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Apr 13, 2017 at 12:58 history edited CommunityBot
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May 30, 2011 at 9:18 comment added Martin Brandenburg Another remark: The category of schemes can be constructed categorically from the category of rings (-> Yves Dier, Categories of Commutative Algebras; the key step is to define localizations of rings with the help of so-called codisjunctors). Rigidity would be answered if there is a converse, i.e. the category of affine schemes can be constructed categorically from the category of schemes.
May 30, 2011 at 9:12 comment added Martin Brandenburg @Ryan: Just because my question cannot be answered within the usual realm of algebraic geometry or other theories, this does not mean that it cannot be answered at all. I know all these basics about schemes which you repeat, but we cannot use them here at all. This does not mean that I don't appreciate them! By the way, I do not want to define every scheme-theoretic notion in categorical terms, but enough in order to show that the category of schemes is rigid. It may be argued that this is a unnatural question, but this may be discussed per E-Mail.
May 30, 2011 at 2:47 comment added Ryan Reich (continued) However, I think that specifying affine schemes may be necessary from the outset. Even the most intrinsic definitions of, say, "open immersion" seem to rely on affine test schemes: either you base change the map in question to all affines and use the Zariski topology, or you use rings with nilpotent ideals to test for formal smoothness of a (categorically defined) monomorphism. Given what I wrote above, I think you should consider this to be an emergent property of using affine schemes as your "fiber functor", along with all other pesky topological notions.
May 30, 2011 at 2:40 comment added Ryan Reich (continued) It seems to me that categorifying schemes will have the same issue, because affine schemes play a similarly definitive role as an undefined primitive concept. If you do want to go this route, you might consider looking for a reconstruction theorem giving back the ring A from the category-plus-other-data Schemes/A, making sure that your category of categories-plus-data is equivalent to that of rings. If this is possible, it will clarify the extent to which categorical information does define schemes. (continued)
May 30, 2011 at 2:34 comment added Ryan Reich (continued) It's not just that Rep(G) minus the vector spaces has automorphisms as a category but also that the collection of all Tannakian categories itself forms a category equivalent to that of affine pro-algebraic groups; if you forget the fiber functors, you don't just get extra automorphisms or 2-automorphisms, you also get extra objects (rigid tensor categories). If you use tensor functors to vector bundles on some scheme X as the fiber functor, you will get some category of sheaves of groups on X locally equivalent to G. (continued)
May 30, 2011 at 2:22 comment added Ryan Reich @Martin: I wonder if your goal of reducing many (all?) concepts in scheme theory to purely categorical terms is unrealizable. Consider Tannakian categories as an analogy: although every bit of the important algebra of (affine algebraic) group representations can be captured by the rigid tensor category formalism, we cannot give an "intrinsic" description of Rep(G), avoiding all mention of vector spaces: the fiber functor is a necessary part of the data. Without it we can't distinguish categories actually equivalent to some Rep(G) from those merely sharing the formal properties. (continued)
May 30, 2011 at 1:06 answer added Akhil Mathew timeline score: 4
May 29, 2011 at 9:10 history edited Martin Brandenburg CC BY-SA 3.0
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May 28, 2011 at 19:54 comment added Martin Brandenburg @Akhil: I know this EGA stuff. This uses affine schemes as test objects and is therefore not categorical. Unless you give me a categorical definition of affine schemes within the category of schemes, which would make me very, very happy ;-)
May 28, 2011 at 18:09 comment added Akhil Mathew @Martin: There's a functorial characterization of "locally of finite presentation" in EGA IV-8, and locally f.p. + formally \'etale + radicial = open immersion.
May 28, 2011 at 12:38 comment added Martin Brandenburg Also remark that the valuative criterion is only categorical if we were able to show that a) quasiseparated schemes, b) spectra of valuation rings are categorical.
May 28, 2011 at 12:37 comment added Martin Brandenburg Open/closed immersions are the étale/proper monomorphisms, but every definition of étale/proper uses some finiteness condition which is - a priori - not categorical.
May 28, 2011 at 11:50 comment added Tom Goodwillie How about this? If $A\to X$ is a closed immersion then the corresponding open immersion is the terminal example of a map $Y\to X$ such that $A\times_XY$ is empty.
May 28, 2011 at 11:32 comment added Kevin Buzzard I remember that the proper monomorphisms are the closed immersions and I thought that there was a similar story for open immersions but I've forgotten it. For properness you might be able to use some valuative criterion, but in this arguably pathological situation where maps aren't of finite type and schemes aren't noetherian perhaps anything can happen.
May 28, 2011 at 11:08 comment added Martin Brandenburg @Qiaochu: The same nlab article shows that even for topological spaces this does not give the correct notion. It is better suited for algebraic categories.
May 28, 2011 at 10:13 comment added Qiaochu Yuan Related: nlab.mathforge.org/nlab/show/compact+object
May 28, 2011 at 9:47 history asked Martin Brandenburg CC BY-SA 3.0