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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 30th 2011
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   Author: zskoda
   Format: MarkdownItexThere was lots of study here of general formalism of schemes in generalized sense like in Lurie. So let $Sch$ be a category of schemes just as an abstract category, without any Grothendieck topology singled out. Can we build on it the stack of qcoh sheaves, or at least what are the global sections of the structure sheaf, just from $Sch$ as an abstract category ? Once we get such sheaves we can get more or less everything by manipulating qcoh sheaves, direct and inverse images and so on.

   There was lots of study here of general formalism of schemes in generalized sense like in Lurie. So let $Sch$ be a category of schemes just as an abstract category, without any Grothendieck topology singled out. Can we build on it the stack of qcoh sheaves, or at least what are the global sections of the structure sheaf, just from $Sch$ as an abstract category ? Once we get such sheaves we can get more or less everything by manipulating qcoh sheaves, direct and inverse images and so on.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2011
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   Author: Urs
   Format: MarkdownItexYes, the stack of quasicoherent sheaves on any category $C$ is the stack presented by the fibered category which is the [[tangent category]] of $C^{op}$. Take $C = Aff$, then $T C^{op} \simeq $ [[Mod]]. That's the ordinary stack of quasicoherent sheaves.

   Yes, the stack of quasicoherent sheaves on any category $C$ is the stack presented by the fibered category which is the tangent category of $C^{op}$.

   Take $C = Aff$, then $T C^{op} \simeq$ Mod. That’s the ordinary stack of quasicoherent sheaves.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2011
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   Author: Urs
   Format: MarkdownItexSo what one could and should consider: take any (small) category $C$ whatsoever. Then ask for coverages/Grothendieck topologies that make $(T C^{op})^{op} \to C$ a stack. This is then a "natural" kind of coverage for generalized geometry parameterized by $C$.

   So what one could and should consider:

   take any (small) category $C$ whatsoever. Then ask for coverages/Grothendieck topologies that make $(T C^{op})^{op} \to C$ a stack. This is then a “natural” kind of coverage for generalized geometry parameterized by $C$.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 30th 2011
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   Author: zskoda
   Format: MarkdownItexThat is somewhat helpful, but it does not work if we modify to rings replaced by generalized rings (commutative finitary monads in $Set$) -- in that case quasicoherent modules are not abelian as algebras over a theory do not make an abelian category, nor additive in general. So already the Durov's commutative case does not work. On the other hand it works for the Grothendieck case. Though in that case, the qcoh sheaves are stack for the descent topology which is in fact much finer than the Zariski topology. The effective descent topology for qcoh sheaves is not one of the standard ones like Zariski. Zariski can be axiomatized in more complex way once we have the correct category of qcoh sheaves. This works in Durov case, where the things are nonadditive. There must be some nonadditive generalization of the tanegnt category...any idea ?

   That is somewhat helpful, but it does not work if we modify to rings replaced by generalized rings (commutative finitary monads in $Set$) – in that case quasicoherent modules are not abelian as algebras over a theory do not make an abelian category, nor additive in general. So already the Durov’s commutative case does not work. On the other hand it works for the Grothendieck case. Though in that case, the qcoh sheaves are stack for the descent topology which is in fact much finer than the Zariski topology. The effective descent topology for qcoh sheaves is not one of the standard ones like Zariski. Zariski can be axiomatized in more complex way once we have the correct category of qcoh sheaves. This works in Durov case, where the things are nonadditive. There must be some nonadditive generalization of the tanegnt category…any idea ?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeMay 31st 2011
   • (edited May 31st 2011)
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   Author: zskoda
   Format: MarkdownItexWait a second. I was asking about $Sch$, not about $Aff$. Your proposal works for $Aff$ where there are other constructions as well. The problem is with my original question about $Sch$ already (not to mention the nonadditive context from 4). Are you sure that for $Sch$ the stack of qcoh modules is obtained via the same construction ? (Remember, we are starting with $Sch$, so no presheaf constructions are allowed, we just know $Sch$ as a category and we do not know, a priori, which part of $Sch$ is $Aff$. As far as 3 I am also wondering -- you see one thing is to take topology which makes it a stack and another is to take a topology which makes quasicoherent presheaves sheaves. Now which one is stronger in general ? (is there a general statement) The second property is more basic.

   Wait a second. I was asking about $Sch$, not about $Aff$. Your proposal works for $Aff$ where there are other constructions as well. The problem is with my original question about $Sch$ already (not to mention the nonadditive context from 4). Are you sure that for $Sch$ the stack of qcoh modules is obtained via the same construction ? (Remember, we are starting with $Sch$, so no presheaf constructions are allowed, we just know $Sch$ as a category and we do not know, a priori, which part of $Sch$ is $Aff$.

   As far as 3 I am also wondering – you see one thing is to take topology which makes it a stack and another is to take a topology which makes quasicoherent presheaves sheaves. Now which one is stronger in general ? (is there a general statement) The second property is more basic.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2011
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   Author: Urs
   Format: MarkdownItexI haven't thought about what happens when you start with $Sch$, form the canonical prestack $(T Sch^{op})^op \to Sch^op$ and then stackify with respect to a topology that makes $Aff$ not be dense in $Sch$. Probably one can get weird results. But why would one want to do this?

   I haven’t thought about what happens when you start with $Sch$, form the canonical prestack $(T Sch^{op})^op \to Sch^op$ and then stackify with respect to a topology that makes $Aff$ not be dense in $Sch$. Probably one can get weird results.

   But why would one want to do this?

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeMay 31st 2011
   • (edited May 31st 2011)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThe main question is if one can define qcoh modules on $Sch$ just from bare $Sch$ as a category. If one has that, one can define almost everything in scheme theory from qcoh modules, hence one can define all the geometry of the category of scheme just from knowing $Sch$ as a category and in particular detect just categorically which schemes are affine. Isn't that a dream of category theory -- to reduce geometry to category theory ? > and then stackify with respect to a topology that makes Aff not be dense in Sch. I do not understand this. What the density for presheaves of sets has to do with sheaf property of qusicoherent presheaves ? Again, this about stackification and so on I see as seperate questions from the original question if we can define the stack of qcoh modules (I really do not care that it is a stack in this question, this is a fact which we do not need to prove independently) on $Sch$ just from knowing $Sch$ as an abstract category. P.S. there is a related MO question [here](http://mathoverflow.net/questions/56887/rigidity-of-the-category-of-schemes) which would be corollary of the answer here, if solved.

   The main question is if one can define qcoh modules on $Sch$ just from bare $Sch$ as a category. If one has that, one can define almost everything in scheme theory from qcoh modules, hence one can define all the geometry of the category of scheme just from knowing $Sch$ as a category and in particular detect just categorically which schemes are affine. Isn’t that a dream of category theory – to reduce geometry to category theory ?

   and then stackify with respect to a topology that makes Aff not be dense in Sch.

   I do not understand this. What the density for presheaves of sets has to do with sheaf property of qusicoherent presheaves ? Again, this about stackification and so on I see as seperate questions from the original question if we can define the stack of qcoh modules (I really do not care that it is a stack in this question, this is a fact which we do not need to prove independently) on $Sch$ just from knowing $Sch$ as an abstract category.

   P.S. there is a related MO question here which would be corollary of the answer here, if solved.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMay 31st 2011
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   Author: Urs
   Format: MarkdownItexI mean if one starts with $T Sch^{op}$ and then stackifies with respect to a topology that makes schemes be sheaves over $Aff$, the result will be the same as the obtained from $T Aff^{op}$. But I think for the kind of question here it is not very useful to start with $Sch$. That is already a subcategory of sheaves on $Aff$, so if we are going to pass to sheaves and stacks anyway, we should just start with $Aff$. Of course that depends on what exactly you want to do.

   I mean if one starts with $T Sch^{op}$ and then stackifies with respect to a topology that makes schemes be sheaves over $Aff$, the result will be the same as the obtained from $T Aff^{op}$.

   But I think for the kind of question here it is not very useful to start with $Sch$. That is already a subcategory of sheaves on $Aff$, so if we are going to pass to sheaves and stacks anyway, we should just start with $Aff$.

   Of course that depends on what exactly you want to do.