extension d'une topologie de $C$ à $C^\wedge$"). So one way to extend a topology to Sch would be to extend first to all presheaves and then restrict to schemes.

   Regarding the second question, the topology of a site always extends canonically to the category of presheaves: a covering morphism is simply a morphism which becomes an epimorphism after sheafification. It is also the coarsest subcanonical supercanonical topology on the category of presheaves which extends the topology of the site. This is discussed in details in SGA 4-1, II, 5 (“Extension d’une topologie de $C$ à $C^\wedge$”).

   So one way to extend a topology to Sch would be to extend first to all presheaves and then restrict to schemes.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 15th 2012
   • (edited Apr 15th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexmaybe check out the answers to this MO question [http://mathoverflow.net/questions/9571/canonical-topology-on-the-category-of-schemes](http://mathoverflow.net/questions/9571/canonical-topology-on-the-category-of-schemes) [edit: ah, no, sorry, I see that the answers all restrict attention to affine schemes]

   maybe check out the answers to this MO question http://mathoverflow.net/questions/9571/canonical-topology-on-the-category-of-schemes

   [edit: ah, no, sorry, I see that the answers all restrict attention to affine schemes]

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexMarc - I know about this, and mentioned in a first version my question (in different words), so I shouldn't have deleted it. The tricky bit is whether the new topology, when restricted to representables, _is the original topology_. This is clearly false for e.g. open covers of spaces, where the topology on the category of sheaves, when restricted to spaces, gives us that covers are local-section-admitting maps. Perhaps what I'm thinking of doesn't exist... EDIT: Or perhaps at least some way of recovering the original topology from the induced topology. This is where things like geometries come in, I think.

   Marc - I know about this, and mentioned in a first version my question (in different words), so I shouldn’t have deleted it. The tricky bit is whether the new topology, when restricted to representables, is the original topology. This is clearly false for e.g. open covers of spaces, where the topology on the category of sheaves, when restricted to spaces, gives us that covers are local-section-admitting maps.

   Perhaps what I’m thinking of doesn’t exist…

   EDIT: Or perhaps at least some way of recovering the original topology from the induced topology. This is where things like geometries come in, I think.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeApr 16th 2012
   • PermaLink
   Author: Marc Hoyois
   Format: MarkdownItexI actually misread the statement, the extension I described is of course <i>finer</i> than the canonical topology on presheaves, so it doesn't really answer anything...

   I actually misread the statement, the extension I described is of course finer than the canonical topology on presheaves, so it doesn’t really answer anything…