Timeline for Commutative rings to algebraic spaces in one jump?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2010 at 21:05 | history | edited | JBorger | CC BY-SA 2.5 |
added 66 characters in body
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| Feb 7, 2010 at 18:35 | vote | accept | Harry Gindi | ||
| Jan 10, 2010 at 5:29 | comment | added | Harry Gindi | I'm reading toen right now, and I see what I was doing wrong. Thanks! | |
| Jan 10, 2010 at 4:31 | comment | added | JBorger | OK, let me try again. The T-V approach just means this: an algebraic space with affine diagonal is the same as a sheaf X on Aff which is covered by affines U_i such that each U_ij := U_i \times_X U_j is affine and etale over U_i and U_j. A general algebraic space is the same except that you only require that each U_ij is an algebraic space with affine diagonal (instead of being affine). | |
| Jan 10, 2010 at 4:06 | comment | added | JBorger | For instance, on just about any affine scheme (e.g. the affine line), the colimit of all nilpotent neighborhoods of the diagonal is ind-algebraic but not algebraic. Therefore the quotient (the so-called de Rham space discussed recently on MO) is also not algebraic. Yet it is locally affine in the sense that it is covered by affine schemes, namely the original one you started with. | |
| Jan 10, 2010 at 4:03 | comment | added | JBorger | I'm not exactly sure what you mean. I would say that what I wrote is the functor of points point of view. The answer to your other question, in its most literal interpretation, no. The reason is that you also need to say something about how you glue the affine pieces together, i.e. about the equivalence relation. You could take the quotient of any affine scheme by some random equivalence relation, which would have no reason to be an algebraic space. cont'd | |
| Jan 10, 2010 at 3:35 | comment | added | Harry Gindi | I wasn't familiar with this "equivalence relation" viewpoint, but after looking it up, I'll accept the answer. The functor of points view is much simpler though. I assume that he means that the category of algebraic spaces are the locally affine sheaves over the etale topology on CommRing^op? | |
| Jan 10, 2010 at 3:35 | vote | accept | Harry Gindi | ||
| Feb 7, 2010 at 18:33 | |||||
| Jan 9, 2010 at 20:28 | comment | added | Chris Schommer-Pries | This answer seems to me to answer the question you ask. Perhaps you can clarify your question a bit? | |
| Jan 9, 2010 at 13:59 | history | answered | JBorger | CC BY-SA 2.5 |