Timeline for Rigidity of the category of schemes
Current License: CC BY-SA 2.5
9 events
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| May 31, 2011 at 15:37 | comment | added | Zoran Skoda | Martin, I tkink that your question is interesting, it would be nice to have at least partial answer. On the other hand, it is to be expected that affine schemes as determined by an algebraic theory be categorically exhaustable in many ways, while schemes as locally such and such spaces are in any definition, sheaf theoretic or not, glued objects and should be viewed relative to the category of local models. For affine schemes, as Urs has reminded me, the cat. of qcoh modules is obtained by the Quillen construction from the category of arrows, then properties of morphisms can be said from Qcoh. | |
| May 12, 2011 at 10:04 | comment | added | Martin Brandenburg | I've started to study Luo's categorical geometry, and also Dier's book. I'm pretty convinced that these approaches won't answer my question: The categories in study are always similar to the category of rings (Dier) resp. affine(!) schemes (Luo); you can define notions of commutative algebra in pure categorical terms, but not the ones of algebraic geometry within the category of schemes. Dier just defines a scheme as a space together with a sheaf of "generalized" rings, and within the category of schemes no pure categorical definitions are made | |
| Feb 28, 2011 at 23:43 | comment | added | John Iskra | Then isn't that a problem you need to overcome anyway, regardless of whether we have an acceptable definition of etale? Maybe I don't understand your strategy. | |
| Feb 28, 2011 at 22:51 | comment | added | Martin Brandenburg | Already the categorical definition of "finitely presentable" makes problems, because in the category of schemes the "correct" notion only uses affine test schemes. | |
| Feb 28, 2011 at 22:06 | comment | added | John Iskra | According to Diers (I switch directions here so we are working on the geometric side): An object $X$ is pre-neat if for all local objects $L$, any two arrows $f$ and $g$ from $L$ to $X$ are either equal or disjoint. Take the definition of local object to be that given in Luo's categorical geometry in \S 3.4, disjoint arrows are those whose pullback is the initial object. An etale arrow is then one which is finitely presentable, neat and coflat. I'll work on the extension of this to schemes. | |
| Feb 28, 2011 at 16:10 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Feb 28, 2011 at 16:06 | comment | added | John Iskra | I don't know that he does so explicitly. Read Categorical Geometry. Toward the end (in 5.3, I think) he provides sufficient conditions for an arrow f to induce an open map Spec(f). The conditions are entirely categorical. It should be possible, given Luo's characterization of schemes to give similar sufficient conditions for open immersions in the category of schemes over affine schemes. My advice is to read cg, if you haven't already. | |
| Feb 28, 2011 at 15:03 | comment | added | Martin Brandenburg | I also found Luo's site recently. But I have not found anything which answery my question. It is very interesting that the category of schemes can be constructed categorically from the category of affine schemes (geometry.net/cg/scheme.html), but I don't need that here. Where are open immersions defined categorically? | |
| Feb 28, 2011 at 14:50 | history | answered | John Iskra | CC BY-SA 2.5 |